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A Lagrangian approach for the compressible Navier-Stokes equations. (Une approche lagrangienne pour le systéme de Navier-Stokes compressible.) (English. French summary) Zbl 1311.35214
The author investigates the Cauchy problem for the barotropic compressible Navier-Stokes equations with variable density in the whole space \(\mathbb R^{n}\), \(n \geq 2\), in critical Besov functional spaces \(\dot{B}_{p,1}^{n/p} (\mathbb R^{n})\), which have the same invariance with respect to time and space dilation as the system itself, at the initial date for the density \(\rho=\rho(t,x) \in ^{\mathbb R_{t}}\) and the velocity \(u=u(x)\), \(a \equiv \rho_{0}(0,x)-\lambda \in \), \(u_{0}(0,x) \in \dot{B}_{p,1}^{n/p-1} (\mathbb R^{n})\).
For any \(p\in[1,2n)\) at the assumption \(\| a_{0}\|_{\dot{B}_{p,1}^{n/p} (\mathbb R^{n})} \leq c\) with small enough constant \(c\) on the previous author’s articles the following facts are proved: the existence of a local solution \((\rho,u)\) such that \(a=(\rho-1) \in C_{b}([0,T];\dot{B}_{p,1}^{n/p} )\), \(u \in C_{b}([0,T];\dot{B}_{p,1}^{n/p-1} )\), \(\partial_{t} u\), \(\nabla^{2} u \in L^{1}([0,T];\dot{B}_{p,1}^{n/p-1})\) with its uniqueness in the above space, if \(p \leq n\).
In this article the author has improved these results: initial velocities in critical Besov spaces with (not too) negative indices generate a unique local solution; apart from (critical) regularity, the initial density just has to be bounded away from \(0\) and to tend to some positive constant at infinity. These results may be extended on density-dependent viscosity coefficients. The usage of Lagrangian coordinates is a key moment for the proof that the conditions for the solution uniqueness are the same as for the existence. Lipschitz continuity of the flow map in Lagrangian coordinates is also established.

MSC:
35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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