Recent zbMATH articles in MSC 35Ahttps://www.zbmath.org/atom/cc/35A2021-04-16T16:22:00+00:00WerkzeugAsymptotic behavior in chemical reaction-diffusion systems with boundary equilibria.https://www.zbmath.org/1456.350382021-04-16T16:22:00+00:00"Pierre, Michel"https://www.zbmath.org/authors/?q=ai:pierre.michel"Suzuki, Takashi"https://www.zbmath.org/authors/?q=ai:suzuki.takashi"Umakoshi, Haruki"https://www.zbmath.org/authors/?q=ai:umakoshi.harukiSummary: We consider the asymptotic behavior for large time of solutions to reaction-diffusion systems modeling reversible chemical reactions. We focus on the case where multiple equilibria exist. In this case, due to the existence of so-called ``boundary equilibria'', the analysis of the asymptotic behavior is not obvious. The solution is understood in a weak sense as a limit of adequate approximate solutions. We prove that this solution converges in \(L^1\) toward an equilibrium as time goes to infinity and that the convergence is exponential if the limit is strictly positive.Some models for the interaction of long and short waves in dispersive media. I: Derivation.https://www.zbmath.org/1456.351572021-04-16T16:22:00+00:00"Nguyen, Nghiem V."https://www.zbmath.org/authors/?q=ai:nguyen.nghiem-v"Liu, Chuangye"https://www.zbmath.org/authors/?q=ai:liu.chuangyeSummary: It is universally accepted that the cubic, nonlinear Schrödinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Korteweg-de Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves, and such a system has been studied over the last few decades. However, questions about the validity of this system in the study of water waves were raised in a previous work of one of us where the analysis was presented using the fifth-order KdV as the starting point. These questions will now be settled unequivocally in a series of papers. In this first part, we show that the NLS-KdV system (or even the \textit{linear} Schrödinger-KdV system) cannot be resulted from the full Euler equations formulated in the study of water waves. In the process of so doing, we also propose a few alternative models for describing the interaction of long and short waves.Positive solutions of singular semilinear elliptic equation in bounded NTA-domains.https://www.zbmath.org/1456.350972021-04-16T16:22:00+00:00"Ben Boubaker, Mohamed Amine"https://www.zbmath.org/authors/?q=ai:ben-boubaker.mohamed-amineSummary: We study the existence, the uniqueness and the sharp estimate of a positive solution of the nonlinear equation
\[\Delta v+\psi(\cdot,v)=0,\]
in a bounded NTA-domain \(\Omega\) in \(\mathbb{R}^n \) (\(n\geq 2\)), when a measurable function \(\psi(\cdot,\cdot)\) is continuous and non-increasing with respect to the second variable.Lyapunov-type inequalities for a nonlinear fractional boundary value problem.https://www.zbmath.org/1456.352152021-04-16T16:22:00+00:00"Kassymov, Aidyn"https://www.zbmath.org/authors/?q=ai:kassymov.aidyn-adilovich"Torebek, Berikbol T."https://www.zbmath.org/authors/?q=ai:torebek.berikbol-tillabayulySummary: In this paper, we obtain a Lyapunov-type and a Hartman-Wintner-type inequalities for a nonlinear fractional hybrid equation with left Riemann-Liouville and right Caputo fractional derivatives of order \(1/2<\alpha \leq 1\), subject to Dirichlet boundary conditions. It is also shown that failure of the Lyapunov-type and Hartman-Wintner-type inequalities, corresponding nonlinear boundary value problem has only trivial solutions. In the case \(\alpha =1\), our results coincide with the classical Lyapunov and Hartman-Wintner inequalities, respectively.Existence of semiclassical solutions for some critical Dirac equation.https://www.zbmath.org/1456.811662021-04-16T16:22:00+00:00"Ding, Yanheng"https://www.zbmath.org/authors/?q=ai:ding.yanheng"Guo, Qi"https://www.zbmath.org/authors/?q=ai:guo.qi"Yu, Yuanyang"https://www.zbmath.org/authors/?q=ai:yu.yuanyangSummary: In this paper, we study the following critical Dirac equation \(- i \varepsilon \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x) u = P(x) f(| u |) u + Q(x) | u | u, x \in \mathbb{R}^3\), where \(\varepsilon > 0\) is a small parameter; \(a > 0\) is a constant; \( \alpha_1, \alpha_2, \alpha_3\), and \(\beta\) are \(4 \times 4\) Pauli-Dirac matrices; and \(V, P, Q\), and \(f\) are continuous but are not necessarily of class \(\mathcal{C}^1\). We prove the existence and concentration of semiclassical solutions under suitable assumptions on the potentials \(V(x), P(x)\), and \(Q(x)\) by using variational methods. We also show the semiclassical solutions \(\omega_\varepsilon\) with maximum points \(x_\varepsilon\) of |\( \omega_\varepsilon\)| concentrating at a special set \(\mathcal{H}_P\) characterized by \(V(x), P(x)\), and \(Q(x)\) and for any sequence \(x_\varepsilon \to x_0 \in \mathcal{H}_P, v_\varepsilon(x) := \omega_\varepsilon(\varepsilon x + x_\varepsilon)\) converges in \(W^{1, q}(\mathbb{R}^3, \mathbb{C}^4)\) for \(q \geq 2\) to a ground state solution \(u\) of \(- i \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x_0) u = P(x_0) f(| u |) u + Q(x_0) | u | u, \text{in} \mathbb{R}^3\). Finally, we estimate the exponential decay properties of solutions.
{\copyright 2021 American Institute of Physics}Generalized Darboux transformation and localized waves in coupled Hirota equations.https://www.zbmath.org/1456.351892021-04-16T16:22:00+00:00"Wang, Xin"https://www.zbmath.org/authors/?q=ai:wang.xin|wang.xin.7|wang.xin.3|wang.xin.6|wang.xin.8|wang.xin.10|wang.xin.4|wang.xin.11|wang.xin.2|wang.xin.12|wang.xin.13|wang.xin.5|wang.xin.1|wang.xin.9"Li, Yuqi"https://www.zbmath.org/authors/?q=ai:li.yuqi"Chen, Yong"https://www.zbmath.org/authors/?q=ai:chen.yongSummary: In this paper, we construct a generalized Darboux transformation to the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and inelastic Raman scattering terms. As application, an \(N\)th-order localized wave solution on the plane backgrounds with the same spectral parameter is derived through the direct iterative rule. In particular, some semi-rational, multi-parametric localized wave solutions are obtained: (1) vector generalization of the first- and the second-order rogue wave solutions; (2) interactional solutions between a dark-bright soliton and a rogue wave, two dark-bright solitons and a second-order rogue wave; (3) interactional solutions between a breather and a rogue wave, two breathers and a second-order rogue wave. The results further reveal the striking dynamic structures of localized waves in complex coupled systems.A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation.https://www.zbmath.org/1456.350872021-04-16T16:22:00+00:00"Khalique, Chaudry Masood"https://www.zbmath.org/authors/?q=ai:khalique.chaudry-masood"Moleleki, Letlhogonolo Daddy"https://www.zbmath.org/authors/?q=ai:moleleki.letlhogonolo-daddySummary: This paper aims to study a generalized first extended (3+1)-dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the \(({G'}/{G})\)-expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.Unconditional convergence for discretizations of dynamical optimal transport.https://www.zbmath.org/1456.650462021-04-16T16:22:00+00:00"Lavenant, Hugo"https://www.zbmath.org/authors/?q=ai:lavenant.hugoSummary: The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or computational fluid dynamics formulation, amounts to writing the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several discretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space.
