Related papers: Large data local solutions for the derivative NLS …

We address the system of partial differential equations modeling motion of an elastic body interacting with an incompressible fluid. The fluid is modeled by the incompressible Navier-Stokes equations while the structure is represented by a…

We consider Nonlinear Schrodinger type equations on $S^1$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of their solutions. The key is to derive an iteration bound based on a frequency decomposition…

In this paper we study the local and global regularity properties of the cubic nonlinear Schr\"odinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and…

We develop inverse scattering for the derivative nonlinear Schrodinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering…

We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…

We propose a priori estimates for a weak solution to the derivative nonlinear Schr\"odinger equation (DNLS) on torus with small $L^2$-norm datum in low regularity Sobolev spaces. These estimates allow us to show the existence of solutions…

We provide the rigorous justification of the NLS approximation, in Sobolev regularity, for a class of quasilinear Hamiltonian Klein Gordon equations with quadratic nonlinearities on large one-dimensional tori $\T_L:=\mathbb{R}/(2\pi L…

We prove Asymptotic Completeness of one dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity.

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s > 1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the…

For any subcritical index of regularity $s>3/2$, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space…

In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates…

This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such…

We consider the NLS hierarchy with the nonzero boundary condition $q(t, x) \rightarrow q_\pm \in \mathbb{S}^1$ as $x \rightarrow \pm \infty$ and prove that it is global well-posedness for initial data of high regularity. Specifically, we…

Recently, Dolbeault-Esteban-Figalli-Frank-Loss [20] established the optimal stability of the first-order $L^2$-Sobolev inequality with dimension-dependent constant. Subsequently, Chen-Lu-Tang [18] obtained the optimal stability for the…

We study a derivative nonlinear Schr\"{o}dinger equation, allowing non-integer powers in the nonlinearity, $|u|^{2\sigma} u_x$. Making careful use of the energy method, we are able to establish short-time existence of solutions with initial…

We consider the viscous $n$-dimensional Camassa-Holm equations, with $n=2,3,4$ in the whole space. We establish existence and regularity of the solutions and study the large time behavior of the solutions in several Sobolev spaces. We first…

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, \…

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…

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger equation (IBNLS) \[iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $\lambda \in \mathbb…

In this study, we consider the nonlinear Sch\"odinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean…