On global solutions of Cauchy problems for compressible Navier-Stokes equations. (English) Zbl 1097.35116
The author considers the initial value problem for the compressible Navier-Stokes equations. He allows the viscosities to be bounded functions of the spatial variable \(x\) and of the density, the pressure can be any stricly increasing function of the density.
Under the assumptions that the initial conditions for the velocity and for the density are sufficiently small in the space \(H^{s_0}\), where \(s_0 = [n/2]+1\), he shows the existence of the unique global strong solution to the compressible Navier-Stokes equations.
The proof is based on showing a prori estimates leading to local existence result, and on the prolongation argument giving the global in time existence.
The main result of this paper is, in comparison to the well known paper by A. Matsumura and T. Nishida [Contemp. Math. 17, 103–116 (1983; Zbl 0511.76063)], that he requires less regularity on the initial conditions (the above mentioned authors used the initial conditions in \(H^{s_0+1}\)).
Under the assumptions that the initial conditions for the velocity and for the density are sufficiently small in the space \(H^{s_0}\), where \(s_0 = [n/2]+1\), he shows the existence of the unique global strong solution to the compressible Navier-Stokes equations.
The proof is based on showing a prori estimates leading to local existence result, and on the prolongation argument giving the global in time existence.
The main result of this paper is, in comparison to the well known paper by A. Matsumura and T. Nishida [Contemp. Math. 17, 103–116 (1983; Zbl 0511.76063)], that he requires less regularity on the initial conditions (the above mentioned authors used the initial conditions in \(H^{s_0+1}\)).
Reviewer: Milan Pokorný (Praha)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
Keywords:
compressible fluid flow; compressible Navier-Stokes equations; strong solutions; global solutionsCitations:
Zbl 0511.76063References:
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