In this paper, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional problem for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by \textit{P. Gladbach} et al. [SIAM J. Math. Anal. 52, No. 3, 2759--2802 (2020; Zbl 1447.49062)], as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien, and Solomon, fit in this framework.On the Cauchy problem for the generalized weak disspative Camassa-Holm equation.https://www.zbmath.org/1456.351792021-04-16T16:22:00+00:00"Chen, Wenxia"https://www.zbmath.org/authors/?q=ai:chen.wenxia"Zhang, Jianmei"https://www.zbmath.org/authors/?q=ai:zhang.jianmei"Deng, Xiaoyan"https://www.zbmath.org/authors/?q=ai:deng.xiaoyan(no abstract)Global existence of weak solutions for the anisotropic compressible Stokes system.https://www.zbmath.org/1456.351602021-04-16T16:22:00+00:00"Bresch, Didier"https://www.zbmath.org/authors/?q=ai:bresch.didier"Burtea, C."https://www.zbmath.org/authors/?q=ai:burtea.cosminThe authors consider the quasi-stationary compressible Stokes equations \(\partial _{t}\rho +div(\rho u)=0\), \(-\operatorname{div}\tau +a\nabla \rho^{\gamma }=f\), \(\gamma >1\), where \(u\) is the fluid velocity field, \(\rho \) the fluid density and \(\tau \) the viscous stress tensor defined through \(\tau _{ij}(t,x,D(u))=A_{ijkl}(t,x)D[u]_{kl}\), for a matrix \(A_{ijkl}(t,x)\in W^{1,\infty }((0,T)\times \mathbb{T}^{3})\). The initial condition \(\rho \mid_{t=0}=\rho _{0}\geq 0\) is imposed. The main result of the paper proves that if \(f\in W^{1,2}(0,T;L^{6/5}(\mathbb{T}^{3})\) satisfies \(\int_{\mathbb{T}^{3}}f(t)dx=0\), if \(\rho _{0}\geq 0\) further satisfies \(0<\int_{\mathbb{T}^{3}}\rho _{0}dx<+\infty \) and \(\int_{\mathbb{T}^{3}}\rho _{0}^{\gamma}dx<+\infty \), there exists a global weak solution \((\rho ,u)\) to the above system with \(\rho \in C([0,T];L_{weak}^{\gamma }(\mathbb{T}^{3}))\cap L^{2\gamma }((0,T)\times \mathbb{T}^{3})\), \(u\in L^{2}((0,T);H^{1}(\mathbb{T}
^{3}))\) and \(\int_{\mathbb{T}^{3}}udx=0\). The authors assume that \(A\) satisfies some symmetry properties and that the tensor \(\tau \) satisfies regularity hypotheses. For the proof, the authors first define a finite-energy weak solution to the problem \(\partial _{t}\rho +\operatorname{div}(\rho u)=0\), \(-\operatorname{div}\tau +\nabla \rho ^{\gamma }=f\), \(\rho \mid_{t=0}=\rho _{0}\geq 0\), as a pair \((\rho ,u)\in L^{\infty }(0,T;L^{\gamma }(\mathbb{T}^{3}))\times L^{2}(0,T;H^{1}(\mathbb{T}^{3}))\) which satisfies this problem in the sense of distributions, the mass conservation identity \(\int_{\mathbb{T}^{3}}\rho (t)=\int_{\mathbb{T}^{3}}\rho _{0}\) and the energy inequality \(\int_{\mathbb{T}^{3}}\rho ^{\gamma }+\int_{0}^{t}\int_{\mathbb{T}^{3}}\tau :\nabla udx\leq \int_{\mathbb{T}^{3}}\rho _{0}^{\gamma
}+\int_{0}^{t}\int_{\mathbb{T}^{3}}u\cdot f\). They prove estimates and a stability property for sequences of finite-energy weak solutions, the limit satisfying an identity which involves a defect measure associated to the pressure. For the construction of solutions to the original problem, the authors introduce a regularized system with diffusion and drag terms in the density equation: \(\partial _{t}\rho +\operatorname{div}(\rho \omega _{\delta }\ast u)=\varepsilon \Delta \rho -\varepsilon \rho ^{2\gamma }-\varepsilon \rho^{2\gamma +1}-\varepsilon \rho ^{3}\), \(\mathcal{A}u+\nabla \omega _{\delta}\ast \rho ^{\gamma }=f\), where \(\omega _{\delta }\) is a regularizing kernel
and \(\mathcal{A}\) is the application \(v\rightarrow -div\tau (t,x,D(v))\). The authors prove the global existence and uniqueness of a strong solution on \((0,T)\) using a fixed point argument. They pass to the limit with respect to the regularization parameter \(\delta \), then with respect to \(\varepsilon \), thanks to estimates and a weak stability result for the perturbed system, to obtain a global solution to the quasi-stationary compressible Stokes system. The paper ends with possible extensions of the main result to close systems.
Reviewer: Alain Brillard (Riedisheim)Global well-posedness of axisymmetric solution to the 3D axisymmetric chemotaxis-Navier-Stokes equations with logistic source.https://www.zbmath.org/1456.351732021-04-16T16:22:00+00:00"Zhang, Qian"https://www.zbmath.org/authors/?q=ai:zhang.qian"Zheng, Xiaoxin"https://www.zbmath.org/authors/?q=ai:zheng.xiaoxinThe system of coupled chemotaxis and incompressible Navier-Stokes equations is studied in the three-dimensional space. The main result is that axisymmetric solutions corresponding to reasonably regular initial data are unique, global-in-time and regular. A novel type apriori inequalities are used in the proofs together with nice symmetry arguments.
Reviewer: Piotr Biler (Wrocław)Oscillatory line source for water waves in shear flow.https://www.zbmath.org/1456.760282021-04-16T16:22:00+00:00"Tyvand, Peder A."https://www.zbmath.org/authors/?q=ai:tyvand.peder-a"Lepperød, Mikkel Elle"https://www.zbmath.org/authors/?q=ai:lepperod.mikkel-elleSummary: The linearized water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. The vorticity is uniform, with zero velocity at the free surface. Then there will be at most two emitted waves, and no Doppler effects. Exact far-field waves are derived, with radiation conditions applied at infinity. An upstream wave will always exist, whereas the downstream wave exists only when the angular frequency of oscillation exceeds the vorticity. The wave radiation problem is solved also for oscillating vortex and dipoles. The amplitudes and energy fluxes are calculated.Finite time blow-up and global existence of weak solutions for pseudo-parabolic equation with exponential nonlinearity.https://www.zbmath.org/1456.351152021-04-16T16:22:00+00:00"Long, Qunfei"https://www.zbmath.org/authors/?q=ai:long.qunfei"Chen, Jianqing"https://www.zbmath.org/authors/?q=ai:chen.jianqing"Yang, Ganshan"https://www.zbmath.org/authors/?q=ai:yang.ganshanSummary: This paper is concerned with the initial boundary value problem of a class of pseudo-parabolic equation \(u_t - \triangle u - \triangle u_t + u = f(u)\) with an exponential nonlinearity. The eigenfunction method and the Galerkin method are used to prove the blow-up, the local existence and the global existence of weak solutions. Moreover, we also obtain other properties of weak solutions by the eigenfunction method.On the use of the rotation minimizing frame for variational systems with Euclidean symmetry.https://www.zbmath.org/1456.829592021-04-16T16:22:00+00:00"Mansfield, E. L."https://www.zbmath.org/authors/?q=ai:mansfield.elizabeth-louise"Rojo-Echeburúa, A."https://www.zbmath.org/authors/?q=ai:rojo-echeburua.aIn this paper, the authors consider variational problems for curves in 3-space for which the Lagrangian is invariant
under the special Euclidean group \(\mathrm{SE}(3)=\mathrm{SO}(3)\ltimes\mathbb{R}^3\) acting linearly in the standard way. They use the rotation minimizing frame, known as the normal, parallel, or Bishop frame. The authors derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants and syzygy operator for variational problems with a Euclidean symmetry. As application the author use variational problems in the study of stands of proteins, nucleid acids, and polymers.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Uniqueness in Calderón's problem for conductivities with unbounded gradient.https://www.zbmath.org/1456.352302021-04-16T16:22:00+00:00"Haberman, Boaz"https://www.zbmath.org/authors/?q=ai:haberman.boazSummary: We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in \({W^{s,p}(\Omega)}\), where \({\Omega \subset \mathbb{R}^{n}}\) is Lipschitz, \({3\leq n \leq 6}\), and \(s\) and \(p\) are such that \({ W^{s,p}(\Omega)\not \subset W^{1,\infty}(\Omega)}\). In particular, we obtain uniqueness for conductivities in \({W^{1,n}(\Omega)}\) (\(n = 3, 4\)). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.The wave based method: an overview of 15 years of research.https://www.zbmath.org/1456.350062021-04-16T16:22:00+00:00"Deckers, Elke"https://www.zbmath.org/authors/?q=ai:deckers.elke"Atak, Onur"https://www.zbmath.org/authors/?q=ai:atak.onur"Coox, Laurens"https://www.zbmath.org/authors/?q=ai:coox.laurens"D'Amico, Roberto"https://www.zbmath.org/authors/?q=ai:damico.roberto"Devriendt, Hendrik"https://www.zbmath.org/authors/?q=ai:devriendt.hendrik"Jonckheere, Stijn"https://www.zbmath.org/authors/?q=ai:jonckheere.stijn"Koo, Kunmo"https://www.zbmath.org/authors/?q=ai:koo.kunmo"Pluymers, Bert"https://www.zbmath.org/authors/?q=ai:pluymers.bert"Vandepitte, Dirk"https://www.zbmath.org/authors/?q=ai:vandepitte.dirk"Desmet, Wim"https://www.zbmath.org/authors/?q=ai:desmet.wimSummary: The Wave Based Method is a deterministic prediction technique to solve steady-state dynamic problems and is developed to overcome some of the frequency limitations imposed by element-based prediction techniques. The method belongs to the family of indirect Trefftz approaches and uses a weighted sum of so-called wave functions, which are exact solutions of the governing partial differential equations, to approximate the dynamic field variables. By minimising the errors on boundary and interface conditions, a system of equations is obtained which can be solved for the unknown contribution factors of each wave function. As a result, the system of equations is smaller and a higher convergence rate and lower computational loads are obtained as compared to conventional prediction techniques. On the other hand, the method shows its full efficiency for rather moderately complex geometries. As a result, various enhancements have been made to the method through the years, in order to extend the applicability of the Wave Based Method. This paper gives an overview of the current state of the art of the Wave Based Method, elaborating on the modelling procedure, a comparison of the properties of the Wave Based Method and element-based prediction techniques, application areas, extensions to the method such as hybrid and multi-level approaches and the most recent developments.Spectral element methods a priori and a posteriori error estimates for penalized unilateral obstacle problem.https://www.zbmath.org/1456.651092021-04-16T16:22:00+00:00"Djeridi, Bochra"https://www.zbmath.org/authors/?q=ai:djeridi.bochra"Ghanem, Radouen"https://www.zbmath.org/authors/?q=ai:ghanem.radouen"Sissaoui, Hocine"https://www.zbmath.org/authors/?q=ai:sissaoui.hocineSummary: The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre-Gauss-Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.On the multidimensional Black-Scholes partial differential equation.https://www.zbmath.org/1456.351002021-04-16T16:22:00+00:00"Guillaume, Tristan"https://www.zbmath.org/authors/?q=ai:guillaume.tristanAuthor's abstract: In this article, two general results are provided about the multidimensional Black-Scholes partial differential equation: its fundamental solution is derived, and it is shown how to turn it into the standard heat equation in whatever dimension. A fundamental connection is established between the multivariate normal distribution and the linear second order partial differential operator of parabolic type. These results allow to compute new closed form formulae for the valuation of multiasset options, with possible boundary crossing conditions, thus partially alleviating the ``curse of dimensionality'', at least in moderate dimension.
