Abstract
In this paper, we are concerned with the local existence of strong solutions to the k-e model equations for turbulent flows in a bounded domain $\Omega\subset\mathbb{R}^{3}$ . We prove the existence of unique local strong solutions under the assumption that the turbulent kinetic energy and the initial density both have lower bounds away from zero.
Highlights
Turbulence is a natural phenomenon, which occurs inevitably when the Reynolds number of flows becomes high enough
In the process of applying the method to the k-ε model equations, we find that the regularity of the solutions should be higher, which is induced by the higher nonlinearity in the compressible Navier-Stokes equations and compressible MHD equations than that in [ ] and [ ]
In Section, we prove the existence theorem of the local strong solution of the original nonlinear problem
Summary
Turbulence is a natural phenomenon, which occurs inevitably when the Reynolds number of flows becomes high enough ( or more). If the initial data have better regularity, the compressible isentropic Navier-Stokes equations will admit a unique local strong solution under various boundary conditions [ – ]. As for compressible MHD equations, the research directions, which mainly contain first the existence of weak and strong solutions and second the condition of weak solutions becoming a strong or even classical one and the local becoming a global one, are similar to that of Navier-Stokes equations. Without loss of generality, we assume throughout this paper that the turbulent kinetic energy k has a positive lower bound away from zero, namely, < m < k with m a constant To conclude this introduction, we give the outline of the rest of this paper: In Section , we consider a linearized problem of the k-ε equations and derive some local-in-time estimates for the solutions of the linearized problem. In Section , we prove the existence theorem of the local strong solution of the original nonlinear problem
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