Related papers: On hyperbolic systems with time dependent H\"older…
In this paper, we propose a globally hyperbolic regularization to the general Grad's moment system in multi-dimensional spaces. Systems with moments up to an arbitrary order are studied. The characteristic speeds of the regularized moment…
In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent H\"older propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian…
It is known that hyperbolic spaces have strict negative type, a condition on the distances of any finite subset of points. We show that they have strong negative type, a condition on every probability distribution of points (with integrable…
Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs…
We study the hyperbolic version of the Prandtl system derived from the hyperbolic Navier-Stokes system with no-slip boundary condition. Compared to the classical Prandtl system, the quasi-linear terms in the hyperbolic Prandtl equation…
We consider the well-posedness of the initial value problem associated to the k-generalized Zakharov-Kuznetsov equation in fractional weighted Sobolev spaces. Our method of proof is based on the contraction mapping principle and it mainly…
We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local well-posedness is known since the work…
This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal…
The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order $2 \times 2$ linear hyperbolic systems. The main technical point is to show that we…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric…
We consider the Cauchy problem for an equation of the form \partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms and no quadratic uu_{xx} term. For a polynomial nonlinearity with no quadratic…
This paper is concerned with the approximation of linear and nonlinearinitial-boundary-value problems of pseudo-parabolic equations with Dirichlet boundary conditions. They are discretized in space by spectral Galerkin and collocation…
We are concerned with the well-posedness of the density-dependent incompressible viscoelastic fluid system. By Schauder-Tychonoff fixed point argument, when $\|{1}/{\rho_0}-1\|_{\dot{B}_{p,1}^{{N}/{p}}}$ is small, local well-posedness is…
We investigate the global hypoellipticity of a class of overdetermined systems with coefficients depending both on time and space variables in the setting of time-periodic Gelfand-Shilov spaces. Our main result provides necessary and…
We study a supersolid in the context of a Gross-Pitaevskii theory with a non-local effective potential. We employ a homogenisation technique which allows us to calculate the elastic moduli, supersolid fraction and other state variables of…
We show that the system is locally wellposed in by establishing a new commutator estimate
In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly…
In this paper, we propose a first order energy stable linear semi-implicit method for solving the Allen-Cahn-Ohta-Kawasaki equation. By introducing a new nonlinear term in the Ohta-Kawasaki free energy functional, all the system forces in…
We study the critical dissipative quasi-geostrophic equations in $\bR^2$ with arbitrary $H^1$ initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable…
We prove the existence and smoothness of density for the solution of a hyperbolic SPDE with free term coefficients depending on time, under hypoelliptic non degeneracy conditions. The result extends those proved in Cattiaux and Mesnager,…