Reviewer: Miklavž Mastinšek (Maribor)Equivalence of sharp Trudinger-Moser inequalities in Lorentz-Sobolev spaces.https://www.zbmath.org/1456.460342021-04-16T16:22:00+00:00"Tang, Hanli"https://www.zbmath.org/authors/?q=ai:tang.hanliSummary: The critical and subcritical Trudinger-Moser inequalities in Lorentz Sobolev space have been studied by \textit{D. Cassani} and \textit{C. Tarsi} [Asymptotic Anal. 64, No. 1--2, 29--51 (2009; Zbl 1184.46034)], \textit{G.-Z. Lu} and \textit{H.-L. Tang} [Adv. Nonlinear Stud. 16, No. 3, 581--601 (2016; Zbl 1359.46035)].
In this paper, we will prove that these critical and subcritical Trudinger-Moser inequalities are actually equivalent and thus extend those equivalence results of \textit{Nguyen Lam} et al. [Rev. Mat. Iberoam. 33, No. 4, 1219--1246 (2017; Zbl 1387.46034)] into Lorentz Sobolev spaces.Space-time transformation acoustics.https://www.zbmath.org/1456.350862021-04-16T16:22:00+00:00"García-Meca, C."https://www.zbmath.org/authors/?q=ai:garcia-meca.carlos"Carloni, S."https://www.zbmath.org/authors/?q=ai:carloni.sante"Barceló, C."https://www.zbmath.org/authors/?q=ai:barcelo.carlos"Jannes, G."https://www.zbmath.org/authors/?q=ai:jannes.gil"Sánchez-Dehesa, J."https://www.zbmath.org/authors/?q=ai:sanchez-dehesa.jose|sanchez-dehesa.jesus"Martínez, A."https://www.zbmath.org/authors/?q=ai:martinez.asael-fabian|martinez.ana-m|martinez.angel-d|martinez.andre-l-m|martinez.alejandro-s|martinez.antonio-b|martinez.amadis|martinez.alvaro|martinez.arturo|martinez.aurea|martinez.alfonso|martinez.antonio-olivas|martinez.angel-r|martinez.alberto-a|martinez.anton|martinez.andrea|martinez.angeles|martinez.alejandro-d|martinez.aleix-m|martinez.alejandro-j|martinez.ana-gabriela|martinez.adam|martinez.andre|martinez.abel|martinez.alexandre-s|martinez.a-perez|martinez.alex-reche|martinez.aitor|martinez.andres-l|martinez.angelo|martinez.alejandra-m|martinez.andrew-bSummary: A recently proposed analogue transformation method has allowed the extension of transformation acoustics to general space-time transformations. We analyze here in detail the differences between this new analogue transformation acoustics (ATA) method and the standard one (STA). We show explicitly that STA is not suitable for transformations that mix space and time. ATA takes as starting point the acoustic equation for the velocity potential, instead of that for the pressure as in STA. This velocity-potential equation by itself already allows for some transformations mixing space and time, but not all of them. We explicitly obtain the entire set of transformations that leave its form invariant. It is for the rest of transformations that ATA shows its true potential, allowing for building a transformation acoustics method that enables the full range of space-time transformations. We provide an example of an important transformation which cannot be achieved with STA. Using this transformation, we design and simulate an acoustic frequency converter via the ATA approach. Furthermore, in those cases in which one can apply both the STA and ATA approaches, we study the different transformational properties of the corresponding physical quantities.Micropolar fluid flows with delay on 2D unbounded domains.https://www.zbmath.org/1456.351682021-04-16T16:22:00+00:00"Sun, Wenlong"https://www.zbmath.org/authors/?q=ai:sun.wenlongSummary: In this paper, we investigate the incompressible micropolar fluid flows on 2D unbounded domains with external force containing some hereditary characteristics. Since Sobolev embeddings are not compact on unbounded domains, first, we investigate the existence and uniqueness of the stationary solution, and further verify its exponential stability under appropriate conditions-essentially the viscosity \(\delta_1:=\min\{\nu, c_a+c_d\}\) is asked to be large enough. Then, we establish the global well-posedness of the weak solutions via the Galerkin method combined with the technique of truncation functions and decomposition of spatial domain.On the stochastic Dullin-Gottwald-Holm equation: global existence and wave-breaking phenomena.https://www.zbmath.org/1456.601612021-04-16T16:22:00+00:00"Rohde, Christian"https://www.zbmath.org/authors/?q=ai:rohde.christian"Tang, Hao"https://www.zbmath.org/authors/?q=ai:tang.haoSummary: We consider a class of stochastic evolution equations that include in particular the stochastic Camassa-Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces \(H^s\) with \(s>3/2\). Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.Interior estimates in the sup-norm for a class of generalized functions with integral representations.https://www.zbmath.org/1456.350512021-04-16T16:22:00+00:00"Ariza, Eusebio"https://www.zbmath.org/authors/?q=ai:ariza.eusebio"Di Teodoro, Antonio"https://www.zbmath.org/authors/?q=ai:di-teodoro.antonio-nicola"Vanegas, Judith"https://www.zbmath.org/authors/?q=ai:vanegas.judith-cSummary: In this paper we construct apriori estimates for the first order derivatives in the sup-norm for first order meta-monogenic functions, generalized monogenic functions satisfying a differential equation with an anti-monogenic right hand side and generalized meta-monogenic functions satisfying a differential equation with an anti-meta-monogenic right hand side. We obtain such estimates through integral representations of these classes of functions and give an explicit expression for the corresponding constants appearing in the estimates. Then we show how initial value problems can be solved in case an interior estimate is true in the function spaces under consideration. All related functions are in a Clifford type algebra.Existence of solutions for a class of nonlinear Choquard equations with critical growth.https://www.zbmath.org/1456.351862021-04-16T16:22:00+00:00"Ao, Yong"https://www.zbmath.org/authors/?q=ai:ao.yongSummary: In this paper, we consider the nonlinear Choquard equation of the form
\[
- \Delta u+u = (I_\alpha * |u|^p)|u|^{p-2}u + |u|^{q-2}u \text{ in } \mathbb{R}^N,
\] where \(I_\alpha\) is a Riesz potential, \(\alpha \in (0,1) \), \(N \geq 4\), \(p=\frac{N+\alpha}{N-2}\), \(2<q<2^*\). We show the existence of a nontrivial solution of the equation. Moreover, we consider the corresponding minimizing problem and obtain a nonnegative minimizer.An \(L^2\) to \(L^\infty\) framework for the Landau equation.https://www.zbmath.org/1456.351962021-04-16T16:22:00+00:00"Kim, Jinoh"https://www.zbmath.org/authors/?q=ai:kim.jinoh"Guo, Yan"https://www.zbmath.org/authors/?q=ai:guo.yan"Hwang, Hyung Ju"https://www.zbmath.org/authors/?q=ai:hwang.hyung-juThe authors consider the Landau equation with Coulomb potential: \(\partial
_{t}F+v\cdot \nabla _{x}F=Q(F,F)=\nabla v\cdot \int_{\mathbb{R}^{3}}\phi
(v-v^{\prime })[F(v^{\prime })\nabla _{v}F(v)-F(v)\nabla _{v}F(v^{\prime
})]dv\), posed in \((0,\infty )\times \mathbb{T}^{3}\), where \(\mathbb{T}^{3}\)
is the 3D torus, \(F(t,x,v)\geq 0\) is the spatially periodic distribution
function for particles, and \(\phi \) is the non-negative matrix defined as \(
\phi ^{ij}(v)=\{\delta _{i,j}-\frac{v_{i}v_{j}}{\left\vert v\right\vert ^{2}}
\}\left\vert v\right\vert ^{-1}\). They introduce the normalized Maxwellian \(
\mu (v)=e^{-\left\vert v\right\vert ^{2}}\)\ and writing \(F(t,x,v)=\mu
(v)+f(t,x,v)\) they observe that \(f\) satisfies \(f_{t}+v\cdot \partial
_{x}f+Lf=\Gamma (f,f)\), where \(L=-A-K\) is the linear operator with \(Af=\mu
^{-1/2}\partial _{i}\{\mu ^{1/2}\sigma ^{ij}[\partial _{j}f+v_{j}f]\}\), \(
Kf=-\mu ^{-1/2}\partial _{i}\{\mu \phi ^{ij}\ast \mu ^{1/2}[\partial
_{j}f+v_{j}f]\}\), and \(\Gamma (g,f)=\partial _{i}[\{\phi ^{ij}\ast \lbrack
\mu ^{1/2}g]\}\partial _{j}f]+\{\phi ^{ij}\ast \lbrack v_{i}\mu
^{1/2}g]\}\partial _{j}f-\partial _{i}[\{\phi ^{ij}\ast \lbrack \mu
^{1/2}\partial _{j}g]\}f]+\{\phi ^{ij}\ast \lbrack v_{i}\mu ^{1/2}\partial
_{j}g]\}f\). The initial condition \(f(0,x,v)=f_{0}(x,v)\) is added, where \(
