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On the solvability of the compressible Navier-Stokes system in bounded domains

Abstract : This paper is dedicated to the well-posedness issue for the barotropic Navier-Stokes system with homogeneous Dirichlet boundary conditions in bounded domains of R(N). We aim at considering data in as large a class as possible. Our main result is that if the initial density is bounded away from zero and belongs to some W(1,r) with r > N, if the initial velocity is in the Besov space B(r,p)(2-(2/p)) (and satisfies a suitable boundary condition), and if the body force is in L(loc)(p)(R(+); Lr) for some p > 1 then the system has a unique local solution. Our regularity assumptions are consistent with a dimensional analysis which shows that critical data would correspond to r = N and p = 1, and improve an old result by Solonnikov (1980 J. Sov. Math. 14 1120-32).
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Raphaël Danchin. On the solvability of the compressible Navier-Stokes system in bounded domains. Nonlinearity, IOP Publishing, 2010, 23 (2), pp.383--407. ⟨10.1088/0951-7715/23/2/009⟩. ⟨hal-00693031⟩

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