Related papers: Low regularity well-posedness for two-dimensional …

We consider the linearized instability of 2D irrotational solitary water waves. The maxima of energy and the travel speed of solitary waves are not obtained at the highest wave, which has a 120 degree angle at the crest. Under the…

In this paper we consider the 1D Green-Naghdi system over a nonflat bottom. This system describes the evolution of water waves over an uneven bottom in the shallow water regime in terms of the water depth $h$ and the horizontal velocity…

In this paper, we first establish a kind of weighted space-time $L^2$ estimate, which belongs to Keel-Smith-Sogge type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the…

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a…

We consider wave functions in the Hilbert space $\mathcal{H}=L^2(\mathbb{R}^3,\mathbb{C}^4)$ of a single Dirac particle, specifically from the positive-energy subspace $\mathcal{H}_+$ of the free Dirac Hamiltonian. Over the decades, various…

We study existence and stability of standing waves for coupled nonlinear Hartree type equations \[ -i\frac{\partial}{\partial t}\psi_j=\Delta \psi_j+\sum_{k=1}^m \left(W\star |\psi_k|^p \right)|\psi_j|^{p-2}\psi_j, \] where…

We present a comprehensive introduction and overview of a recently derived model equation for waves of large amplitude in the context of shallow water waves and provide a literature review of all the available studies on this equation.…

We study a model of internal waves under periodic forcing in an effectively 2-dimensional aquarium. When the underlying classical dynamics has sufficiently irrational rotation number, we prove that the solution to the internal waves…

This study analyzes steady periodic hydroelastic waves propagating on the water surface of finite depth beneath nonlinear elastic membranes. Unlike previous work \cite{BaldiT,BaldiT1,Toland,Toland1}, our formulation accommodates rotational…

The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space…

In the dynamics generated by the suspension bridge equation, traveling waves are an essential feature. The existing literature focuses primarily on the idealized one-dimensional case, while traveling structures in two spatial dimensions…

This article concerns the water wave problem in a three-dimensional domain of infinite depth and examines the modulational regime for weakly nonlinear wavetrains. We use the method of normal form transformations near the equilibrium state…

By establishing a sharp Strichartz estimate for the velocity and density, we prove the local well-posedness of solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial velocity, density, and…

We give an elementary proof for the interior double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of the Euler Equations in a bounded $C^2$-domain $\Omega \subset \mathbb{R}^d$; $d\geq 3$. That is, for velocity $u…

The local well-posedness problem is considered for the Dirac-Klein-Gordon system in two space dimensions for data in Fourier-Lebesgue spaces $\hat{H}^{s,r}$ , where $\|f\|_{\hat{H}^{s,r}} = \| \langle \xi \rangle^s \hat{f}\|_{L^{r'}}$ and…

This paper is concerned with the initial value problem for a system of one-dimensional fourth-order dispersive partial differential equations on the torus with nonlinearity involving derivatives up to second order. This paper gives…

A noteworthy property of many parabolic stochastic PDEs is that they locally linearize (Foondun, Khoshnevisan and Mahboubi (2015), Hairer (2013, 2014), Hairer and Pardoux (2015), Khoshnevisan, Swanson, Xiao and Zhang (2013)). We prove that,…

We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant…

We investigate the well-posedness in the generalized Hartree equation $iu_t + \Delta u + (|x|^{-(N-\gamma)} \ast |u|^p)|u|^{p-2}u=0$, $x \in \mathbb{R}^N$, $0<\gamma<N$, for low powers of nonlinearity, $p<2$. We establish the local…

We consider the Maxwell-Klein-Gordon equation in 2D in the Coulomb gauge. We establish local well-posedness for $s=\frac 14+\epsilon$ for data for the spatial part of the gauge potentials and for $s=\frac 58+\epsilon$ for the solution…