Related papers: Global $H^2$-solutions for the generalized derivat…
We consider the incompressible Navier-Stokes (NS) equations on a torus, in the setting of the spaces L^2 and H^1; our approach is based on a general framework for semi- or quasi-linear parabolic equations proposed in the previous work [9].…
We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schr\"{o}dinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear…
The nonlinear Schr\"odinger equations with nonlinearities $|u|^{2k}u$ on the $d$-dimensional torus are considered for arbitrary positive integers $k$ and $d$. The solution of the Cauchy problem is shown to be unique in the class $C_tH^s_x$…
A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the…
We show that the derivative nonlinear Schr\"odinger (DNLS) equation is globally well-posed in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. Our result exploits the complete integrability of DNLS and removes certain spectral conditions…
We consider the Cauchy problem for the kinetic derivative nonlinear Schr\"odinger equation on the torus: \[ \partial_t u - i \partial_x^2 u = \alpha \partial_x \big( |u|^2 u \big) + \beta \partial_x \big[ H \big( |u|^2 \big) u \big] , \quad…
In this paper, we consider the Cauchy problem of the cubic nonlinear Schr\"{o}dinger equation with derivative in $H^s(\R)$. This equation was known to be the local well-posedness for $s\geq \frac12$ (Takaoka,1999), ill-posedness for…
We prove the existence of weak solutions in the space of energy for a class of non-linear Schroedinger equations in the presence of a external rough magnetic potential. Under our assumptions it is not possible to study the problem by means…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq…
By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Sch\"odinger equation (DNLS). These transformations are shown to lead to two solution formulas for DNLS in…
In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of…
We consider the Cauchy problem to the general defocusing and focusing $p\times q$ matrix nonlinear Schr\"{o}dinger (NLS) equations with initial data allowing arbitrary-order poles and spectral singularities. By establishing the…
We consider the nonlinear biharmonic Schr\"odinger equation $$i\partial_tu+(\Delta^2+\mu\Delta)u+f(u)=0,\qquad (\text{BNLS})$$ in the critical Sobolev space $H^s(\R^N)$, where $N\ge1$, $\mu=0$ or $-1$, $0<s<\min\{\fc N2,8\}$ and $f(u)$ is a…
We consider the Cauchy problem for the (derivative) nonlocal NLS in super-critical function spaces $E^s_\sigma$ for which the norms are defined by $$ \|f\|_{E^s_\sigma} = \|\langle\xi\rangle^\sigma 2^{s|\xi|}\hat{f}(\xi)\|_{L^2}, \ s<0, \…
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.
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…
We use a method developed by Strauss to obtain global wellposedness results in the mild sense for the small data Cauchy problem in modulation spaces $M_{p,q}^s(\mathbb{R}^d)$, where $q=1$ and $s\geq0$ or $q\in(1,\infty]$ and…
The spatially periodic initial problem and Cauchy problem for nonlinear Schr\"odinger equations are considered. The existence and uniqueness of global solution with infinite smooth initial data $u_0$, i.e. $u_0,\;|u_0|^{2p}u_0\in…
We consider the Cauchy problem for nonlinear Schr\"odinger equations in a general domain $\Omega\subset\mathbb{R}^N$. Construction of solutions has been only done by classical compactness method in previous results. Here, we construct…
We prove that the 1D Schr\"odinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The result follows from an application of the ``I-method''. This method allows to…