Optimal Gradient Estimates and Asymptotic Behaviour for the Vlasov–Poisson System with Small Initial Data

<p style='text-indent:20px;'>In this paper, we obtain the uniform estimates with respect to the Knudsen number <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> for the fluctuations <inline-formula><tex-math id="M2">\begin{document}$ g^{\pm}_{\varepsilon} $\end{document}</tex-math></inline-formula> to the two-species Vlasov-Poisson-Boltzmann (in briefly, VPB) system. Then, we prove the existence of the global-in-time classical solutions for two-species VPB with all <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon \in (0,1] $\end{document}</tex-math></inline-formula> on the torus under small initial data and rigorously derive the convergence to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (in briefly, NSFP) system as <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> go to 0.</p>

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Decay estimates of solutions to the N ‐species Vlasov–Poisson system with small initial data

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.7611 ◽

2021 ◽

Author(s):

Yichun Wang

Keyword(s):

Initial Data ◽

Decay Estimates ◽

Poisson System ◽

Small Initial Data ◽

Estimates Of Solutions

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Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500019399 ◽

1999 ◽

Vol 129 (6) ◽

pp. 1291-1307 ◽

Cited By ~ 17

Author(s):

S. Snoussi ◽

S. Tayachi ◽

F. B. Weissler

Keyword(s):

Parabolic Equation ◽

Initial Data ◽

Asymptotic Behaviour ◽

Global Solutions ◽

Gradient Term ◽

Semilinear Parabolic Equation ◽

Small Initial Data ◽

Self Similar ◽

Semilinear Parabolic

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q > 1, p > 1. Forq=2p/(p+1) and ½ 1(p-1)>1 (equivalently, q > (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.

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Remarks on nonlinear elastic waves with null conditions

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020039 ◽

2020 ◽

Vol 26 ◽

pp. 121

Author(s):

Dongbing Zha ◽

Weimin Peng

Keyword(s):

Initial Data ◽

Global Existence ◽

Elastic Waves ◽

Wave Equations ◽

Alternative Proof ◽

Classical Solutions ◽

Small Initial Data ◽

Hyperelastic Materials ◽

Variational Structure ◽

Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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Global classical solutions to the elastodynamic equations with damping

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02608-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Mengmeng Liu ◽

Xueyun Lin

Keyword(s):

Initial Data ◽

Periodic Boundary ◽

Energy Estimate ◽

Periodic Boundary Conditions ◽

Deformation Tensor ◽

Classical Solutions ◽

Small Initial Data ◽

Weighted Energy ◽

Global Classical Solutions ◽

Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

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Singularities of solutions for first order quasilinear hyperbolic systems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500016218 ◽

1983 ◽

Vol 94 (1-2) ◽

pp. 137-147 ◽

Cited By ~ 1

Author(s):

Lee Da-tsin(Li Ta-tsien) ◽

Shi Jia-hong

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Hyperbolic Systems ◽

Smooth Solutions ◽

Small Initial Data ◽

Quasilinear Hyperbolic Systems ◽

First Order ◽

Global Smooth Solutions ◽

The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

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