Related papers: Global well-posedness and limit behavior for the m…
We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-\Delta)^{\sigma}u+(-\Delta)^{\delta}\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with…
In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an rotational angular momentum term in the space $\Real^2$.
We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schr\"odinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of…
The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begin{eqnarray*}…
We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent…
The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof…
We consider the following $p$ order nonlinear half wave Schr{\"o}dinger equations$$\left(i \partial\_{t}+\partial\_{x }^2-\left|D\_{y}\right|\right) u=\pm|u|^{p-1} u$$on the plane $\mathbb{R}^2$ with $1<p\leq 2$. This equation is considered…
In this paper, we mainly study the Cauchy problem for the full compressible Navier-Stokes equations in Sobolev spaces. We establish the global well-posedness of the equations with small initial data by using Friedrich's method and…
In this paper we study the Cauchy problem associated to the Maxwell-Schr\"odinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic…
We consider the Cauchy problem for coupled system of Vlasov and non-Newtonian fluid equations. We establish local well--posedness of the strong solutions, provided that the initial data are regular enough. Global existence of unique strong…
In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$…
This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key…
A linear equation Au=f (1) with a bounded, injective, but not boundedly invertible linear operator in a Hilbert space H is studied. A new approach to solving linear ill-posed problems is proposed. The approach consists of solving a Cauchy…
We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global…
It is shown that the Cauchy problem for the DNLS equation in the spatially periodic setting is locally well-posed in Sobolev spaces H^s(T) for s \geq 1/2. Moreover, global well-posedness is shown for s \geq 1 and data with small L^2 norm.
This paper concerns the Cauchy problem of two-dimensional (2D) full compressible magnetohydrodynamic (MHD) equations in the whole plane $\mathbb{R}^2$ with zero density at infinity. By spatial weighted energy method, we derive the local…
We prove that the Zakharov-Kuznetsov equation on cylindrical spaces is globally well-posed below the energy norm. As is known, local well-posedness below energy space was obtained by the first author. We adapt I-method to extend the…
In a recent work, Ionescu and Kenig proved that the Cauchy problem associatedto the Benjamin-Ono equation is well-posed in $L^2(\mathbb R)$. In this paper we give a simpler proof of Ionescu and Kenig's result, which moreover provides…
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu (-\Delta)^2 u_t$, where $\nu >0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion…
We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q}(\mathbb{{R}})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map…