f_{0}\) satisfies the conservation laws \(\int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}f_{0}(x,v)\sqrt{\mu }=\int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}v_{i}f_{0}(x,v)\sqrt{\mu }=\int \int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}\left\vert v\right\vert ^{2}f_{0}(x,v)\sqrt{\mu }=0\). The authors
define the notion of weak solution to this problem as a function \(
f(t,x,v)\in L^{\infty }((0,\infty )\times \mathbb{T}^{3}\times \mathbb{R}
^{3},w^{\vartheta }(v)dtdxdv)\), which satisfies \(\int_{0}^{T}\left\Vert
f(s)\right\Vert _{\sigma ,\vartheta }^{2}ds<+\infty \) and a variational
formulation issued from the above equation. Here \(\left\Vert f(s)\right\Vert
_{\sigma ,\vartheta }^{2}=\int \int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}w^{2\vartheta }[\sigma ^{ij}\partial _{i}f\partial _{j}f+\sigma
^{ij}v_{i}v_{j}f^{2}]dvdx\). The main result of the paper proves the
existence of a unique weak solution to this problem, if the initial data \(
f_{0}\) satisfies \(\left\Vert f_{0}\right\Vert _{\infty ,\vartheta }^{2}\leq
\varepsilon _{0}\) and \(\left\Vert -v\cdot \nabla _{v}f_{0}+\overline{A}
_{f_{0}}f_{0}\right\Vert _{\infty ,\vartheta }+\left\Vert
D_{v}f_{0}\right\Vert _{\infty ,\vartheta }<\infty \) for some \(\varepsilon
_{0}\in (0,1]\) and some positive \(\vartheta \). This weak solution satisfies
different estimates. For the proof, the authors first consider the
linearized Landau equation \(\partial _{t}f+v\cdot \partial _{x}f+Lf=\Gamma
(g,f)\), for some bounded function \(g\). They establish a uniform \(L^{2}\)
-estimate on a\ classical solution to the original problem and to this
linearized problem if \(\left\Vert g\right\Vert _{\infty }\) is small enough,
from which they then deduce a uniform \(L^{\infty }\)-estimate and a \(%
C^{0,\alpha }\)-estimate, through \(L^{2}-L^{\infty }\) estimates for the
solution of auxiliary linear problems. This allows deriving an Hölder
estimate and a \(S^{p}\)-estimate for the solution to the linearized problem,
where \(\left\Vert f\right\Vert _{S^{p}(\Omega )}=\left\Vert f\right\Vert _{L^{p}(\Omega )}+\left\Vert
D_{v}f\right\Vert _{L^{p}(\Omega )}+\left\Vert D_{vv}f\right\Vert
_{L^{p}(\Omega )}+\left\Vert (-\partial _{t}-v\cdot \nabla _{x})f\right\Vert
_{L^{p}(\Omega )}\), with \(\Omega =(0,\infty )\times \mathbb{T}^{3}\times
\mathbb{R}^{3}\).
Reviewer: Alain Brillard (Riedisheim)Hypocoercivity and sub-exponential local equilibria.https://www.zbmath.org/1456.828112021-04-16T16:22:00+00:00"Bouin, E."https://www.zbmath.org/authors/?q=ai:bouin.emeric"Dolbeault, J."https://www.zbmath.org/authors/?q=ai:dolbeault.jean"Lafleche, L."https://www.zbmath.org/authors/?q=ai:lafleche.laurent"Schmeiser, C."https://www.zbmath.org/authors/?q=ai:schmeiser.christianSummary: Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincaré inequalities compared to the more classical weak Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.Backward problems in time for fractional diffusion-wave equation.https://www.zbmath.org/1456.352052021-04-16T16:22:00+00:00"Floridia, Giuseppe"https://www.zbmath.org/authors/?q=ai:floridia.giuseppe"Yamamoto, Masahiro"https://www.zbmath.org/authors/?q=ai:yamamoto.masahiroDetermination of the reaction coefficient in a time dependent nonlocal diffusion process.https://www.zbmath.org/1456.650922021-04-16T16:22:00+00:00"Ding, Ming-Hui"https://www.zbmath.org/authors/?q=ai:ding.minghui"Zheng, Guang-Hui"https://www.zbmath.org/authors/?q=ai:zheng.guanghuiStability and existence of stationary solutions to the Euler-Poisson equations in a domain with a curved boundary.https://www.zbmath.org/1456.828882021-04-16T16:22:00+00:00"Suzuki, Masahiro"https://www.zbmath.org/authors/?q=ai:suzuki.masahiro"Takayama, Masahiro"https://www.zbmath.org/authors/?q=ai:takayama.masahiroSummary: The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of walls immersed in a plasma, and to analyze qualitative information of such a sheath layer. In the case of planar wall, Bohm proposed a criterion on the velocity of the positive ion for the formation of sheath, and several works gave its mathematical validation. It is of more interest to analyze the criterion for the nonplanar wall. In this paper, we study the existence and asymptotic stability of stationary solutions for the Euler-Poisson equations in a domain of which boundary is drawn by a graph. The existence and stability theorems are shown by assuming that the velocity of the positive ion satisfies the Bohm criterion at infinite distance. What most interests us in these theorems is that the criterion together with a suitable necessary condition guarantees the formation of sheaths as long as the shape of walls is drawn by a graph.Lie symmetry analysis and similarity solutions for the Camassa-Choi equations.https://www.zbmath.org/1456.351662021-04-16T16:22:00+00:00"Paliathanasis, Andronikos"https://www.zbmath.org/authors/?q=ai:paliathanasis.andronikosSummary: The method of Lie symmetry analysis of differential equations is applied to determine exact solutions for the Camassa-Choi equation and its generalization. We prove that the Camassa-Choi equation is invariant under an infinity-dimensional Lie algebra, with an essential five-dimensional Lie algebra. The application of the Lie point symmetries leads to the construction of exact similarity solutions.Some singular equations modeling MEMS.https://www.zbmath.org/1456.351932021-04-16T16:22:00+00:00"Laurençot, Philippe"https://www.zbmath.org/authors/?q=ai:laurencot.philippe"Walker, Christoph"https://www.zbmath.org/authors/?q=ai:walker.christophThis paper can be seen as a review about the derivation and the known (and unknown) mathematical results associated to the two dimensional version of microelectromechanical systems. The systems of equations describing these problems are determined by different models and parameters. Results about locally well-posedness, global solutions and stable stationary solutions are recalled depending on the values of these parameters and the precise models. The authors give a good description of the known (and unkown) results in a table where local and global existence and finite time singularity for the evolution problem as well as about existence of steady states are recalled.
Reviewer: Ramón Quintanilla De Latorre (Barcelona)Staggered DG method for coupling of the Stokes and Darcy-Forchheimer problems.https://www.zbmath.org/1456.651722021-04-16T16:22:00+00:00"Zhao, Lina"https://www.zbmath.org/authors/?q=ai:zhao.lina"Chung, Eric T."https://www.zbmath.org/authors/?q=ai:chung.eric-t"Park, Eun-Jae"https://www.zbmath.org/authors/?q=ai:park.eun-jae"Zhou, Guanyu"https://www.zbmath.org/authors/?q=ai:zhou.guanyuExistence results for a super-Liouville equation on compact surfaces.https://www.zbmath.org/1456.580162021-04-16T16:22:00+00:00"Jevnikar, Aleks"https://www.zbmath.org/authors/?q=ai:jevnikar.aleks"Malchiodi, Andrea"https://www.zbmath.org/authors/?q=ai:malchiodi.andrea"Wu, Ruijun"https://www.zbmath.org/authors/?q=ai:wu.ruijunLet \((M,g)\) be a closed Riemannian surface endowed with a genus bigger than one and \(K_g\) stands for the Gauss curvature of \(M\). The authors consider the functional energy defined by
\[\displaystyle J_\rho(u,\psi)=\int_M\left(|\nabla_g u|^2+2K_gu+\exp(2u)+2\langle({D}_g-\rho\exp(u))\psi,\psi\rangle\right)dv_g,\]
such that \(u\in C^\infty(M)\), \(\rho\) is a positive parameter, \(\psi\) is a spinor field on \(M\), and \({D_g}\) represents the Dirac operator on spinors. The Euler-Lagrange equation associated to \(J_\rho\) is defined by
\[ (*):\ \Delta_gu=\exp(2u)+K_g-\rho\exp(u)|\psi|^2\text{ and }{D}_g\psi=\rho\exp(u)\psi.\]
Then the authors state that \((*)\) has a non-zero solution whenever zero and \(\rho\) do not belong to the spectrum of \({D}_{g_0}\) (where \(g_0\) is a conformal metric to \(g\)) and \(K_{g_0}=-1\) (Theorem 1.1). The proof is essentially based on looking for a critical point of \(J_\rho\).
