Related papers: Global solvability of massless Dirac-Maxwell syste…
The Lagrange-d'Alembert equations with constraints belonging to $H^{1,\infty}$ have been considered. A concept of weak solutions to these equations has been built. Global existence theorem for Cauchy problem has been obtained.
This paper investigates the Cauchy problem for the barotropic compressible Navier-Stokes equations in $\mathbb{R}^2$ with the constant state as far field, which may be vacuum or non-vacuum. Under the assumption of a sufficiently large bulk…
We shall be concerned with the Cauchy problem for quasilinear systems in three space dimensions of the form \label{i.1} \partial^2_tu^I-c^2_I\Delta u^I = C^{IJK}_{abc}\partial_c u^J\partial_a\partial_b u^K + B^{IJK}_{ab}\partial_a…
In this paper, we study the Cauchy problem of the isentropic compressible magnetohydrodynamic equations in $\mathbb{R}^{3}$. When $(\gamma-1)^{\frac{1}{6}}E_{0}^{\frac{1}{2}}$, together with the $\|H_{0}\|_{L^{2}}$, is suitably small, a…
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schr\"odinger equation, with certain nonlinearities, are not unique. For any $s<0$ there exist nonzero generalized solutions varying continuously in the…
We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass…
This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in…
In this paper, we investigate Cauchy problem of the two-dimensional full Maxwell-Navier-Stokes system, and prove the global-in-time existence and uniqueness of solution in the borderline space which is very close to $L^2$-energy space by…
In this paper, we mainly consider the global solvability of smooth solutions for the Cauchy problem of the three-dimensional Landau-Lifshitz-Slonczewski equation in the Morrey space. We derive the covariant complex Ginzburg-Landau equation…
In this paper, we consider the Cauchy-type problem for a nonlinear differential equation involving $\Psi$-Hilfer fractional derivative and prove the existence and uniqueness of solutions in the weighted space of functions. The Ulam--Hyers…
We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in…
We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that…
We prove that the Cauchy problem for the Dirac-Klein-Gordon equations in two space dimensions is locally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor, and an associated range of spaces of positive index for…
In this monograph we prove that the nonlinear Lie algebra representation given by the manifestly covariant Maxwell-Dirac (M-D) equations is integrable to a global nonlinear representation $U$ of the Poincar\'e group ${\cal P}_0$ on a…
In this paper, we consider the Cauchy problem to the heat conductive compressible Navier-Stokes equations in the presence of vacuum and with vacuum far field. Global well-posedness of strong solutions is established under the assumption,…
We consider the Cauchy problem for the electron magnetohydrodynamics model in the supercritical regime. For rough initial data in $\mathcal H^{-s}(\mathbb T^n)$ with $s>0$, we obtain global in time weak solutions almost surely via an…
We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove that there exists…
We investigate the Cauchy problem for the Einstein - scalar field equations in asymptotically flat spherically symmetric spacetimes, in the standard 1+3 formulation. We prove the local existence and uniqueness of solutions for initial data…
The purpose of this paper is twofold. In the first part, we provide a new proof of the global existence of the solutions to the Vlasov-Maxwell system with a small initial distribution function. Our approach relies on vector field methods,…
In this paper, we investigate the Cauchy problem of the nonhomogeneous incompressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field density and prove that the 2D Cauchy problem has a unique local strong solution provided that…