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Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity. (English. French summary) Zbl 1383.35160

Summary: This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in \(\mathbb R^N\) with \(N\geq 2\). We address the question of well-posedness for large and small initial data having critical Besov regularity in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by M. Cannone et al. [Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math. Laurent Schwartz, Palaiseau , Exp. No. 8, 10 p. (1994; Zbl 0882.35090)] (where \(u_0\in B_{p,r}^{\frac{N}{p}-1}\) with \(1\leq p<+\infty\), \(1\leq r\leq+\infty\)). This improves the classical analysis where \(u_0\) is considered belonging in \(B_{p,1}^{\frac{N}{p}-1}\) such that the velocity \(u\) remains Lipschitz. Our result relies on a new a priori estimate for transport equation introduce by H. Bahouri et al. [Fourier analysis and nonlinear partial differential equations. Berlin: Heidelberg (2011; Zbl 1227.35004)] when the velocity \(u\) is not necessary Lipschitz but only log Lipschitz. Furthermore it gives a first kind of answer to the problem of self-similar solution.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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References:

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