相关论文: Well-posedness for the surface quasi-geostrophic f…
In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u…
We establish the local well-posedness of the generalized Benjamin-Ono equation $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$ in $H^s(\R)$, $s>1/2-1/k$ for $k\geq 12$ and without smallness assumption on the initial data. The…
This paper is a continuation of \cite{zhang}, in which we established the wellposedness result and a comparison theorem for a class of one dimensional Forward-Backward SDEs. In this paper we extend the wellposedness result to high…
Proving local well-posedness for quasilinear problems in pde's presents a number of difficulties, some of which are universal and others of which are more problem specific. While a common standard, going back to Hadamard, has existed for a…
In this paper, we consider the following modified quasi-geostrophic equations $\partial_t\theta +\Lambda^\alpha\theta +u\vec\nabla\theta =0$, $u=\Lambda ^{\alpha-1}\mathcal{R}^\perp(\theta)$ where $\alpha \in ]0,1[$ is a fixed parameter.…
A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that…
In this paper, we employ a Schauder-type estimate method, as developed in \cite{CHN}, to establish critical well-posedness result for the Fractional Fokker-Planck Equation. This equation serves as a fundamental model in kinetic theory and…
We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schr\"odinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove…
We study patch solutions of a family of transport equations given by a parameter $\alpha$, $0< \alpha <2$, with the cases $\alpha =0$ and $\alpha =1$ corresponding to the Euler and the surface quasi-geostrophic equations respectively. In…
In this paper, we study the global well-posedness problem for the 1d compressible Navier-Stokers system (cNSE) in gas dynamics with rough initial data. Frist, Liu- Yu (2022) established the global well-posedness theory for the 1d isentropic…
In this paper we prove local well-posedness for Quasi-linear Scrh\"odinger equations with initial data in unweighted Sobolev Spaces. For small initial data with minimal smoothness this has addressed by J. Marzuola, J. Metcalfe and D.…
We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k>4$ is an integer number and $\mu=\pm1$. We give an alternative proof of the Kenig, Ponce, and Vega result in…
In this paper we prove an optimal local well-posedness result for the 1+2 dimensional system of nonlinear wave equations (NLW) with quadratic null-form derivative nonlinearities $Q_{\mu\nu}$. The Cauchy problem for these equations is known…
We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u|^{\alpha-1}u$, where $\alpha\in\left[1,\frac{d}{d-2}\right]$, in dimension $d\geq3$ with initial data in $M_{p, p'}^{1}(\mathbb{R}^d)\times…
We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations $$ \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,\ \ v(x,0)=\phi(x), $$ $$ \partial_tw + \alpha\partial_x^3w +…
In this paper, we study the probabilistic local well-posedness of the cubic Schr\"odinger equation (cubic NLS): \[ (i\partial_{t} + \Delta) u = \pm |u|^{2} u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener…
We consider the wave and Klein-Gordon equations on the real hyperbolic space $\mathbb{H}^{n}$ ($n \geq2$) in a framework based on weak-$L^{p}$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of…
Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The…
We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space…
We show the local in time well-posedness of the Prandtl equation for data with Gevrey $2$ regularity in $x$ and $H^1$ regularity in $y$. The main novelty of our result is that we do not make any assumption on the structure of the initial…