Related papers: The Cauchy Problem for the Vibrating Plate Equatio…
An ill-posed Cauchy problem for the wave equation is considered: the solution is to be determined by the Cauchy data on some part of the time-space boundary. By means of Fourier method we obtain a regularization algorithm for this problem,…
We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the modulation space $M_{p,q}^{s}(\mathbb R)$ where $1\leq q\leq2$, $2\leq p<\frac{10q'}{q'+6}$ and…
The Cauchy problem for the inhomogeneous Helmholtz equation with non-uniform refraction index is considered. The ill-posedness of this problem is tackled by means of the variational form of mollification. This approach is proved to be…
We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large times under appropriate scaling…
We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove weak convergence to a…
Mixed Convection inside a cavity resulting from thermal buoyancy force under local modulation via rotating flat plate has been investigated. The present model consists of a square cavity with the left and right vertical walls fixed at…
We consider the Cauchy problem for (energy-subcritical) nonlinear Schr\"odinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum…
The Cauchy problem for a nonlinear elastic wave equations with viscoelastic damping terms is considered on the 3 dimensional whole space. Decay and smoothing properties of the solutions are investigated when the initial data are…
A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the…
We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and…
First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation…
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is…
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 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…
The Cauchy problem for the Vlasov-Maxwell-Boltzmann equations (VMB) is considered. First the renormalized solution to the Vlasov equation with the Lorentz force is discussed and the difficulty on the partial differentiability of the…
In this paper we study the Cauchy problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of…
Under certain conditions, imposed on the viscosity of the fluid, initial data and the class of contours under consideration, the Cauchy problem with finite values of time for the loop equation in turbulence with Gaussian random forces is…
We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the…
In this paper, we study the solvability of a Cauchy- Dirichlet problem for nonlinear parabolic equation with non standard growths and nonlocal terms. We show the existence of weak solutions of the considered problem under more general…