The global Cauchy problem for compressible Euler equations with a nonlocal dissipation. (English) Zbl 1414.35155
Summary: This paper studies the global existence and uniqueness of strong solutions and its large-time behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic Cucker-Smale flocking equation with strong local alignment forces and diffusions through the hydrodynamic limit based on the relative entropy argument. In a perturbation framework, we establish the global existence of a unique strong solution for the system under suitable smallness and regularity assumptions on the initial data. We also provide the large-time behavior of solutions showing the fluid density and the velocity converge to its averages exponentially fast as time goes to infinity.
MSC:
| 35Q31 | Euler equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A09 | Classical solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35D35 | Strong solutions to PDEs |
| 81V70 | Many-body theory; quantum Hall effect |
| 35B44 | Blow-up in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 92C15 | Developmental biology, pattern formation |
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