Related papers: Local Well-posedness for dispersion generalized Be…

This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal…

The Cauchy problem for a unified family of integrable $U(1)$-invariant peakon equations from the NLS hierarchy is studied. As main results, local well-posedness is proved in Besov spaces, and blow-up is established through use of an $L^1$…

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.

In this note we report local well-posedness results for the Cauchy problems associated to generalized KdV type equations with dissipative perturbation for given data in the low regularity $L^2$-based Sobolev spaces. The method of proof is…

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 complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $ H^{-1}(\R) $ with a solution-map that is analytic from $H^{-1}(\R) $ to…

We study the Cauhcy problem for space-time fractional nonlinear Schr\"odinger equation with a general nonlinearity. We prove the local well-posedness of it in fractional Sobolev spaces based on the decay estimates and H\"older type…

We study the generalized derivative nonlinear Schr\"odinger equation $i\partial_t u+\Delta u = P(u,\overline{u},\partial_x u,\partial_x \overline{u})$, where $P$ is a polynomial, in Sobolev spaces. It turns out that when $\text{deg } P\geq…

We consider a higher dimensional version of the Benjamin--Ono equation, $\partial_t u -\mathcal{R}_1\Delta u+u\partial_{x_1} u=0$, where $\mathcal{R}_1$ denotes the Riesz transform with respect to the first coordinate. We first establish…

In this paper, we investigate the one-dimensional derivative nonlinear Schr\"odinger equations of the form $iu_t-u_{xx}+i\lambda\abs{u}^k u_x=0$ with non-zero $\lambda\in \Real$ and any real number $k\gs 5$. We establish the local…

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…

This paper is dedicated to the study of the derivative nonlinear Schr\"odinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global…

In this paper, we consider the Cauchy problem for the $b$-equation. Firstly, for $s>\frac32,$ if $u_{0}(x)\in H^{s}(\mathbb{R})$ and $m_{0}(x)=u_{0}(x)-u_{0xx}(x)\in L^{1}(\mathbb{R}),$ the global solutions of the $b$-equation is…

We prove that the Benjamin Ono equation is globally well-posed in $H^s(\mathbb{R})$ for $s > 1/2$. Our approach does not rely on the global gauge transformation introduced by Tao (arXiv:math/0307289). Instead, we employ a modified version…

We study the well-posedness of Cauchy problems on the upper half space $\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u =(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly elliptic coefficients.…

We consider the Cauchy problem of the nonlinear Schr\"odinger equation with the modulated dispersion and power type nonlinearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk-Gubinelli [K. Chouk and M,…

We establish local well-posedness in Sobolev spaces $H^s(\mathbb{T})$, with $s\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+\eta Lu + uu_x=0;\; x\in \mathbb{T},\; t\geq0, $$ where $\eta >0$,…

We study the one dimensional nonlinear Schr\"odinger equation with power nonlinearity $|u|^{\alpha - 1} u$ for $\alpha \in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy…

We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data…

The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schr\"odinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the…