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

2021 ◽  
Author(s):  
Yichun Wang
Keyword(s):  
Initial Data ◽  
Decay Estimates ◽  
Poisson System ◽  
Small Initial Data ◽  
Estimates Of Solutions

DECAY ESTIMATES OF SOLUTIONS TO A SEMI-LINEAR DISSIPATIVE PLATE EQUATION

2010 ◽  
Vol 07 (03) ◽  
pp. 471-501 ◽  
Author(s):  
YOUSUKE SUGITANI ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Linear Problem ◽  
Dimensional Space ◽  
Weighted Sobolev Spaces ◽  
Decay Estimates ◽  
Plate Equation ◽  
Estimates Of Solutions ◽  
Optimal Decay ◽  
Additional Regularity

We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

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

2011 ◽  
Vol 200 (1) ◽  
pp. 313-360 ◽  
Author(s):  
Hyung Ju Hwang ◽  
Alan D. Rendall ◽  
Juan J. L. Velázquez
Keyword(s):  
Initial Data ◽  
Asymptotic Behaviour ◽  
Gradient Estimates ◽  
Poisson System ◽  
Small Initial Data

scholarly journals Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhendong Fang ◽  
Hao Wang
Keyword(s):  
Initial Data ◽  
Knudsen Number ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Poisson System ◽  
Small Initial Data ◽  
Uniform Estimates ◽  
Poisson Boltzmann ◽  
Two Fluid

<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>

DECAY PROPERTY OF REGULARITY-LOSS TYPE FOR DISSIPATIVE TIMOSHENKO SYSTEM

2008 ◽  
Vol 18 (05) ◽  
pp. 647-667 ◽  
Author(s):  
KENTARO IDE ◽  
KAZUO HARAMOTO ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Initial Data ◽  
Decay Property ◽  
General Situation ◽  
Decay Estimates ◽  
Timoshenko System ◽  
Linear Diffusion ◽  
Diffusion Wave ◽  
Estimates Of Solutions ◽  
Whole Space ◽  
The One

We study the decay property of the dissipative Timoshenko system in the one-dimensional whole space. We derive the L2decay estimates of solutions in a general situation and observe that this decay structure is of the regularity-loss type. Also, we give a refinement of these decay estimates for some special initial data. Moreover, under enough regularity assumption on the initial data, we show that the solution approaches the linear diffusion wave expressed in terms of the heat kernels as time tends to infinity. The proof is based on the detailed pointwise estimates of solutions in the Fourier space.

Remarks on nonlinear elastic waves with null conditions

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.

scholarly journals $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

scholarly journals Global classical solutions to the elastodynamic equations with damping

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.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
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.

Decay estimates of solutions for a hybrid system of flexible structures

1993 ◽  
Vol 4 (3) ◽  
pp. 303-319 ◽  
Author(s):  
Bopeng Rao
Keyword(s):  
Rigid Body ◽  
Hybrid System ◽  
Flexible Structures ◽  
Decay Estimates ◽  
Uniform Decay ◽  
Estimates Of Solutions ◽  
Dissipative Boundary

We consider a hybrid system consisting of a cable linked at its end to a rigid body. It is proved that such a hybrid system can be asymptotically stabilized by means of dissipative boundary feedbacks. Uniform decay estimates of energy are also established.

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