Related papers: Global rough solutions to the critical generalized…

Considering the Cauchy problem for the modified finite-depth-fluid equation $\partial_tu-\G_\delta(\partial_x^2u)\mp u^2u_x=0, u(0)=u_0$, where $\G_\delta f=-i \ft ^{-1}[\coth(2\pi \delta \xi)-\frac{1}{2\pi \delta \xi}]\ft f$, $\delta\ges…

In this article, we prove that the equation \begin{equation*} \left\{\begin{split} &(\partial^2_t-\Delta)u+|u|^{p-1}u=0,\ \ \ 3\leq p<5 &\big(u(0),\partial_tu(0)\big)=(u_0,u_1)\in H^{s}(\mathbb{T}^3)\times…

In this article, we consider a special class of initial data to the 3D Navier-Stokes equations on the torus, in which there is a certain degree of orthogonality in the components of the initial data. We showed that, under such conditions,…

We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove…

We prove that the KP-I initial-value problem \begin{eqnarray*} \begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 {on}{\R}^2_{x,y}\times {\R}_t; u(x,y,0)=\phi(x,y), \end{cases} \end{eqnarray*} is…

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 +…

We propose a numerical method based on physics-informed Random Projection Neural Networks for the solution of Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs) with a focus on stiff problems. We address an Extreme…

We prove the global well-posedness for a $L^2$-critical defocusing cubic higher-order Schr\"odinger equation, namely \[ i\partial_t u + \Lambda^k u = -|u|^2 u, \] where $\Lambda=\sqrt{-\Delta}$ and $k\geq 3, k \in \mathbb{Z}$ in…

We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem…

This paper studies the initial-boundary-value problem (IBVP) of a nonlinear Schr\"odinger equation posed on a strip domain $\mathbb{R}\times[0,1]$ with non-homogeneous Dirichlet boundary conditions. For any $s\ge0$, if the initial data…

We prove pointwise-in-time dispersive estimates for solutions to the generalized Korteweg--de Vries (gKdV) equation. In particular, for solutions to the mass-critical model, we assume only that initial data lie in $\dot{H}^{\frac{1}{4}}…

In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the…

We study the behavior of the solution of a generalized damped KdV equation $u_t + u_x + u_{xxx} + u^p u_x + \mathscr{L}_{\gamma}(u)= 0$. We first state results on the local well-posedness. Then when $p \geq 4$, conditions on…

In this paper we prove that the defocusing, $d$-dimensional mass critical nonlinear Schr{\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$ and $d \geq 3$. To do this, we will prove…

In this paper, the initial boundary value problem of the Korteweg-de Vries Burger equation on the negative half-plane is analyzed. Initially, the well-posedness on $H^s(\R^-)$ for $s\geq 0$ of the IBVP is established to concentrate on the…

In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian…

We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in $H^{s,0}({\mathbb R}^2)$ for $s>3/4$ and unconditional global well-posedness in the energy space. We also prove the global existence…

The Cauchy problem for the Kadomtsev-Petviashvili-II equation (u_t+u_{xxx}+uu_x)_x+u_{yy}=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space \dot…

In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear…

We prove that the solution-map $ u_0 \mapsto u $ associated with the KdV equation cannot be continuously extended in $ H^s(\R) $ for $ s<-1 $. The main ingredients are the well-known Kato smoothing effect for the mKdV equation as well as…