Degenerate Cauchy Research Articles

Degenerate Cauchy–Goursat problem for 2‐D steady isentropic Euler system with van der Waals gas

AbstractThis study concerns with the existence–uniqueness of local classical sonic‐supersonic solution to a degenerate Cauchy–Goursat problem that arises in transonic phenomena. The flow is governed by 2‐D steady isentropic Euler system with a polytropic van der Waals gas. The idea of characteristic decomposition has been used to convert the Euler system into a new system involving the angle variables . To overcome the parabolic degeneracy caused at the sonic curve, the partial hodograph transformation and a variety of dependent–independent variables have been introduced to transform the nonlinear system into a linear one with explicit singularity–regularity structure. The uniform convergence of the sequences has been discussed by employing the mathematical induction. Eventually, the inversion of the solution from partial hodograph plane to the original plane has been established.

On the degenerate Cauchy problem for a nonlinear variational wave system

On the degenerate Cauchy problem for a nonlinear variational wave system

Minimum energy control of degenerate Cauchy problem with skew-Hermitian pencil

Abstract The present research paper deals with the effectiveness of the control of an infinite-dimensional degenerate Cauchy problem with variable operator coefficients, skew-Hermitian pencil and bounded input condition. This study explores the minimum energy control problem. The investigation follows a set of methods to examine the procedure for developing a new result to solve the problem. Indeed, by the use of decomposition transformation of the considered system and the application of the Gramian operator, the formula of the process for controlling the system with minimum energy is obtained. Afterwards, a procedure to compute the optimal input for minimizing the performance index is then proposed. In a nutshell, the obtained results indicate that optimal control for minimizing the performance index ensures the solution of the minimum energy control of an infinite-dimensional degenerate Cauchy problem.

Sonic–supersonic solutions to a degenerate Cauchy–Goursat problem for 2D relativistic Euler equations

Sonic–supersonic solutions to a degenerate Cauchy–Goursat problem for 2D relativistic Euler equations

Two types of hypergeometric degenerate Cauchy numbers

Abstract In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

Differential Equations Associated with Degenerate Cauchy Numbers

In this paper, we introduce the degenerate Cauchy numbers as a degenerate version of the Cauchy numbers and derive a family of nonlinear differential equations satisfied by the generating function for those numbers.

The Solution Of Nonhomogen Abstract Cauchy Problem by Semigroup Theory of Linear Operator

In this article we will investigate how to solve nonhomogen degenerate Cauchy problem via theory of semigroup of linear operator. The problem is formulated in Hilbert space which can be written as direct sum of subset Ker M and Ran M*. By certain assumptions the problem can be reduced to nondegenerate Cauchy problem. And then by composition between invers of operator M and the nondegenerate problem we can transform it to canonic problem, which is easier to solve than the original problem. By taking assumption that the operator A is infinitesimal generator of semigroup, the canonic problem has a unique solution. This allow to define special operator which map the solution of canonic problem to original problem. ©2016 JNSMR UIN Walisongo. All rights reserved.

Singular perturbation degenerate problems occurring in atmospheric dispersion of pollutants

Abstract The boundary value problems for the degenerate differential-operator equations with small parameters generated on all boundary are studied. Several conditions for the separability and the fredholmness in Banach-valued L p -spaces of are given. In applications, maximal regularity of degenerate Cauchy problem for parabolic equation arising in atmospheric dispersion of pollutants studied.

Nonlinear Degenerate Evolution Equations in Mixed Formulation

We develop the theory of degenerate and nonlinear evolution systems in mixed formulation. It will be shown that many of the well-known results for the stationary problem extend to the nonlinear case and that the dynamics of a degenerate Cauchy problem is governed by a nonlinear semigroup. The results are illustrated by a Darcy–Stokes coupled system with multiple nonlinearities.

Existence for a degenerate Cauchy problem

We prove the existence of a solution to the degenerate parabolic Cauchy problem with a possibly unbounded Radon measure as an initial data. To accomplish this, we establish a priori estimates and derive a compactness result. We also show that the result is optimal in the Euclidian setting.

Higher-order degenerate Cauchy problems in locally convex spaces

We establish a criterion for the C-well-posedness of higher order degenerate Cauchy problems (DCP n ) in a sequentially complete locally convex space by introducing a C-propagation family { P( t)} t≥0 for (DCP n ). This result is applied then to a class of (DCP 2) whose coefficient operators are differential operators on various function spaces with Fréchet topologies and two C-well-posedness Theorems are obtained.

