Related papers: Well-posedness for the Majda-Biello system on the …

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…

This paper presents a new approach to the local well-posedness of the $1d$ compressible Navier-Stokes systems with rough initial data. Our approach is based on establishing some smoothing and Lipschitz-type estimates for the $1d$ parabolic…

We consider the cubic non-linear Schr\"odinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows…

This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium…

We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter-Shu-Zhang [9] established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation's…

We establish local well-posedness for the higher-order nonlinear Schr\"odinger equation, formulated on the half-line. We consider the scenario of associated coefficients such that only one boundary condition is required, which is assumed to…

We prove the local well posedness of the Benjamin-Ono equation and the generalized Benjamin-Ono equation in $ H^1(\T) $. This leads to a global well-posedness result in $ H^1(\T)$ for the Benjamin-Ono equation.

We consider the motion of an inextensible hanging string of finite length under the action of the gravity. The motion is governed by nonlinear and nonlocal hyperbolic equations, which is degenerate at the free end of the string. We show…

We establish global well-posedness for both the defocusing and focusing complex-valued modified Korteweg--de Vries equations on the real line in modulation spaces $M_p^{s,2}(\mathbb{R})$, for all $1\leq p<\infty$ and $0\leq s<3/2-1/p$. We…

We prove that the Benjamin-Ono equation is well-posed in $ H^{1/2}(\T) $. This leads to a global well-posedness result in $ H^{1/2}(\T) $ thanks to the energy conservation.

We study the well-posedness of the initial-value problem for the periodic nonlinear "good" Boussinesq equation. We prove that this equation is local well-posed for initial data in Sobolev spaces \textit{$H^s(\T)$} for $s>-1/4$, the same…

The initial-boundary value problem for the generalized Korteweg-de Vries equation on a half-line is studied by adapting the initial value techniques developed by Kenig, Ponce and Vega and Bourgain to the initial-boundary setting. The…

We consider the fourth-order Schr\"odinger equation $$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u=0, $$ where $\alpha>0,\mu=\pm1$ or $0$ and $\lambda\in\mathbb{C}$. Firstly, we prove local well-posedness in…

This paper investigates the well-posedness of linear elliptic equations, focusing on the divergence-free transformation introduced in the author's recent work [J. Math. Anal. Appl. 548 (2025), 129425]. By comparing this approach with…

This article is concerned with the unconditional well-posedness for the Kawahara equation on the real line and shows that this holds true for initial data in $L^2(\mathbb{R})$. This is achieved by applying an infinite iteration scheme of…

We study well-posedness of a first-order-in-time model for nonlinear acoustics with nonhomogeneous boundary conditions in fractional Sobolev spaces. The analysis proceeds by first establishing well-posedness of an abstract parabolic-type…

We propose a necessary and sufficient condition for the well-posedness of the linear non-homogeneous Grad moment equations in half-space. The Grad moment system is based on Hermite expansion and regarded as an efficient reduction model of…

This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schr\"odinger equations posed in a half plane $\mathbb{R} \times \mathbb{R}^+$ with nonhomogeneous Dirichlet boundary conditions. For any given $s \ge 0$, if the…

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…

The initial value problem of the Zakharov system on two dimensional torus with general period is shown to be locally well-posed in the Sobolev spaces of optimal regularity, including the energy space. Proof relies on a standard iteration…