×

The scaling hypothesis for Smoluchowski’s coagulation equation with bounded perturbations of the constant kernel. (English) Zbl 1456.35065

When the coagulation kernel \(K\) is homogeneous with a degree strictly smaller than one, it is expected that solutions to the coagulation equation \[ \partial_\tau \phi(\tau,\xi) = \frac{1}{2} \int_0^\xi K(\xi-\eta,\eta) \phi(\tau,\xi-\eta) \phi(\tau,\eta)\ d\eta - \int_0^\infty K(\xi,\eta) \phi(\tau,\xi) \phi(\tau,\eta)\ d\eta \] where \((\tau,\xi)\in (0,\infty)\times (0,\infty)\), with non-negative initial condition \(\phi_0\in L^1((0,\infty),\xi d\xi)\), behave in a self-similar way for large values of \(\tau\). This conjecture is up to now known to be true for the so-called solvable kernels \(K(\xi,\eta)=2\) and \(K(\xi,\eta)=\xi+\eta\), see [G. Menon and R. L. Pego, Commun. Pure Appl. Math. 57, No. 9, 1197–1232 (2004; Zbl 1049.35048)]. Its validity is extended here to small perturbations of the constant kernel with homogeneity zero. In addition, a temporal decay rate is derived. More precisely, let \(W\in C((0,\infty)^2)\) be a symmetric function satisfying \[ 0 \le W(\xi,\eta) \leq 1\text{ and } W(\lambda\xi,\lambda\eta) = W(\xi,\eta), \qquad (\lambda,\xi,\eta)^3, \] and set \(K_\varepsilon = 2 + \varepsilon W\) for \(\varepsilon \ge 0\). It is shown that, for \(\varepsilon>0\) sufficiently small, there is a unique self-similar solution \((\tau,\xi) \mapsto (1+\tau)^{-2} G_\varepsilon(x(1+\tau)^{-1})\) such that \(\|G_\varepsilon\|_{L^1((0,\infty),\xi d\xi)}=1\) and \(G_\varepsilon\in L^1((0,\infty),\xi^k d\xi)\) for all \(k\ge 0\). It is further proved that this self-similar solution is stable in the following sense: given \(R>0\), \(k>2\), and a non-negative initial condition \(\phi_0\) satisfying \[ \int_0^\infty \xi \phi_0(\xi)\ \mathrm{d}\xi = 1\,, \quad \int_0^\infty |\phi_0(\xi) -G_\varepsilon(\xi)| (1+\xi)^k\ \mathrm{d}\xi \le R\,, \] there are \(M>0\) and \(C>0\) depending only on \(R\) and \(k\) such that \begin{align*} & \int_0^\infty |(1+\tau)^2 \phi(\tau,x (1+\tau)) - G_\varepsilon(x)| (1+x)^k\ \mathrm{d}x \cr & \qquad\qquad \le C (1+\tau)^{(1-2M\varepsilon)/2} \int_0^\infty |\phi_0(\xi) -G_\varepsilon(\xi)| (1+\xi)^k\ \mathrm{d}\xi \end{align*} for \(\tau\ge 0\). The proof relies on a refined study of the dynamics of the coagulation equation with constant kernel \(K_0\), building upon previous works on this particular case. In particular, a spectral gap for the linearised operator around the explicit self-similar profile \(G_0(x) = e^{-x}\) is obtained. Also, the stability of the self-similar profiles \((G_\varepsilon)\) with respect to \(\varepsilon\) is established.

MSC:

35C06 Self-similar solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
45M05 Asymptotics of solutions to integral equations
35B20 Perturbations in context of PDEs

Citations:

Zbl 1049.35048
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ackleh, Azmy S., Parameter estimation in a structured algal coagulation-fragmentation model, Nonlinear Anal., 28, 5, 837-854 (1997) · Zbl 0869.35025
[2] Ackleh, Azmy S.; Fitzpatrick, Ben G., Modeling aggregation and growth processes in an algal population model: analysis and computations, J. Math. Biol., 35, 4, 480-502 (1997) · Zbl 0867.92024
[3] Allen, Eric J.; Bastien, Pierre, On coagulation and the stellar mass spectrum, Astrophys. J., 452, 652 (Oct 1995)
[4] Cañizo, José A.; Lods, Bertrand, Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations, J. Differ. Equ., 255, 5, 905-950 (September 2013)
[5] Cañizo, José A.; Lods, Bertrand, Exponential Trend to Equilibrium for the Inelastic Boltzmann Equation Driven by a Particle Bath (July 2015)
[6] Cañizo, José A.; Mischler, Stéphane, Regularity, local behavior and partial uniqueness for Smoluchowski’s coagulation equation, Rev. Mat. Iberoam., 27, 3, 803-839 (2011) · Zbl 1242.82031
[7] Cañizo, José A.; Mischler, Stéphane; Mouhot, Clément, Rate of convergence to self-similarity for Smoluchowski’s coagulation equation with constant coefficients, SIAM J. Math. Anal., 41, 6, 2283-2314 (2010) · Zbl 1206.82055
[8] Engel, Klaus-Jochen; Nagel, Rainer, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194 (2000), Springer-Verlag: Springer-Verlag New York, with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt · Zbl 0952.47036
[9] Escobedo, M.; Mischler, S., Dust and self-similarity for the Smoluchowski coagulation equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 23, 3, 331-362 (2006) · Zbl 1154.82024
[10] Escobedo, M.; Mischler, S.; Perthame, B., Gelation in coagulation and fragmentation models, Comm. Math. Phys., 231, 1, 157-188 (2002) · Zbl 1016.82027
[11] Escobedo, M.; Mischler, S.; Rodriguez Ricard, M., On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22, 1, 99-125 (2005) · Zbl 1130.35025
[12] Fournier, Nicolas; Laurençot, Philippe, Existence of self-similar solutions to Smoluchowski’s coagulation equation, Comm. Math. Phys., 256, 3, 589-609 (2005) · Zbl 1084.82006
[13] Fournier, Nicolas; Laurençot, Philippe, Local properties of self-similar solutions to Smoluchowski’s coagulation equation with sum kernels, Proc. R. Soc. Edinb., Sect. A, Math., 136, 3, 485-508 (2006) · Zbl 1229.45013
[14] Friedlander, Sheldon K., Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, Topics in Chemical Engineering (2000), Oxford University Press
[15] Gualdani, Maria P.; Mischler, Stéphane; Mouhot, Clément, Factorization for non-symmetric operators and exponential H-theorem, Mém. Soc. Math. Fr., 153 (2017)
[16] Laurençot, Philippe, Uniqueness of mass-conserving self-similar solutions to Smoluchowski’s coagulation equation with inverse power law kernels, J. Stat. Phys., 171, 3, 484-492 (2018) · Zbl 1394.35506
[17] Laurençot, Philippe; Niethammer, Barbara; Velázquez, Juan J. L., Oscillatory dynamics in Smoluchowski’s coagulation equation with diagonal kernel, Kinet. Relat. Models, 11, 4, 933-952 (2018) · Zbl 1407.82039
[18] Menon, Govind; Pego, Robert L., Approach to self-similarity in Smoluchowski’s coagulation equations, Commun. Pure Appl. Math., 57, 9, 1197-1232 (2004) · Zbl 1049.35048
[19] Menon, Govind; Pego, Robert L., Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence, SIAM J. Math. Anal., 36, 5, 1629-1651 (2005) · Zbl 1130.35128
[20] Mischler, S.; Mouhot, C., Cooling process for inelastic Boltzmann equations for hard spheres. II. Self-similar solutions and tail behavior, J. Stat. Phys., 124, 2-4, 703-746 (2006) · Zbl 1135.82030
[21] Mischler, S.; Mouhot, C., Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Commun. Math. Phys., 288, 2, 431-502 (June 2009)
[22] Mischler, S.; Mouhot, C., Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal., 221, 2, 677-723 (Aug 2016)
[23] Mischler, S.; Mouhot, C.; Rodríguez Ricard, M., Cooling process for inelastic Boltzmann equations for hard spheres, part I: the Cauchy problem, J. Stat. Phys., 124, 2, 655-702 (2006) · Zbl 1135.82325
[24] Mouhot, Clément, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Commun. Math. Phys., 261, 3, 629-672 (February 2006)
[25] Niethammer, B.; Throm, S.; Velázquez, J. J.L., A revised proof of uniqueness of self-similar profiles to Smoluchowski’s coagulation equation for kernels close to constant (October 2015), Preprint
[26] Niethammer, B.; Throm, S.; Velázquez, J. J.L., Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with singular kernels, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 5, 1223-1257 (2016) · Zbl 1357.35073
[27] Niethammer, B.; Throm, S.; Velázquez, J. J.L., A uniqueness result for self-similar profiles to Smoluchowski’s coagulation equation revisited, J. Stat. Phys., 164, 2, 399-409 (Jun 2016)
[28] Niethammer, B.; Velázquez, J. J.L., Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels, Comm. Math. Phys., 318, 2, 505-532 (2013) · Zbl 1267.82086
[29] Niethammer, B.; Velázquez, J. J.L., Exponential tail behavior of self-similar solutions to Smoluchowski’s coagulation equation, Commun. Partial Differ. Equ., 39, 12, 2314-2350 (2014) · Zbl 1307.45011
[30] Niethammer, B.; Velázquez, J. J.L., Uniqueness of self-similar solutions to Smoluchowski’s coagulation equations for kernels that are close to constant, J. Stat. Phys., 157, 1, 158-181 (2014) · Zbl 1302.82136
[31] Pruppacher, H. R.; Klett, J. D., Microphysics of Clouds and Precipitation (2010), Springer: Springer Netherlands
[32] Silk, J.; Takahashi, T., A statistical model for the initial stellar mass function, Astrophys. J., 229, 242 (Apr 1979)
[33] Srinivasan, R., Rates of convergence for Smoluchowski’s coagulation equations, SIAM J. Math. Anal., 43, 4, 1835-1854 (2011) · Zbl 1385.45004
[34] Throm, Sebastian, Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel, Mem. AMS (2017), in press
[35] Throm, Sebastian, Stability and uniqueness of self-similar profiles in \(L^1\) spaces for perturbations of the constant kernel in Smoluchowski’s coagulation equation (2019), Preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.