scholarly journals A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data

2020 ◽  
Vol 19 (2) ◽  
pp. 835-882
Author(s):  
Seung-Yeal Ha ◽  
◽  
Bingkang Huang ◽  
Qinghua Xiao ◽  
Xiongtao Zhang ◽  
...  
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Fluid Model ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Large Initial Data

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 Vanishing shear viscosity limit and boundary layer study for the planar MHD system

2019 ◽  
Vol 29 (06) ◽  
pp. 1139-1174 ◽  
Author(s):  
Xulong Qin ◽  
Tong Yang ◽  
Zheng-an Yao ◽  
Wenshu Zhou
Keyword(s):  
Boundary Layer ◽  
Initial Data ◽  
Global Existence ◽  
Shear Viscosity ◽  
Initial Boundary ◽  
Heat Conductivity Coefficient ◽  
Large Initial Data ◽  
Initial Boundary Problem ◽  
Viscosity Limit ◽  
Mhd System

We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity coefficient that depends on both the temperature and the density. Firstly, the global existence of strong solution for large initial data is obtained, and then the limit of the vanishing shear viscosity is justified. In addition, the [Formula: see text] convergence rate is obtained together with the estimation on the thickness of the boundary layer.

Sign in / Sign up

Export Citation Format

Share Document