Reviewer: Mohammed El Aïdi (Bogotá)Global conservative solutions for a modified periodic coupled Camassa-Holm system.https://www.zbmath.org/1456.350572021-04-16T16:22:00+00:00"Chen, Rong"https://www.zbmath.org/authors/?q=ai:chen.rong"Pan, Shihang"https://www.zbmath.org/authors/?q=ai:pan.shihang"Zhang, Baoshuai"https://www.zbmath.org/authors/?q=ai:zhang.baoshuaiSummary: In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.All the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function.https://www.zbmath.org/1456.350822021-04-16T16:22:00+00:00"Fujita, Yasuhiro"https://www.zbmath.org/authors/?q=ai:fujita.yasuhiro"Hamamuki, Nao"https://www.zbmath.org/authors/?q=ai:hamamuki.nao"Yamaguchi, Norikazu"https://www.zbmath.org/authors/?q=ai:yamaguchi.norikazuSummary: We determine all the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function, which is everywhere continuous and nowhere differentiable. This result clarifies how singularities of the solution propagate along generalized characteristics. Moreover it turns out that the Takagi function still keeps the validity of the recent results in [\textit{P. Albano} et al., J. Differ. Equations 268, No. 4, 1412--1426 (2020; Zbl 1437.35153)], in which locally Lipschitz continuous initial data are handled.Green's function for the Schrödinger equation with a generalized point interaction and stability of superoscillations.https://www.zbmath.org/1456.811652021-04-16T16:22:00+00:00"Aharonov, Yakir"https://www.zbmath.org/authors/?q=ai:aharonov.yakir"Behrndt, Jussi"https://www.zbmath.org/authors/?q=ai:behrndt.jussi"Colombo, Fabrizio"https://www.zbmath.org/authors/?q=ai:colombo.fabrizio"Schlosser, Peter"https://www.zbmath.org/authors/?q=ai:schlosser.peterSummary: In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the \(\delta\) and \(\delta^\prime\)-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.Global existence and decay in multi-component reaction-diffusion-advection systems with different velocities: oscillations in time and frequency.https://www.zbmath.org/1456.351092021-04-16T16:22:00+00:00"de Rijk, Björn"https://www.zbmath.org/authors/?q=ai:de-rijk.bjorn"Schneider, Guido"https://www.zbmath.org/authors/?q=ai:schneider.guido.1|schneider.guidoSummary: It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e. nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture the difference in velocities. In this paper we present an alternative method, which is better applicable to multiple components. Our method involves a nonlinear iteration scheme that employs \(L^1-L^p\) estimates in Fourier space and exploits oscillations in time and frequency, which arise due to differences in transport. Under the assumption that each component exhibits different velocities, we establish global existence and decay for small, algebraically localized initial data in multi-component reaction-diffusion-advection systems allowing for cubic mixed-terms and nonlinear terms of Burgers' type.Existence of a solution to a stationary quasi-variational inequality in a multi-connected domain.https://www.zbmath.org/1456.350042021-04-16T16:22:00+00:00"Aramaki, Junichi"https://www.zbmath.org/authors/?q=ai:aramaki.junichiSummary: We consider a stationary quasi-variational inequality in a multi-connected domain and show the existence of a solution to the inequality. The problem is related to the Bean critical-state model in type II superconductors. Mathematically, we are concerned with a quasi-variational inequality containing a \(p\)-curl inequality with a curl constraint by a function of the solution.Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients.https://www.zbmath.org/1456.350722021-04-16T16:22:00+00:00"Chen, Si-Jia"https://www.zbmath.org/authors/?q=ai:chen.sijia"Lü, Xing"https://www.zbmath.org/authors/?q=ai:lu.xing"Tang, Xian-Feng"https://www.zbmath.org/authors/?q=ai:tang.xian-fengSummary: A generalized Burgers equation with variable coefficients is introduced based on the (2+1)-dimensional Burgers equation. Using the test function method combined with the bilinear form, we obtain the lump solutions to the generalized Burgers equation with variable coefficients. The amplitude and velocity of the extremum point are derived to analyze the propagation of the lump wave. Moreover, we derive and study the mixed solutions including lump-one-kink and lump-two-kink cases. With symbolic computation, two cases of relations among the parameters are yielded corresponding to the solutions. Different and interesting interaction phenomena arise from assigning abundant functions to the variable coefficients. Especially, we find that the shape of kink waves might be parabolic type, and one lump wave can be decomposed into two lump waves. The test function method is applicable for the generalized Burgers equation with variable coefficients, and it will be applied to some other variable-coefficient equations in the future.An interpolated time-domain equivalent source method for modeling transient acoustic radiation over a mass-like plane based on the transient half-space Green's function.https://www.zbmath.org/1456.761212021-04-16T16:22:00+00:00"Pan, Siwei"https://www.zbmath.org/authors/?q=ai:pan.siwei"Jiang, Weikang"https://www.zbmath.org/authors/?q=ai:jiang.weikang"Xiang, Shang"https://www.zbmath.org/authors/?q=ai:xiang.shang"Liu, Xiujuan"https://www.zbmath.org/authors/?q=ai:liu.xiujuanSummary: Interpolated time-domain equivalent source method (ITDESM) is based on the assumption of free space, which makes it not suitable for reconstructing the transient acoustic quantities in the half space. Here, a half-space ITDESM is proposed to model the transient acoustic radiation over a mass-like plane. In this method, the free transient Green's function existing in the conventional ITDESM is replaced by a closed-form transient half-space Green's function for a mass-like plane. Such transient Green's function enables one to take the reflection effect of the mass-like plane into consideration. Modeling acoustic radiation from three transient monopoles above an infinite plane with mass-like behavior is studied by numerical simulations to demonstrate the feasibility of the half-space ITDESM. The proposed method is also examined by comparing the reconstruction accuracy among a free-field model, a rigid plane model and a mass-like plane model. An experiment with an impacted steel plate lying above a table plate is conducted in the semi-anechoic room, and the results further verify the effectiveness of the proposed method.WKB analysis for the three coupled long wave-short wave interaction equations.https://www.zbmath.org/1456.811912021-04-16T16:22:00+00:00"Muslu, Gulcin M."https://www.zbmath.org/authors/?q=ai:muslu.gulcin-m"Lin, Chi-Kun"https://www.zbmath.org/authors/?q=ai:lin.chikunSummary: This paper is devoted to the WKB analysis of the three coupled long wave-short wave interaction (LSI for short) equations. We consider the zero-dispersion limit of the LSI for initial data with Sobolev regularity, before the shocks appear in the limit system. For the smooth solution, the limit system is given by the two fluids equations. The split-step Fourier method is also employed to justify the numerical simulation of the small dispersion limit.Distributional solutions for damped wave equations.https://www.zbmath.org/1456.350032021-04-16T16:22:00+00:00"Nualart, Marc"https://www.zbmath.org/authors/?q=ai:nualart.marcSummary: This work presents results on solutions to the one-dimensional damped wave equation, also called telegrapher's equation, when the initial conditions are general distributions. We make a complete deduction of its fundamental solutions, both for positive and negative times. To obtain them we only use self-similarity arguments and distributional calculus, making no use of Fourier or Laplace transforms. We next use these fundamental solutions to prove both the existence and the uniqueness of solutions to the distributional initial value problem. As applications we recover the semi-group property for initial data in classical function spaces, and we find the probability distribution function for a recent financial model of evolution of prices.A facile method to realize perfectly matched layers for elastic waves.https://www.zbmath.org/1456.740692021-04-16T16:22:00+00:00"Chang, Zheng"https://www.zbmath.org/authors/?q=ai:chang.zheng"Guo, Dengke"https://www.zbmath.org/authors/?q=ai:guo.dengke"Feng, Xi-Qiao"https://www.zbmath.org/authors/?q=ai:feng.xiqiao"Hu, Gengkai"https://www.zbmath.org/authors/?q=ai:hu.gengkaiSummary: In perfectly matched layer (PML) technique, an artificial layer is introduced in the simulation of wave propagation as a boundary condition which absorbs all incident waves without any reflection. Such a layer is generally thought to be unrealizable due to its complicated material formulation. In this paper, on the basis of transformation elastodynamics and complex coordinate transformation, a novel method is proposed to design PMLs for elastic waves. By applying the conformal transformation technique, the proposed PML is formulated in terms of conventional constitutive parameters and then can be easily realized by functionally graded viscoelastic materials. We perform numerical simulations to validate the material realization and performance of this PML.Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms.https://www.zbmath.org/1456.350532021-04-16T16:22:00+00:00"Li, Fengbai"https://www.zbmath.org/authors/?q=ai:li.fengbai"Wang, Weike"https://www.zbmath.org/authors/?q=ai:wang.weike"Wang, Yutong"https://www.zbmath.org/authors/?q=ai:wang.yutongSummary: This article concerns the Cauchy problem of conservation laws with nonlocal dissipation-type terms in \(\mathbb{R}^3\). By using Green's function and the time-frequency decomposition method, we study global classical solutions and their long time behavior including pointwise estimates for large initial data, for solutions near the nontrivial equilibrium state.Two-dimensional singular splash pulses.https://www.zbmath.org/1456.350052021-04-16T16:22:00+00:00"Zlobina, E. A."https://www.zbmath.org/authors/?q=ai:zlobina.ekaterina-a"Kiselev, A. P."https://www.zbmath.org/authors/?q=ai:kiselev.aleksei-pSummary: It is proved that a certain simple specification of the 2D Bateman-type complexified solution having a singularity at a running point satisfies the homogeneous wave equation, whereas the respective noncomplexified function does not.Existence and uniqueness results for time-inhomogeneous time-change equations and Fokker-Planck equations.https://www.zbmath.org/1456.351982021-04-16T16:22:00+00:00"Döring, Leif"https://www.zbmath.org/authors/?q=ai:doring.leif"Gonon, Lukas"https://www.zbmath.org/authors/?q=ai:gonon.lukas"Prömel, David J."https://www.zbmath.org/authors/?q=ai:promel.david-j"Reichmann, Oleg"https://www.zbmath.org/authors/?q=ai:reichmann.olegSummary: We prove existence and uniqueness of solutions to Fokker-Planck equations associated with Markov operators multiplicatively perturbed by degenerate time-inhomogeneous coefficients. Precise conditions on the time-inhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be either globally bounded or bounded away from zero. The approach is based on constructing random time-changes and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.Well-posedness and finite element approximation for the stationary magneto-hydrodynamics problem with temperature-dependent parameters.https://www.zbmath.org/1456.651662021-04-16T16:22:00+00:00"Qiu, Hailong"https://www.zbmath.org/authors/?q=ai:qiu.hailongSummary: In this article we study a well-posedness and finite element approximation for the non-isothermal incompressible magneto-hydrodynamics flow subject to a generalized Boussinesq problem with temperature-dependent parameters. Applying some similar hypotheses in [\textit{R. Oyarzúa} et al., IMA J. Numer. Anal. 34, No. 3, 1104--1135 (2014; Zbl 1301.76052)], we prove the existence and uniqueness of weak solutions and discrete weak solutions, and derive optimal error estimates for small and smooth solutions. Finally, we provide some numerical results to confirm the rates of convergence.A variational Lagrangian scheme for a phase-field model: a discrete energetic variational approach.https://www.zbmath.org/1456.651182021-04-16T16:22:00+00:00"Liu, Chun"https://www.zbmath.org/authors/?q=ai:liu.chun"Wang, Yiwei"https://www.zbmath.org/authors/?q=ai:wang.yiweiIn this paper, a variational Lagrangian scheme to a phase-field model is proposed, which can compute equilibrium states of the original Allen-Cahn type phase-field model with a proper choice of the initial condition. A modified Allen-Cahn type model is obtained, by adding energy-dissipation law for a general phase-field model. The additional dissipation term can be viewed as a regularization term, that imposes a mechanism to minimize the total free energy in terms of the Lagrangian map. Based on the energy-dissipation law, a variational structure-preserving Lagrangian scheme is constructed by employing a discrete energetic variational approach. Several numerical tests on a quasi-1D problem shrinkage of a circular domain, phase-field model with the volume constraint, and on a slightly compressible flow, show that the scheme can capture the thin diffuse interface in equilibria with a small number of mesh points.
