Recent zbMATH articles in MSC 35A35 https://www.zbmath.org/atom/cc/35A35 2021-04-16T16:22:00+00:00 Werkzeug An $$L^2$$ to $$L^\infty$$ framework for the Landau equation. https://www.zbmath.org/1456.35196 2021-04-16T16:22:00+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-ju The 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) Shear-wave resonances in a fluid-solid-solid layered structure. https://www.zbmath.org/1456.35133 2021-04-16T16:22:00+00:00 "Martin, P. A." https://www.zbmath.org/authors/?q=ai:martin.paul-andrew|martin.paulo-a|martin.philippa-a Summary: 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.