Related papers: Sharp local well-posedness for the Hirota-Satsuma …

We prove that the Benjamin Ono equation is globally well-posed in $H^s(\mathbb{R})$ for $s > 1/2$. Our approach does not rely on the global gauge transformation introduced by Tao (arXiv:math/0307289). Instead, we employ a modified version…

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of…

We establish the well-posedness of a coupled micro-macro parabolic-elliptic system modeling the interplay between two pressures in a gas-liquid mixture close to equilibrium that is filling a porous media with distributed microstructures.…

In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional $H^s$ scaling subcritical case with $1 \leq s \leq 2$, the local well-posedness follows…

We consider the defocusing inhomogeneous nonlinear Schr\"{o}dinger equation $i\partial_tu+\Delta u= |x|^{-b}|u|^{\alpha}u,$ where $0<b<1$ and $0<\alpha<\infty$. This problem has been extensively studied for initial data in $H^1(\R^N)$ with…

We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which…

We consider the local well-posedness problem of a one-parameter family of coupled KdV-type systems both in the periodic and non-periodic setting. In particular, we show that certain resonances occur, closely depending on the value of a…

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in [4]. It is numerically shown to be stable and a good approximation to the…

We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove…

The Maxwell-Dirac equations in one space dimension are proved to be well posed in the charge class, that is, with $L^2$ data for the spinor. We also prove that this result is sharp, in the sense that well-posedness fails for spinor data in…

We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.

We study semilinear local well-posedness of the two-dimensional periodic cubic hyperbolic nonlinear Schr\"odinger equation (HNLS) in Fourier-Lebesgue spaces. By employing the Fourier restriction norm method, we first establish sharp…

We consider the two-dimensional nonlinear Schr\"odinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We…

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s…

The study of resonances (and well-posedness) for complex systems under time-periodic loading is of broad interest in application. The work of Galdi et al.~(2014) connects asymptotic stability of solutions to an unforced Cauchy problem to…

We study the mKdV equation with periodic boundary conditions. We establish low regularity well -posedness in $H^{\frac{1}{4}+}(T)$. The proof involves a non-linear, solution dependent gauge transformation, similar to the one considered in…

The work is devoted to the global well-posedness in W^{1, (4, 2)}(R\times R^{+}) of the integro-differential problem involving the square of the one dimensional Laplace operator along with the drift term. Our proof is based on a fixed point…

In this paper, we study cubic and quintic nonlinear Schr\"odinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space $H^s_{\underline{\lambda}},$ and all of our results hold on rational and irrational rectangular,…

In this paper, the distribution dependent stochastic differential equation in a separable Hilbert space with a Dini continuous drift is investigated. The existence and uniqueness of weak and strong solutions are obtained. Moreover, some…

We study the Cauchy problem for a generalized derivative nonlinear Schr\"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed…