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 Space-time L2 estimates, regularity and almost global existence for elastic waves

2018 ◽  
Vol 30 (5) ◽  
pp. 1291-1307 ◽  
Author(s):  
Kunio Hidano ◽  
Dongbing Zha
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Differential Equations ◽  
Elastic Waves ◽  
Wave Equations ◽  
Space Time ◽  
Existence Theory ◽  
Small Initial Data ◽  
Nonlinear Elastic ◽  
Appl Math

AbstractIn this paper, we first establish a kind of weighted space-time {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in {H^{3}\times H^{2}}.

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.

scholarly journals Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Jincheng Shi ◽  
Shengzhong Xiao
Keyword(s):  
Global Existence ◽  
Life Span ◽  
General Model ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Conditions ◽  
Short Wave ◽  
Classical Solutions ◽  
Global Classical Solutions

We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. Under suitable initial conditions, the existence of the global classical solutions for the viscosity long-short wave equations is proved. If it does not exist globally, the life span which is the largest time where the solutions exist is also obtained.

scholarly journals Global existence and asymptotic behavior for a generalized Boussinesq type equation without dissipation

2021 ◽  
Author(s):  
Guowei Liu ◽  
Wei Wang ◽  
Qiuju Xu
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Initial Data ◽  
Global Existence ◽  
Linear Group ◽  
Type Equation ◽  
Small Initial Data ◽  
Dispersive Estimate ◽  
The Cauchy Problem

In this paper, we study the Cauchy problem for a generalized Boussinesq type equation in $\mathbb{R}^n$. We establish a dispersive estimate for the linear group associated with the generalized Boussinesq type equation. As applications, the global existence, decay and scattering of solutions are established for small initial data.

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