Fujita–Kato theorem for the 3-D inhomogeneous Navier–Stokes equations

Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density

“…In this paper, we will get a similar result to [9] for the 3-D inhomogeneous incompressible MHD system. We establish the global existence and uniqueness of the solution, under the condition that the initial date (u 0 , H 0 ) is small in the critical spaceḢ 1 2 .…”

“…When H = 0, (1) turns into the well-known inhomogeneous incompressible Navier-Stokes system, which has been studied by many researchers (see [1], [2], [9], [12], [13], [14], [28], [31], [32], [35], [38]). When the density ρ is a constant, (1) reduces to be the classical MHD system which has been studied also by many researchers (see [6], [7], [10], [16], [21], [22], [29], [30], [33]).…”

“…An important idea in [23], [24] and [25] is that by using an appropriate time weight, the energy estimate for space derivatives of the velocity field can be closed although the initial date has low regularity (even only in L 2 (R d )). By using a similar idea, Chen, Zhang and Zhao in [9] obtained the Fujita-Kato solution for the 3-D inhomogeneous Navier-Stokes equations.…”

“…We establish the global existence and uniqueness of the solution, under the condition that the initial date (u 0 , H 0 ) is small in the critical spaceḢ 1 2 . We use the idea of time weight in [9], [23], [24] and [25] to deal with less regular initial velocity field and magnetic field, and the Lagrangian idea in [13], [14] to deal with the proof of the uniqueness of the solution. At last, we also get the decay estimate of the solution by the dual method as a generalization of the result in [9].…”

Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system

“…As in [5,26], with a little bit more regularity assumption on the initial velocity field, we can also prove the uniqueness of such weak solutions of (1.1) constructed in Theorem 1.2.…”