Abstract Degenerate Cauchy Problems in Locally Convex Spaces

This paper is concerned with the abstract degenerate Cauchy problem DCP,ddtBu(t)=Au(t)(t≥0),Bu(0)=Bu0, where A and B are closed linear operators in a sequentially complete locally convex space. A C-propagation family for (DCP) is introduced (C is an operator), leading to a general C-wellposedness result about (DCP). Moreover, conditions are given ensuring the existence of C-propagation families for those (DCP) with differential operators, on various function spaces with Fréchet topologies, as coefficient operators. These results are new even in the case of Banach spaces.

Semigroup Theory of Degenerate Linear Cauchy Problems

In this paper we present a straightforward application of semigroup theory to degenerate Cauchy problems given by a pair of linear operators (A,M) defined in a Banach space. We prove existence and uniqueness of strict solutions for initial values contained in suitable subspaces under very mild conditions on the operators A and M . Furthermore, we describe the factorization with respect to KerM . A canonical example for the necessity of this procedure is provided by the Dirac equation in the nonrelativistic limit. The relation between the factorized problem and the degenerate problem is given by a mapping Z. We give simple examples showing that, in general, Z is unbounded and not even closable.

THE CAUCHY PROBLEM FOR A DIFFERENTIAL INCLUSION IN BANACH SPACES AND DISTRIBUTION SPACES

We study well-posedness of degenerate Cauchy problems treated as Cauchy problems for a differential inclusion with a multivalued linear operator. Using a new approach to the definition of degenerate integrated semigroups and their generators in a Banach space, we obtain a well-posedness criterion for the problem. Moreover, we consider the Cauchy problem for a differential inclusion in the space of abstract distributions and give necessary and sufficient conditions for well-posedness in the distribution space.

Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit

Let u α be the solution of the Ito stochastic parabolic Cauchy problem $\partial u/\partial t-L_\a u = \xi \cdot \nabla u, u\v_{t=0} = f$ , where ξ is a space—time noise. We prove that u α depends continuously on α , when the coefficients in L α converge to those in L 0 . This result is used to study the diffusion limit for the Cauchy problem in the Stratonovich sense: when the coefficients of L α tend to 0 the corresponding solutions u α converge to the solution u 0 of the degenerate Cauchy problem $\partial u_0/ \partial t =\xi\circ \nabla u_0, u_0\v_{t=0} = f$ . These results are based on a criterion for the existence of strong limits in the space of Hida distributions (S) * . As a by-product it is proved that weak solutions of the above Cauchy problem are in fact strong solutions.

Stability of degenerate Cauchy horizons in black hole spacetimes

In the multihorizon black hole spacetimes, it is possible that there are degenerate Cauchy horizons with vanishing surface gravities. We investigate the stability of the degenerate Cauchy horizon in black hole spacetimes. Despite the asymptotic behavior of spacetimes (flat, anti-de Sitter, or de Sitter), we find that the Cauchy horizon is stable against the classical perturbations, but unstable quantum mechanically.

Properties of an abstract pseudoresolvent and well-posedness of the degenerate Cauchy problem

The degenerate Cauchy problem in a Banach space is studied on the basis of properties of an abstract analytical function, satisfying the Hilbert identity, and a related pair of operators A, B.

Perturbations of skew-Hermitian bundles and the degenerate cauchy problem

Perturbations of skew-Hermitian bundles and the degenerate cauchy problem

A class of one-dimensional degenerate parabolic equations

Abstract : Degenerate parabolic equations arise in the description of melting processes, gas dynamics and certain biological models. The interfaces corresponding to degeneracies in the constitutive function usually separate different media in the physical problem. Problem (P) stated in the abstract is related to nonlinear diffusion equations with nonmonotone constitutive functions as has been discussed in previous documents. This report obtains explicit self-similar solutions for (P) corresponding to a class of model initial data and determine the free boundary explicitly. The qualitative behavior of these solutions, in particular of their interfaces, is typical of the situation in more general problems. For general initial data, the author then uses these self-similar solutions as comparison functions to study the regularity and the behavior of the free boundary for small time. Keywords: One dimensional degenerate Cauchy problem.

The finite element method for a degenerate hyperbolic partial differential equation

Cahlon has recently considered finite difference methods for the degenerate Cauchy problem (tut)t=Δu,u(x, 0)=u0(x) withu0 analytic. In this paper existence of a unique solution is shown for a more general class of initial data. A smoothing property of the solution operator is then exhibited. The usual semidiscrete finite element method is considered. The approximation is shown to be stable and superconvergent with orderO(h2μ−2) inl2 andl∞, whereμ−1 is the degree of the polynomials used. OptimalL2 andL∞ estimates are also derived.