Related papers: Low regularity global well-posedness for the two-d…
We show an improved global well-posedness result for the Zakharov system in two space dimensions with minimal regularity assumptions for the data. Especially we are able to allow Schroedinger and wave data, which do not belong to H^1 and…
The 1-dimensional Zakharov system is shown to have a unique global solution for data without finite energy. The proof uses the " I-method " introduced by Colliander, Keel, Staffilani, Takaoka, and Tao in connection with a refined bilinear…
We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be…
We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schr\"odinger system, which are systems in two variables $u:\mathbb{R}_x^d\times \mathbb{R}_t \to \mathbb{C}$ and $n:\mathbb{R}^d_x\times…
The Klein-Gordon-Schr\"odinger system in 3D is shown to be locally well-posed for Schr\"odinger data in H^s and wave data in H^{\sigma} \times H^{\sigma -1}, if s > - 1/4, \sigma > - 1/2, \sigma -2s > 3/2 and \sigma -2 < s < \sigma +1 .…
The Cauchy problem for the 1-dimensional Zakharov system is shown to be globally well-posed for large data which not necessarily have finite energy. The proof combines the local well-posedness result of Ginibre, Tsutsumi, Velo and a general…
We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space…
We study the local and global well-posedness for the coupled system of Schr\"odinger and Kawahara equations on the real line. The Sobolev space $L^{2} \times H^{-2}$ is the space where the lowest regularity local solutions are obtained. The…
This work is devoted to the stochastic Zakharov system in dimension four, which is the energy-critical dimension. First, we prove local well-posedness in the energy space $H^1\times L^2$ up to the maximal existence time and a blow-up…
The Zakharov system in dimension $d=2,3$ is shown to have a local unique solution for any initial values in the energy space $H^{s} \times H^{l} \times H^{l-1}$, where the range of regularity $(s, l)$ is extended, especially at $s=l-1$. The…
The sharp range of Sobolev spaces is determined in which the Cauchy problem for the classical Zakharov system is well-posed, which includes existence of solutions, uniqueness, persistence of initial regularity, and real-analytic dependence…
We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a…
The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth…
In this paper we study the local and global regularity properties of the Zakharov system on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the…
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…
In this paper we study the Zakharov system on the upper half--plane $U=\{(x ,y)\in \R^2: y>0\}$ with non-homogenous boundary conditions. In particular we obtain low regularity local well--posedness using the restricted norm method of…
We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in…
In this paper, we'll prove a L^2-concentration result of Zakharov system in space dimension two, with radial initial data (u_0,n_0,n_1)\in H^s\times L^2\times H^{-1} ({16/17}<s<1), when blow up of the solution happens by I-method. In…
The Cauchy problem for the Zakharov system in the energy-critical dimension $d=4$ is considered. We prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground…
It is shown that the spatial Sobolev norms of regular global solutions of the (2+1),(3+1) and (4+1)-dimensional Klein-Gordon-Schroedinger system and the (2+1) and (3+1)-dimensional Zakharov system grow at most polynomially with the bound…