Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity

By Jingchi Huang, Marius Paicu, Ping Zhang. Archive for Rational Mechanics and Analysis, 2012

92 citations recorded by the source corpus.

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$$\end{document}, which satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mu \| a_0 \|_{L^\infty} + \|u_0^h\|_{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\|u_0^d\|_{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$$\end{document} for some positive constants c0, Cr and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal Lp(Lq) regularity theorem for the heat kernel plays an essential role in this context.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5876b3a265c471fe5c39f5ee0cd9aa027bc628e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Ping Zhang
2. Zhifei Zhang
3. Marius Paicu
4. F. Lin
5. Nader Masmoudi
6. Zhen Lei
7. Frédéric Rousset
8. Helge Holden

In Computational Mechanics

1. Ping Zhang
2. Zhifei Zhang
3. Marius Paicu
4. F. Lin

In Mathematical Physics

1. Ping Zhang
2. Zhifei Zhang
3. Marius Paicu
4. F. Lin
5. Nader Masmoudi
6. Zhen Lei
7. Frédéric Rousset
8. Helge Holden

Citations in the corpus

• Global Unique Solvability of Inhomogeneous Navier-Stokes Equations with Bounded Density (2013, background): “Again in the scaling invariant framework, the authors [13] proved the global existence of weak solutions to (1.”