Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity

Boris Haspot[1]

  • [1] Université Paris Dauphine Ceremade UMR CNRS 7534 Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16 (France) Karls Ruprecht Universität HeidelBerg Institut for Applied Mathematics Im Neuenheimer Feld 294 D-69120 Heildelberg (Germany)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 5, page 1717-1763
  • ISSN: 0373-0956

Abstract

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This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in ℝ N with N ≥ 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 Cannone, Meyer and Planchon (where u 0 ∈ B p , r N p - 1 with 1 ≤ p < + ∞ , 1 ≤ r ≤ + ∞ ). This improves the classical analysis where u 0 is considered belonging in B p , 1 N p - 1 such that the velocity u remains Lipschitz. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin 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.

How to cite

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Haspot, Boris. "Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity." Annales de l’institut Fourier 62.5 (2012): 1717-1763. <http://eudml.org/doc/251028>.

@article{Haspot2012,
abstract = {This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in $\mathbb\{R\}^\{N\}$ with $N\ge 2$. 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 Cannone, Meyer and Planchon (where $u_\{0\}\in B^\{\{\frac\{N\}\{p\}\}-1\}_\{p,r\}$ with $1\le p&lt;+\infty ,1\le r\le +\infty $). This improves the classical analysis where $u_\{0\}$ is considered belonging in $B^\{\{\frac\{N\}\{p\}\}-1\}_\{p,1\}$ such that the velocity $u$ remains Lipschitz. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin 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.},
affiliation = {Université Paris Dauphine Ceremade UMR CNRS 7534 Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16 (France) Karls Ruprecht Universität HeidelBerg Institut for Applied Mathematics Im Neuenheimer Feld 294 D-69120 Heildelberg (Germany)},
author = {Haspot, Boris},
journal = {Annales de l’institut Fourier},
keywords = {Navier-Stokes equations Cauchy problem; Littlewood-Paley theory; losing estimates for the transport equation},
language = {eng},
number = {5},
pages = {1717-1763},
publisher = {Association des Annales de l’institut Fourier},
title = {Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity},
url = {http://eudml.org/doc/251028},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Haspot, Boris
TI - Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 5
SP - 1717
EP - 1763
AB - This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in $\mathbb{R}^{N}$ with $N\ge 2$. 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 Cannone, Meyer and Planchon (where $u_{0}\in B^{{\frac{N}{p}}-1}_{p,r}$ with $1\le p&lt;+\infty ,1\le r\le +\infty $). This improves the classical analysis where $u_{0}$ is considered belonging in $B^{{\frac{N}{p}}-1}_{p,1}$ such that the velocity $u$ remains Lipschitz. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin 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.
LA - eng
KW - Navier-Stokes equations Cauchy problem; Littlewood-Paley theory; losing estimates for the transport equation
UR - http://eudml.org/doc/251028
ER -

References

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