Related papers: Local well-posedness for the Maxwell-Dirac system …

In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in $\mathcal{H}^m$ for $m \geq 3$. The Maxwell equations are equipped with instantaneous nonlinear material laws…

This paper is a continuation of the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as…

The Zakharov system in dimension $d\leqslant 3$ is shown to be locally well-posed in Sobolev spaces $H^s \times H^l$, extending the previously known result. We construct new solution spaces by modifying the $X^{s,b}$ spaces, specifically by…

We prove local well-posedness of the Benjamin-Ono equation for a class of bounded initial data including periodic and bore-like functions. As a consequence, we obtain local well-posedness in $H^s(\mathbb{R})+H^\sigma(\mathbb{T})$ for…

We prove that the "good" Boussinesq model with the periodic boundary condition is locally well-posed in the space $H^{s}\times H^{s-2}$ for $s > -3/8$. In the proof, we employ the normal form approach, which allows us to explicitly extract…

The space-time monopole equation on $\R^{2+1}$ can be derived by a dimensional reduction of the anti-self-dual Yang Mills equations on $\R^{2+2}$. It can be also viewed as the hyperbolic analog of Bogomolny equations. We uncover null forms…

In this paper, we consider the one-dimensional generalized Benjamin--Bona--Mahony (gBBM) equation \[(1-\partial_x^2)u_t+(u+u^p)_x=0,\qquad p=2,3,4,\dots,\] posed either on the real line $\mathbb R$ or on the torus $\mathbb T$. This equation…

We consider the Cauchy problem for semi-linear Schr\"odinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space…

The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions…

In this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. The problem is…

In this paper, we present a partial result on the global well-posedness of the Cauchy problem for the Einstein-Yang-Mills system in the constant mean extrinsic curvature spatial harmonic and generalized Coulomb gauges as introduced in…

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and…

For $p\geq 2$, we prove local wellposedness for the nonlinear Schr\"odinger equation $(i\partial_t + \Delta)u = \pm|u|^pu$ on $\mathbb{T}^3$ with initial data in $H^{s_c}(\mathbb{T}^3)$, where $\mathbb{T}^3$ is a rectangular irrational…

This work studies the local well-posedness of the initial-value problem for the nonlinear sixth-order Boussinesq equation $u_{tt}=u_{xx}+\beta u_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$, where $\beta=\pm1$. We prove local well-posedness with initial…

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…

Studied in this paper is the sixth-order Boussinesq equation. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the ``bad'' fourth term…

We consider the Cauchy problem for a system of quadratic derivative nonlinear Schr\"odinger equations introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. Under the condition that the flow map fails to be twice…

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution…

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like --…

In this paper, we consider the Jordan--Moore--Gibson--Thompson with a time-fractional damping term of the type $\delta \textup{D}_t^{1-\alpha} \Delta \psit$ where we allow the challenging so-called critical case ($\delta=0$). This equation…