Global well-posedness for 2D fractional inhomogeneous Navier-Stokes equations with rough density

Abstract

The paper concerns with the global well-posedness issue of the 2D incompressible inhomogeneous Navier-Stokes (INS) equations with fractional dissipation and rough density. We first establish the $L_t^q(L^p)$-maximal regularity estimate for the generalized Stokes system with fractional dissipation, and then we employ it to obtain the global existence of solution for the 2D fractional INS equations with large velocity field, provided that the $L^2\cap L^{\infty }$-norm of density minus constant 1 is small enough. Moreover, by additionally assuming that the density minus 1 is sufficiently small in the norm of some multiplier spaces, we prove the uniqueness of the constructed solution by using the Lagrangian coordinates approach. We also consider the density patch problem for the 2D fractional INS equations, and show the global persistence of $C^{1,\gamma }$-regularity of the density patch boundary when the piecewise jump of density is small enough.

arXiv comment43 pages

References in corpus (7)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations
• On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework
• Incompressible flows with piecewise constant density
• Global wellposedness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity
• The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system
• Forward self-similar solutions of the fractional Navier-Stokes Equations
• Between homogeneous and inhomogeneous Navier-Stokes systems: the issue of stability