Related papers: Cauchy problem with data on a characteristic cone …
Nowadays the Euler-Poisson-Darboux equation is extensively studied in several settings. The main questions on avery spaces are explicit solutions for the classical Cauchy problems with the second data null. In this note we will generalize…
In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove…
We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model $-1\leq \gamma< 0$ with the small initial data in three dimensional space. Thus our…
The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by…
In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In…
We show the short time existence and uniqueness of solutions to the Cauchy problem for fully nonlinear systems of arbitrary even order on closed manifolds which are strongly parabolic at the initial values. The proof uses a linearization…
We present a simple method to obtain vacuum solutions of Einstein's equations in parabolic coordinates starting from ones with cylindrical symmetries. Furthermore, a generalization of the method to a more general situation is given together…
We consider the Cauchy problem in ${\bf R}^{n}$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted $L^{1,1}({\bf R}^{n})$ data by using a method introduced in [10].
We present new results concerning the solvability, of lack thereof, in the Cauchy problem for the debar operator, with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples.
In this article, we initiate the study of the Cauchy problem for the two-dimensional relativistic Euler equations in a low-regularity setting. By introducing good variables--a rescaled velocity, logarithmic enthalpy, and an appropriately…
In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic…
We solve the Einstein constraint equations for a first-order causal viscous relativistic hydrodynamic theory in the case of a conformal fluid. For such a theory, a direct application of the conformal method does not lead to a decoupling of…
The method of obtaining of Vlasov-type equations for systems of interacting massive charged particles from the general relativistic Einstein-Hilbert action is considered. An effective approach to synchronizing the proper times of various…
Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects…
A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that…
As a first step in exploring time-periodic solutions of the Einstein equations with a negative cosmological constant, we study the cubic conformal wave equation on the Einstein cylinder. Using a combination of numerical and perturbative…
In this paper, the existence, the uniqueness and estimates of solution to the integral Cauchy problem for linear and nonlinear abstract wave equations are proved. The equation includes a linear operator A defined in a Banach space E, in…
The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution…
In this note, we show a classical result on the local existence and uniqueness of a solution to an initial value problem subject to a Lipschitz condition. We use only elementary tools from mathematical analysis, without involving any…
We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy…