Related papers: Weakly hyperbolic equations with non-analytic coef…
This paper is concerned with the Cauchy problem of the quadratic nonlinear Schr\"{o}dinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $\eta |u|^2$ where $\eta \in \mathbb{C} \setminus \{0\}$ and low regularity…
An initial-boundary value problem for a viscoelastic wave equation subject to a strong time-localized delay in a Kelvin & Voigt-type material law is considered. Transforming the equation to an abstract Cauchy problem on the extended phase…
In this paper, we investigate the well-posedness of weak solutions to the time-fractional Fokker-Planck equation. Its dynamics is governed by anomalous diffusion, and we consider the most general case of space-time dependent forces.…
We prove that the Cauchy problem is well-posed in a strong sense and in a general setting. Our main result is the construction of an abstract semi-flow for the Hele-Shaw problem within general fluid domains (enabling, for instance, changes…
We investigate the Westervelt equation with several versions of nonlinear damping and lower order damping terms and Neumann as well as absorbing boundary conditions. We prove local in time existence of weak solutions under the assumption…
This paper concerns the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial…
In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces $B^{s}_{p,r}$ with $s>1+\frac 1 p$ and $s=1+\frac 1 p , r=1,p\in…
In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational…
This paper considers the two-dimensional Cauchy problem of the full compressible Navier-Stokes equations with far-field vacuum in $\mathbb{R}^2$, where the viscosity and heat-conductivity coefficients depend on the absolute temperature…
We consider the Cauchy problem for a generalized KdV equation \begin{eqnarray*} u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0, \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem…
Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove…
In this paper we study the Cauchy problem for the Landau Hamiltonian wave equation, with time dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a `very…
We consider the following Cauchy problem for weakly coupled systems of semi-linear damped elastic waves with a power source non-linearity in three-dimensions: \begin{equation*} U_{tt}-a^2\Delta U-\big(b^2-a^2\big)\nabla\text{div }…
We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic…
Ivrii's conjecture asserts that the Cauchy problem is $C^{\infty}$ well-posed for any lower order term if every critical point of the principal symbol is effectively hyperbolic. Effectively hyperbolic critical point is at most triple…
The Cauchy problem is studied for very general systems of evolution equations, where the time derivative of solution is written by Fourier multipliers in space and analytic nonlinearity, with no other structural requirement. We construct a…
In this paper, the local well-posedness of strong solutions to the Cauchy problem of the isentropic compressible Navier-Stokes equations is proved with the initial date being allowed to have vacuum. The main contribution of this paper is…
These notes are devoted to the notion of well-posedness of the Cauchy problem for nonlinear dispersive equations. We present recent methods for proving ill-posedness type results for dispersive PDE's. The common feature in the analysis is…
We review curvature-based hyperbolic forms of the evolution part of the Cauchy problem of General Relativity that we have obtained recently. We emphasize first order symmetrizable hyperbolic systems possessing only physical characteristics.
We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \fw) \in H^s\times H^s\times H^{s'}$, $2<s'<s$. The classical…