Local and global well-posedness results for flows of inhomogeneous viscous fluids

By Raphaël Danchin. Advances in Differential Equations, 2004

143 citations recorded by the source corpus.

This paper is devoted to the the study of density-dependent, incompressible Navier-Stokes equations with periodic boundary condi- tions, or in the whole space. We aim at stating well-posedness in func- tional spaces as close as possible to the ones imposed by the scaling of the equations. Preliminary results have been obtained in (5) under the assumption that the density is close to a constant. Getting rid of this assumption (by allowing smoother data if necessary) is the main motiva- tion of the present paper. Local well-posedness is stated for data (ρ0 ,u 0) such that (ρ0 − cste) ∈ H N 2 +α and inf ρ0 > 0, and u0 ∈ H N 2 −1+β . The indices α, β > 0 may be taken arbitrarily small. We further derive a blow-up criterion which entails global well-posedness in dimension N =2 if there is no vacuum initially.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a95a4ca2ae63b4325252f21eae580ced6b1d5fa3

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In Applied Mathematics

1. Ping Zhang
2. Marius Paicu
3. Zhifei Zhang
4. Hammadi Abidi
5. Raphaël Danchin
6. Nader Masmoudi
7. F. Lin
8. Taoufik Hmidi

In Control and Systems Engineering

1. Marius Paicu
2. Ping Zhang
3. Zhifei Zhang
4. Hammadi Abidi
5. Raphaël Danchin
6. Nader Masmoudi
7. F. Lin
8. Dehua Wang

In Mathematical Physics

1. Ping Zhang
2. Marius Paicu
3. Zhifei Zhang
4. Hammadi Abidi
5. Raphaël Danchin
6. F. Lin
7. Taoufik Hmidi
8. Yue Liu

Citations in the corpus

• Global solutions to the 3-D incompressible inhomogeneous Navier–Stokes system (2012, result): “Similar results were obtained by Danchin [11] in RN with initial data in the almost critical Sobolev spaces.”
• Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients (2005, methodology): “This latter result will be used in a forthcoming paper to prove global well-posedness for two-dimensional incompressible density-dependent viscous fluids (see [11]).”
• Global Unique Solvability of Inhomogeneous Navier-Stokes Equations with Bounded Density (2013, background): “[8] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv.”
• Equation de Navier-Stokes avec densité et viscosité variables dans l'espace critique (2007, background): “Danchin’s result for the case where viscosity is constant (see [7]).”