Global Well-Posedness of Free Interface Problems for the Incompressible Inviscid Resistive MHD

Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations

The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier-Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (Arch. Ration. Mech. Anal. 204 (1):189–230, 2012, and J. Math. Pures Appl. 100 (1):166–203, 2013) to a more lower regularity index a...

Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u+ (u)x = 0, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is globally well-posed in Hs (s > sα), and uniformly globally well-posed in H s (s > −3/4) for all ǫ ∈ (0, 1). Moreover, we prove that for any T > 0, its solution converges in C([0, T ]; Hs) to that of the KdV equa...

Global Well-posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. None smallness conditions are imposed on the data. The global existence issues for infinit...

Global well-posedness and inviscid limit for the modified Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u = 2(u)x, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is uniformly globally well-posed in Hs (s ≥ 1) for all ǫ ∈ (0, 1]. Moreover, we prove that for any s ≥ 1 and T > 0, its solution converges in C([0, T ]; Hs) to that of the MKdV equation if ǫ tends to 0.

Well-posedness for Hyperbolic Problems