Reviewer: Bülent Karasözen (Ankara)Existence, multiplicity and regularity of solutions for the fractional \(p\)-Laplacian equation.https://www.zbmath.org/1456.352162021-04-16T16:22:00+00:00"Kim, Yun-Ho"https://www.zbmath.org/authors/?q=ai:kim.yunho.1|kim.yunhoSummary: : We are concerned with the following elliptic equations:
\[
\begin{cases} (-\Delta)_p^su=\lambda f(x,u) \quad \text{in }\Omega\\u= 0\quad\text{on }\mathbb{R}^N\backslash\Omega,\end{cases}
\]
where \(\lambda\) are real parameters, \((-\Delta)_p^s\) is the fractional \(p\)-Laplacian operator, \(0 < s < 1 < p < +\infty, sp < N\), and \(f:\Omega\times\mathbb R\to\mathbb R\) satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters \(\lambda\) for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity \(f\) has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in \(L^{\infty}(\Omega)\) of any possible weak solution by applying the bootstrap argument.Local-in-time well-posedness for compressible MHD boundary layer.https://www.zbmath.org/1456.351622021-04-16T16:22:00+00:00"Huang, Yongting"https://www.zbmath.org/authors/?q=ai:huang.yongting"Liu, Cheng-Jie"https://www.zbmath.org/authors/?q=ai:liu.cheng-jie"Yang, Tong"https://www.zbmath.org/authors/?q=ai:yang.tongSummary: In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work [the second author et al., Commun. Pure Appl. Math. 72, No. 1, 63--121 (2019; Zbl 1404.35492)], we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.Categorical localization for the coherent-constructible correspondence.https://www.zbmath.org/1456.140472021-04-16T16:22:00+00:00"Ike, Yuichi"https://www.zbmath.org/authors/?q=ai:ike.yuichi"Kuwagaki, Tatsuki"https://www.zbmath.org/authors/?q=ai:kuwagaki.tatsukiKontsevich's homological mirror symmetry(HMS) conjecture states that two categories associated to
a mirror pair are equivalent. For a Calabi-Yau(CY) variety, a mirror is also Calabi-Yau and the conjecture is a
quasi-equivalence between the dg category of coherent sheaves over one and the derived Fukaya category of
the other. For non-CY's, mirrors do not need to be varieties. For a Fano toric variety, its mirror is a
Landau-Ginzburg (LG) model, which is a holomorphic function on \((\mathbb C^\times)^n\) which can be read from
the defining fan of the toric variety which is in fact the specialization of Lagrangian potential
function of the toric \(A\)-model that is the generating function of open Gromov-Witten invariants of
a toric fiber [\textit{C.-H. Cho} and \textit{Y.-G. Oh}, Asian J. Math. 10, No. 4, 773--814 (2006; Zbl 1130.53055); \textit{K. Fukaya} et al., Duke Math. J. 151, No. 1, 23--175 (2010; Zbl 1190.53078)].
For a smooth Fano, it has been proven for many special cases that the dg category of coherent sheaves
\(\mathbf{coh}\, X_\Sigma\) over the toric variety \(X_\Sigma\) associated to the fan \(\Sigma\) is quasi-equivalent to
the Fukaya-Seidel category \(\mathfrak{Fuk}(W_\Sigma)\) of the associated Laurent polynomial \(W_\Sigma\).
When a variety is not complete, \(\mathbf{coh}\, X_\Sigma\) is of infinite dimensional nature and its Fukaya-type
category also should have an infinite-dimensional nature. Such a construction is known to be
(partially) wrapped Fukaya categories. In this regard, the main theme of the present paper in review is
to establish a quasi-isomorphism \(\mathbf{coh}(X\setminus D)\cong \mathbf{coh} X/\mathbf{coh}_D X\) in some special cases
in the microlocal world: Here
\(X\setminus D\) is the complement of a divisor \(D\) and \(\mathbf{coh} X/\mathbf{coh}_D X\) is the dg category of
sheaves supported in \(D\) by relating the isomorphism to a similar isomorphism
\[
W_{\mathbf{s}\setminus\mathbf{r}}(M) \cong W_{\mathbf{s}}(M)/\mathfrak B_{\mathbf{r}}
\]
of \textit{Z. Sylvan} [J. Topol. 12, No. 2, 372--441 (2019; Zbl 1430.53097)] in the Fukaya-Seidel side: Here \(\mathbf{s}\) is a collection
of symplectic stops and \(\mathbf{r} \subset\mathbf{s}\) is a sub-collection thereof, and
\(\mathfrak B_{\mathbf{r}}\) is the full subcategory spanned by Lagrangians near the sub-stops \(\mathbf{r}\).
The paper extends a version of coherent-constructible correspondence [\textit{B. Fang} et al., Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011); \textit{K. Bongartz} et al., Adv. Math. 226, No. 2, 1875--1910 (2011; Zbl 1223.16004)] to the dg category of
\emph{quasi-coherent shaves} over \(X_\Sigma\) in dimension 2.
Reviewer: Yong-Geun Oh (Pohang)Persistence time of solutions of the three-dimensional Navier-Stokes equations in Sobolev-Gevrey classes.https://www.zbmath.org/1456.351512021-04-16T16:22:00+00:00"Biswas, Animikh"https://www.zbmath.org/authors/?q=ai:biswas.animikh"Hudson, Joshua"https://www.zbmath.org/authors/?q=ai:hudson.joshua"Tian, Jing"https://www.zbmath.org/authors/?q=ai:tian.jingSummary: In this paper, we study existence times of strong solutions of the three-dimensional Navier-Stokes equations in time-varying analytic Gevrey classes based on Sobolev spaces \(H^s, s > \frac{1}{2}\). This complements the seminal work of \textit{C. Foias} and \textit{R. Temam} on \(H^1\) based Gevrey classes [J. Funct. Anal. 87, No. 2, 359--369 (1989; Zbl 0702.35203)], thus enabling us to improve estimates of the analyticity radius of solutions for certain classes of initial data. The main thrust of the paper consists in showing that the existence times in the much stronger Gevrey norms (i.e. the norms defining the analytic Gevrey classes which quantify the radius of real-analyticity of solutions) match the best known persistence times in Sobolev classes. Additionally, as in the case of persistence times in the corresponding Sobolev classes, our existence times in Gevrey norms are optimal for \(\frac{1}{2} < s < \frac{5}{2}\).Nonexistence of positive supersolutions of nonlinear biharmonic equations without the maximum principle.https://www.zbmath.org/1456.352392021-04-16T16:22:00+00:00"Ghergu, Marius"https://www.zbmath.org/authors/?q=ai:ghergu.marius"Taliaferro, Steven D."https://www.zbmath.org/authors/?q=ai:taliaferro.steven-dSummary: We study classical positive solutions of the biharmonic inequality
\[
-\Delta^2 r\geq f(v)
\]
in exterior domains in \(\mathbb{R}^n\) where \(f: (0,\infty) \to (0,\infty)\) is continuous function. We give lower bounds on the growth of \(f(s)\) at \(s=0\) and/or \(s=\infty\) such that inequality \((0.1)\) has no \(C^{4}\) positive solution in any exterior domain of \(\mathbb{R}^n\). Similar results were obtained by Armstrong and Sirakov for \(-\Delta v \geq f(v)\) using a method which depends only on properties related to the maximum principle. Since the maximum principle does not hold for the biharmonic operator, we adopt a different approach which relies on a new representation formula and an a priori pointwise bound for nonnegative solutions of \(-\Delta^{2}u \geq 0\) in a punctured neighborhood of the origin in \(\mathbb{R}^n\).Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation.https://www.zbmath.org/1456.350072021-04-16T16:22:00+00:00"He, Xue-Jiao"https://www.zbmath.org/authors/?q=ai:he.xue-jiao"Lü, Xing"https://www.zbmath.org/authors/?q=ai:lu.xing"Li, Meng-Gang"https://www.zbmath.org/authors/?q=ai:li.menggangSummary: With the Hirota bilinear method and symbolic computation, we investigate the \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation. Based on its bilinear form, the bilinear Bäcklund transformation is constructed, which consists of four equations and five free parameters. The Pfaffian, Wronskian and Grammian form solutions are derived by using the properties of determinant. As an example, the one-, two- and three-soliton solutions are constructed in the context of the Pfaffian, Wronskian and Grammian forms. Moreover, the triangle function solutions are given based on the Pfaffian form solution. A few particular solutions are plotted by choosing the appropriate parameters.Maximal-in-time existence and uniqueness of strong solution of a 3D fluid-structure interaction model.https://www.zbmath.org/1456.351652021-04-16T16:22:00+00:00"Maity, Debayan"https://www.zbmath.org/authors/?q=ai:maity.debayan"Raymond, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:raymond.jean-pierre"Roy, Arnab"https://www.zbmath.org/authors/?q=ai:roy.arnabThe authors examine the interaction between a viscous, Newtonian, incompressible fluid and an elastic structure modeled by a nonlinear
damped shell equation. They prove the existence of a unique maximal strong solution. The main tools in the proof are, among many, a geometric change of variables, appropriate weighted Sobolev spaces and Banach fixed point theorem.
