We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order operators, with L_1 non-degenerate and L_2 elliptic.
The aim of this paper is to investigate well-posedness of the Cauchy problem for the degenerate Zakharov system. Local well-posedness holds for anisotropic Sobolev data by applying $U^2, V^2$ type spaces. We give the Schrödinger initial data $H^{s_k, s'}$ and the wave data $H^{s_l, s'}$ where $s_k > (d-1)/2, s_l > (d-2)/2, s_k - s_l = 1/2$ and $s' > 1/2$.
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 up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in $L^2({\mathbb{R}}^2)$.
The Cauchy problem for the classical Zakharov system is shown to be ill-posed in the sense of norm inflation in a range of Sobolev spaces $H^s(\mathbb{R}^d)\times H^l(\mathbb{R}^d)$ for all dimensions $d$. This proves several results on well-posedness, which includes existence of solutions, uniqueness and continuous dependence on the initial data, to be sharp up to endpoints.
The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces and it is proved that solutions scatter to free solutions as $t \to \pm \infty$. The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $d=4$…
On the 2d Zakharov system with L^2 Schrödinger data
我们证明了L^2 x H^{-1/2} x H^{-3/2}中具有大初始数据的二维Zakharov系统的局部时间适定性。这是最优正则性的空间，因为在基于L^2的索伯列夫尺度下，对于任何更粗糙的空间对，数据到解的映射在原点处都不平滑。此外，它是一个自然的空间柯西问题的亚音速极限方程，即聚焦立方非线性薛定谔方程。我们得到的存在时间仅取决于初始数据的相应范数--这一结果对于二维的三次非线性薛定谔方程来说是错误的--并且它是最优的，因为Glangetas-Merle的解在那时爆炸。
We establish well-posedness for the Cauchy problem associated to the dispersion generalized Zakharov-Kutnesov equation in the cylinder. Our main ingredient is a localized Strichartz estimate and an argument of compactness
We establish well-posedness conclusions for the Cauchy problem associated to the dispersion generalized Zakharov-Kutnesov equation in bi-periodic Sobolev spaces $H^{s}\left(\mathbb{T}^{2}\right)$, $s>(\frac{3}{2}-\frac{1}{2^{α+2}})(\frac{3}{2}-\fracβ{4})$.
We consider a model that describes electromigration in nanoconductors known as surface electromigration (SEM) equation. Our purpose here is to establish local well-posedness for the associated initial value problem in Sobolev spaces from two different points of view. In the first one, we study the pure Cauchy problem and establish local well-posedness in $H^s(\mathbb{R}^2)$, $s>1/2$. In the second one, we study the Cauchy problem on the background of a Korteweg-de Vries solitary traveling wave in a less regular space. To obtain our results we make use of the smoothing properties of solutions f…
研究一阶拟线性偏微分方程组的Cauchy问题。当初始主象征的频谱不包括在实线中时，即，当双曲性在初始时刻被破坏时，则柯西问题在Hadamard意义下是强不稳定的。这一现象推广了线性Lax-Mizohata定理，G. Métivier in [{ \it Remarks on the well-posedness of the nonlinear Cauchy problem}，Contemp.~数学~ 2005年]。在本文中，我们感兴趣的是从双曲性到非双曲性的过渡，即双曲性在初始时刻成立，但在正时刻被破坏的极限情况：在这样的假设下，我们推广了N.勒纳Y。Morimoto和C.-徐俊[{ \it一类非线性系Cauchy-Kovalevskaya解的不稳定性}，美国J.~数学~ 2010]，因为我们证明，即使是一个弱缺陷的双曲性意味着一个强大的阿达玛不稳定性。我们的例子包括Burgers系统，Van der Waals气体动力学和Klein-Gordon-Zakharov系统。我们的分析依赖于B.~ [{ \it伪微分流的近似}]，印第安纳大学数学J。~ 2016年]。
We obtain the the existence of global solutions to the Cauchy problem of the Fokas-Lenells (FL) equation on the line \begin{align} &u_{xt}+αβ^2u-2iαβu_x-αu_{xx}-iαβ^2|u|^2u_x=0,\nonumber \\ &u(x,t=0)=u_0(x), \nonumber \end{align} where without the small-norm assumption on initial data $u_0(x)\in H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$. Our main technical tool is the inverse scattering transform method based on the representation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem. The existence and the uniqueness of the RH problem is shown via a general vanishing lemma.…
