Related papers: Low regularity solutions to the Chern-Simons-Dirac…

We consider the Cauchy problem for a generalized KdV equation \begin{eqnarray*} u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0, \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem…

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…

In this paper we consider the supercritical generalized Korteweg-de Vries equation $\partial_t\psi + \partial_{xxx}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5\leq p\in\R$. We prove a local well-posedness result in the homogeneous…

In [12], we proved that $1$-d periodic fractional Schr\"odinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s>\frac{1-\alpha}{2}$ and globally well-posed for $s>\frac{5\alpha-1}{6}$. In this paper we define an…

The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. In high dimensions $n\geq4$, global wellposedness for data which is small in the critical…

In this paper, complex Ginzburg-Landau (CGL) equations with superlinear growth terms are studied. We discuss the local well-posedness in the energy space H1 for the initial-boundary value problem of the equations in general domains. The…

We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the…

Local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in the fully periodic case with initial data in Sobolev spaces $H^s$, $s>1$, is proved. Frequency dependent time localization is utilized to control the derivative…

We consider the Cauchy problem for a quadratic derivative nonlinear Schr\"odinger equation whose nonlinearity is a linear combination of $\partial_x (u^2)$ and $\partial_x (|u|^2)$. We prove the local well-posedness in the $L^2$-based…

Time local well-posedness for the Maxwell-Schr\"odinger equation in the coulomb gauge is studied in Sobolev spaces by the contraction mapping principle. The Lorentz gauge and the temporal gauge cases are also treated by the gauge transform.

In this article we present local well-posedness results in the classical Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also…

We study the derivative nonlinear wave equation \( - \partial_{tt} u + \Delta u = |\nabla u|^2 \) on \( \mathbb{R}^{1+3} \). The deterministic theory is determined by the Lorentz-critical regularity \( s_L = 2 \), and both local…

In this work we prove that the initial value problem associated to the Schr\"odinger-Benjamin-Ono type system \begin{equation*} \left\{ \begin{array}{ll} \mathrm{i}\partial_{t}u+ \partial_{x}^{2} u= uv+ \beta u|u|^{2},…

We prove local well-posedness of partially periodic and periodic modified KP-I equations, namely for $\partial_t u+(-1)^{\frac{l+1}{2}}\partial^l_x u-\partial_x^{-1}\partial_y^2 u+u^2\partial_x u=0$ in the anisotropic Sobolev space…

We consider the defocusing inhomogeneous nonlinear Schr\"{o}dinger equation $i\partial_tu+\Delta u= |x|^{-b}|u|^{\alpha}u,$ where $0<b<1$ and $0<\alpha<\infty$. This problem has been extensively studied for initial data in $H^1(\R^N)$ with…

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and…

We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional…

In this paper, we investigate the Cauchy problem for the $H^s$-critical inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t}\pm \Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\]…

We consider the Cauchy problem of the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} =…

We extend recent results of S. Machihara and H. Pecher on low regularity well-posedness of the Dirac-Klein-Gordon (DKG) system in one dimension. Our proof, like that of Pecher, relies on the null structure of DKG, recently completed by…