Reviewer: Mohamed Majdoub (Dammam)Uniform boundedness for reaction-diffusion systems with mass dissipation.https://www.zbmath.org/1456.351052021-04-16T16:22:00+00:00"Cupps, Brian P."https://www.zbmath.org/authors/?q=ai:cupps.brian-p"Morgan, Jeff"https://www.zbmath.org/authors/?q=ai:morgan.jeff-j"Tang, Bao Quoc"https://www.zbmath.org/authors/?q=ai:tang.bao-quocA fully-mixed formulation for the steady double-diffusive convection system based upon Brinkman-Forchheimer equations.https://www.zbmath.org/1456.651552021-04-16T16:22:00+00:00"Caucao, Sergio"https://www.zbmath.org/authors/?q=ai:caucao.sergio"Gatica, Gabriel N."https://www.zbmath.org/authors/?q=ai:gatica.gabriel-n"Oyarzúa, Ricardo"https://www.zbmath.org/authors/?q=ai:oyarzua.ricardo"Sánchez, Nestor"https://www.zbmath.org/authors/?q=ai:sanchez.nestor-eSummary: We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman-Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman-Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in \(\text{L}^4\). As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable \(\text{H}(\operatorname{div})\)-type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram and Banach-Nečas-Babuška theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart-Thomas spaces of order \(k\geq 0\) for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree \(\leq k+1\) are employed for the velocity, and piecewise polynomials of degree \(\leq k\) for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.Shear-wave resonances in a fluid-solid-solid layered structure.https://www.zbmath.org/1456.351332021-04-16T16:22:00+00:00"Martin, P. A."https://www.zbmath.org/authors/?q=ai:martin.paul-andrew|martin.paulo-a|martin.philippa-aSummary: An inhomogeneous solid layer is bounded on one side by a fluid half-space and on the other by a homogeneous solid half-space. An acoustic wave in the fluid is incident on the layer. Experiments suggest that some kind of shear-wave resonance of the layer exists. Here, the layer is modeled with exponential variations of the material properties (Epstein model). Solutions in terms of hypergeometric functions are found. Genuine resonances are found but only when the layer is not bonded to the solid half-space; these are analogous to Jones frequencies in fluid-solid interaction problems. When the solid half-space is present, the resonances become complex: they are scattering frequencies. Simple but accurate asymptotic approximations are found using known estimates for hypergeometric functions with large parameters.Time-fractional Allen-Cahn equations: analysis and numerical methods.https://www.zbmath.org/1456.650622021-04-16T16:22:00+00:00"Du, Qiang"https://www.zbmath.org/authors/?q=ai:du.qiang"Yang, Jiang"https://www.zbmath.org/authors/?q=ai:yang.jiang"Zhou, Zhi"https://www.zbmath.org/authors/?q=ai:zhou.zhiSummary: In this work, we consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order \(\alpha \in (0,1)\). First, the well-posedness and (limited) smoothing property are studied, by using the maximal \(L^p\) regularity of fractional evolution equations and the fractional Grönwall's inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall's inequality and maximal \(\ell^p\) regularity, we prove that the convergence rates of those time-stepping schemes are \(O(\tau^\alpha)\) without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen-Cahn dynamics.Global analysis of quasilinear wave equations on asymptotically de Sitter spaces.https://www.zbmath.org/1456.351412021-04-16T16:22:00+00:00"Hintz, Peter"https://www.zbmath.org/authors/?q=ai:hintz.peterSummary: We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity \(L^2\)-based function spaces and using iterative arguments for the nonlinear problems. The linear analysis is accomplished in two parts: Firstly, a regularity theory is developed by means of a calculus for pseudodifferential operators with non-smooth coefficients, similar to the one developed by Beals and Reed, on manifolds with boundary. Secondly, the asymptotic behavior of solutions to linear equations is studied using resonance expansions, introduced in this context by Vasy using the framework of Melrose's \(b\)-analysis.Dynamics and stability of sessile drops with contact points.https://www.zbmath.org/1456.351542021-04-16T16:22:00+00:00"Tice, Ian"https://www.zbmath.org/authors/?q=ai:tice.ian"Wu, Lei"https://www.zbmath.org/authors/?q=ai:wu.lei.1Authors' abstract: The authors consider the dynamics of a two-dimensional droplet of incompressible viscous fluid evolving above a one-dimensional flat surface under the influence of gravity. This is a free boundary problem: the interface between the fluid on the surface and the air above is free to move and experience capillary forces. A mathematical model of this problem is formulated and some a priori estimates are obtained. These estimates are used to show that for initial data sufficiently close to equilibrium, there exist global solutions of the model that decay to a shifted equilibrium exponentially fast.
Reviewer: Gheorghe Moroşanu (Cluj-Napoca)Classical solutions to the initial-boundary value problems for nonautonomous fractional diffusion equations.https://www.zbmath.org/1456.352182021-04-16T16:22:00+00:00"Mu, Jia"https://www.zbmath.org/authors/?q=ai:mu.jia"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.6|liu.yang.3|liu.yang.2|liu.yang.17|liu.yang.18|liu.yang.11|liu.yang.8|liu.yang.21|liu.yang.14|liu.yang.22|liu.yang.19|liu.yang.9|liu.yang.4|liu.yang.5|liu.yang.20|liu.yang.12|liu.yang.13|liu.yang|liu.yang.23|liu.yang.10|liu.yang.15|liu.yang.16|liu.yang.1"Zhang, Huanhuan"https://www.zbmath.org/authors/?q=ai:zhang.huanhuanSummary: In this paper, we investigate a class of nonautonomous fractional diffusion equations (NFDEs). Firstly, under the condition of weighted Hölder continuity, the existence and two estimates of classical solutions are obtained by virtue of the properties of the probability density function and the evolution operator family. Secondly, it focuses on the continuity and an estimate of classical solutions in the sense of fractional power norm. The results generalize some existing results on classical solutions and provide theoretical support for the application of NFDE.On generalized diffusion and heat systems on an evolving surface with a boundary.https://www.zbmath.org/1456.351952021-04-16T16:22:00+00:00"Koba, Hajime"https://www.zbmath.org/authors/?q=ai:koba.hajimeThe author considers a bounded domain \(U\subset \mathbb{R}^{2}\) with a
piecewise Lipschitz continuous boundary and an evolving surface \(\Gamma
(t)\subset \mathbb{R}^{3}\), \(0 < t < T\leq +\infty \), defined through \(
\Gamma (t)=\{x=^{t}(x_{1},x_{2},x_{3})\in \mathbb{R}^{3};\) \(x=\widehat{x}
(X,t)\), \(X\in U\}\) with a piecewise Lipschitz continuous boundary. The
motion velocity \(w\) of the evolving surface \(\Gamma (t)\) is defined through \(
w(\widehat{x}(X,t),t)=\frac{\partial \widehat{x}}{\partial t}(X,t)\). The
first main purpose of the paper is to derive and study the generalized
diffusion system on the evolving surface \(\cup _{0 < t < T}\Gamma (t)\times \{t\}
\): \(\partial _{t}f+(w\cdot \nabla )f+(\mathrm{div}_{\Gamma })C=\mathrm{div}
_{\Gamma }\{e_{1}^{\prime }\left\vert \mathrm{grad}_{\Gamma }C\right\vert ^{2}
\mathrm{grad}_{\Gamma }C\}\). The proof is based on a surface divergence
theorem in this case: for every \(\varphi =^{t}(\varphi _{1},\varphi
_{2},\varphi _{3})\in C^{1}(\overline{\Gamma (t)})\), one has: \(\int_{\Gamma
(t)} \mathrm{div}_{\Gamma }\varphi d\mathcal{H}_{x}^{2}=-\int_{\Gamma (t)}
\mathrm{div}_{\Gamma }n(n\cdot \varphi )d\mathcal{H}_{x}^{2}+\int_{\partial \Gamma
(t)}\nu \cdot \varphi d\mathcal{H}_{x}^{1}\), where \(\nu \) is the unit outer
co-normal vector. The proof also uses a formula for the variation of a
dissipation energy: if \(e_{D}\) is a \(C^{1}\)-function, \(0 < t < T\), \(f\in
C^{2}(\Gamma (t))\), \(-1<\varepsilon <1\), \(\psi \in C_{0}^{1}(\Gamma (t))\),
and \(E_{D}[f+\varepsilon \psi ](t)=-\int_{\Gamma (t)}\frac{1}{2}
e_{D}(\left\vert \nabla _{\Gamma }(f+\varepsilon \psi )\right\vert ^{2})d
\mathcal{H}_{x}^{2}\), then \(\frac{d}{d\varepsilon }\mid _{\varepsilon
=0}E_{D}[f+\varepsilon \psi ]=\int_{\Gamma (t)} \mathrm{div}_{\Gamma
}\{e_{D}^{\prime }(\left\vert \mathrm{grad}_{\Gamma }f\right\vert ^{2})
\mathrm{grad}_{\Gamma }f\}\psi d\mathcal{H}_{x}^{2}\). Then the author proves that if
\(\frac{\partial C}{\partial \nu }\mid _{\Gamma (t)}=0\), any solution to the
diffusion system satisfies the conservation law \(\int_{\Gamma
(t_{2})}C(x,t_{2})d\mathcal{H}_{x}^{2}=\int_{\Gamma (t_{1})}C(x,t_{1})d
\mathcal{H}_{x}^{2}\) and the energy law \(\int_{\Gamma (t_{2})}\frac{1}{2}
\left\vert C(x,t_{2})\right\vert ^{2}d\mathcal{H}_{x}^{2}+
\int_{t_{1}}^{t_{2}}\int_{\Gamma (\tau )}e_{1}^{\prime }(\left\vert
\mathrm{grad}_{\Gamma }C\right\vert ^{2})\left\vert \mathrm{grad}_{\Gamma
}C\right\vert ^{2}d\mathcal{H}_{x}^{2}d\tau =\int_{\Gamma (t_{1})}\frac{1}{2}
\left\vert C(x,t_{1})\right\vert ^{2}d\mathcal{H}_{x}^{2}\) (\(t_{1}<t_{1}\)).
