Abstract
In this paper, the compressible Euler system with velocity alignment and damping is considered, where the influence matrix of velocity alignment is not positive definite. Sound speed is used to reformulate the system into symmetric hyperbolic type. The global existence and uniqueness of smooth solution for small initial data is provided.
Highlights
We study the following Cauchy problem
The global existence of classical solutions for the hydrodynamic model with linear pressure term and non-local velocity alignment was given in [7], where the shock wave was prevented by velocity alignment
The influence function of velocity alignment Γ is a matrix which corresponds to a linear projection of the velocity field
Summary
The global existence of classical solutions for the hydrodynamic model with linear pressure term and non-local velocity alignment was given in [7], where the shock wave was prevented by velocity alignment. The influence function of velocity alignment Γ is a matrix which corresponds to a linear projection of the velocity field It is non-constant and not positive definite, which reflects the anisotropic non-local interaction within the system. After a detailed analysis of the relationship between velocity alignment and damping, the anisotropic non-local interaction is overcome by using damping, and the existence of the global classical solution of problem (1.1)–(1.2) is obtained.
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