Related papers: The Cauchy problem for a higher order shallow wate…
In this paper, we investigate the Cauchy problem for the shallow water type equation \begin{eqnarray*} u_{t}+\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2})+…
In this paper, we investigate the Cauchy problem for the shallow water type equation \[ u_{t}+\partial_{x}^{3}u + \frac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0,x\in {\mathbf…
In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{eqnarray*} u_{t}+(-1)^{j+1}\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2}) = 0,j\in N^{+},x\in\mathbf{T}= [0,2\pi \lambda) \end{eqnarray*}…
The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $\hat{H}^r_s(\R)$ defined by the norm $$\n{v_0}{\hat{H}^r_s(\R)} := \n{< \xi > ^s\hat{v_0}}{L^{r'}_{\xi}},\quad < \xi…
We present a comprehensive introduction and overview of a recently derived model equation for waves of large amplitude in the context of shallow water waves and provide a literature review of all the available studies on this equation.…
In this work we study Cauchy problem for a high-order differential equation $\frac{\partial u(y,x)}{\partial y}+P(\frac{\partial}{\partial x})u(y,x)=\gamma\frac{\partial}{\partial x}(u^2(y,x))+F(y,x)$. We prove that the problem is…
We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating…
In this paper we consider the Cauchy problem for 2D viscous shallow water system in $H^s(\mathbb{R}^2)$, $s>1$. We first prove the local well-posedness of this problem by using the Littlewood-Paley theory, the Bony decomposition, and the…
The Cauchy problem is investigated for the parabolic type in the some finite part $[t_0, t_1] \subset [0, \infty)$ of the semi axis $t \in [0, \infty)$ and degenarated to Schrodinger type in the remain part of the same semi axes the second…
We consider the Cauchy problem of the KdV-type equation \[ \partial_t u + \frac{1}{3} \partial_x^3 u = c_1 u \partial_x^2u + c_2 (\partial_x u)^2, \quad u(0)=u_0. \] Pilod (2008) showed that the flow map of this Cauchy problem fails to be…
The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation $$u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,$$ is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines…
The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begin{eqnarray*}…
In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(\mathbb{R}^2)$, $s>max\{1,\frac{2}{p}\}$, $1\leq p,r\leq \infty$ by…
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…
This paper is devoted to studying the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-\beta \partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -\gamma u=0, \end{eqnarray*} with positive $\beta$ and…
We study the global well-posedness and asymptotic behavior of solutions for the Cauchy problem of three-dimensional sixth order Cahn-Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a…
We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.
This article concerns the Cauchy problem for the gravity-capillary water waves system in general dimensions. We establish local well-posedness for initial data in $H^s$, with $s > \frac{d}{2} + 2 - \mu$, with $\mu = \frac{3}{14}$ and $\mu =…
This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for data $\frac{1}{4}$ derivatives less…
This paper focuses on Cauchy problem for the three-dimensional two-fluid type model, in which the presence of vacuum is permitted. Under some assumptions that the initial data satisfy appropriate regularity conditions and a compatibility…