The author also derives and studies the generalized heat system on the
evolving surface \(\cup _{0<t<T}\Gamma (t)\times \{t\}\): \(\partial _{t}\rho
+(w\cdot \nabla )\rho +(\mathrm{div}_{\Gamma }w)\rho =0\), \(\rho \partial
_{t}\theta +\rho (w\cdot \nabla )\theta =e_{2}^{\prime }(\left\vert
\mathrm{grad}_{\Gamma }\theta \right\vert ^{2})\mathrm{grad}_{\Gamma }\theta \). For
the proof, the author first derives a transport equation for every \(f\in
C^{1}(\Gamma (t))\) and \(\Omega (t)\subset \Gamma (t)\), \(\frac{d}{
d\varepsilon }\int_{\Omega (t)}f(x,t)d\mathcal{H}_{x}^{2}=\int_{\Omega
(t)}\{\partial _{t}f+(w\cdot \nabla )f+(\mathrm{div}_{\Gamma }w)\rho \}(x,t)d
\mathcal{H}_{x}^{2}\). In the two last parts of his paper, the author
considers the case of an evolving double bubble. He here proves divergence
and transport theorems and he proposes a mathematical model for a diffusion
process on an evolving double bubble from an energetic point of view.
Reviewer: Alain Brillard (Riedisheim)The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications.https://www.zbmath.org/1456.350982021-04-16T16:22:00+00:00"Lewin, Mathieu"https://www.zbmath.org/authors/?q=ai:lewin.mathieu"Rota Nodari, Simona"https://www.zbmath.org/authors/?q=ai:nodari.simona-rotaSummary: In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form \(\Delta u+g(u)=0\). Our result applies in particular to the double power non-linearity where \(g(u)=u^q-u^p-\mu u\) for \(p>q>1\) and \(\mu >0\), which we discuss with more details. In this case, the non-degeneracy of the unique solution \(u_\mu\) allows us to derive its behavior in the two limits \(\mu \rightarrow 0\) and \(\mu \rightarrow \mu_*\) where \(\mu_*\) is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the \(L^2\) mass of \(u_\mu\) in terms of \(\mu\), which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of \(u_\mu\).On critical Schrödinger-Kirchhoff-type problems involving the fractional \(p\)-Laplacian with potential vanishing at infinity.https://www.zbmath.org/1456.352232021-04-16T16:22:00+00:00"Van Thin, Nguyen"https://www.zbmath.org/authors/?q=ai:thin.nguyen-van"Xiang, Mingqi"https://www.zbmath.org/authors/?q=ai:xiang.mingqi"Zhang, Binlin"https://www.zbmath.org/authors/?q=ai:zhang.binlinSummary: The aim of this paper is to study the existence of solutions for critical Schrödinger-Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem:
\[
M\left( \iint_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\text{dxdy}+\int_{{\mathbb{R}}^N}V(x)|u(x)|^p\text{{d}}x\right) ((-\Delta )_p^su(x)+V(x)|u|^{p-2}u) =K(x)(\lambda f(x,u)+|u|^{p_s^*-2}u),
\]
where \(M:[0, \infty)\rightarrow [0, \infty)\) is a continuous function, \((-\Delta)_p^s\) is the fractional \(p\)-Laplacian, \(0<s<1<p<\infty\) with \(sp<N\), \(p_s^*=Np/(N-ps)\), \(K\), \(V\) are nonnegative continuous functions satisfying some conditions, and \(f\) is a continuous function on \(\mathbb{R}^N\times \mathbb{R}\) satisfying the Ambrosetti-Rabinowitz-type condition, \(\lambda >0\) is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space \(W\). For this, we first study the properties of the embedding from \(W\) into \(L_K^\alpha (\mathbb{R}^N), \alpha \in [p,p_s^*]\). Then, we obtain the differentiability of energy functional with some suitable conditions on \(f\). To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional \(p\)-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of \textit{J. M. Do Ó.} et al. [Commun. Contemp. Math. 18, No. 6, Article ID 1550063, 20 p. (2016; Zbl 1348.35152)] and \textit{Q. Li} et al. [Mediterr. J. Math. 14, No. 2, Paper No. 80, 14 p. (2017; Zbl 1371.35047)].Global well-posedness for the 2D fractional Boussinesq equations in the subcritical case.https://www.zbmath.org/1456.351742021-04-16T16:22:00+00:00"Zhou, Daoguo"https://www.zbmath.org/authors/?q=ai:zhou.daoguo"Li, Zilai"https://www.zbmath.org/authors/?q=ai:li.zilai"Shang, Haifeng"https://www.zbmath.org/authors/?q=ai:shang.haifeng"Wu, Jiahong"https://www.zbmath.org/authors/?q=ai:wu.jiahong"Yuan, Baoquan"https://www.zbmath.org/authors/?q=ai:yuan.baoquan"Zhao, Jiefeng"https://www.zbmath.org/authors/?q=ai:zhao.jiefengThe paper deals with the global in time well-posedness problem for the 2D Boussinesq equations with fractional dissipation in the subcritical regime. Namely, in the velocity dissipation dominated case for the largest possible range of the exponent of \(-\Delta^{\frac{\alpha}{2}}\), \(\alpha\in(0,1)\), the existence of a unique regular global in time solution is proved.
Reviewer: Georg V. Jaiani (Tbilisi)Symmetric vortices for two-component \(p\)-Ginzburg-Landau systems.https://www.zbmath.org/1456.351902021-04-16T16:22:00+00:00"Duan, Lipeng"https://www.zbmath.org/authors/?q=ai:duan.lipeng"Yang, Jun"https://www.zbmath.org/authors/?q=ai:yang.jun.1|yang.jun.2|yang.jun|yang.jun.3Summary: Given \(p > 2\) for the following coupled \(p\)-Ginzburg-Landau model in \(\mathbb{R}^2\)
\[
\begin{aligned}
-\Delta_p u^+ + \left[A_+ (|u^+|^2 - t^{+^2}) + A_0(|u^-|^2 - t^{-^2})\right] u^+ = 0, \\
-\Delta_p u^- + \left[A_- (|u^-|^2 - t^{-^2}) + A_0(|u^+|^2 - t^{+^2})\right] u^- = 0,
\end{aligned}
\]
with the constraints
\[
A_+, A_- > 0, A_0^2 < A_+ A_- \quad \text{and} \quad t^+, t^- > 0,
\]
we consider the existence of symmetric vortex solutions \(u(x) = (U_p^+(r) e^{in^+ \theta}, U_p^-(r) e^{in^- \theta})\) with given degree \((n^+, n^-) \in \mathbb{Z}^2\), and then prove the uniqueness and regularity results for the vortex profile \((U_p^+, U_p^-)\) under more constraint of the parameters. Moreover, we also establish the stability result for second variation of the energy around this vortex profile when we consider the perturbations in a space of radial functions.