scholarly journals Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yumi Yahagi
Keyword(s):  
Initial Data ◽  
Mild Solution ◽  
Small Initial Data ◽  
Unique Mild Solution

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 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.

scholarly journals A remark on global solutions to random 3D vorticity equations for small initial data

2019 ◽  
Vol 24 (8) ◽  
pp. 4021-4030 ◽  
Author(s):  
Michael Röckner ◽  
◽  
Rongchan Zhu ◽  
Xiangchan Zhu ◽  
◽  
...  
Keyword(s):  
Initial Data ◽  
Global Solutions ◽  
Small Initial Data ◽  
Vorticity Equations

EXISTENCE, UNIQUENESS AND REGULARITY w.r.t. THE INITIAL CONDITION OF MILD SOLUTIONS OF SPDEs DRIVEN BY POISSON NOISE

2010 ◽  
Vol 13 (01) ◽  
pp. 133-163 ◽  
Author(s):  
CLAUDIA INGRID PRÉVÔT
Keyword(s):  
Mild Solution ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Separable Hilbert Space ◽  
Mild Solutions ◽  
Random Measure ◽  
Initial Condition ◽  
Poisson Random Measure ◽  
Unique Mild Solution ◽  
Compensated Poisson Random Measure

In this paper we investigate stochastic partial differential equations in a separable Hilbert space driven by a compensated Poisson random measure. Our interest is directed towards the existence and uniqueness of mild solutions and their regularity w.r.t. the inital condition. We show the existence of a unique mild solution and prove the Gâteaux differentiability of the mild solution w.r.t. the initial condition. As a consequence, we obtain a gradient estimate for the Gâteaux derivative of the mild solution and for the resolvent associated to the mild solution.

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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