The Cauchy Problem and Propagation of Singularities

1984 ◽  
pp. 185-201 ◽  
Author(s):  
Richard Melrose
Keyword(s):  
Cauchy Problem ◽  
The Cauchy Problem ◽  
Propagation Of Singularities

scholarly journals Precise propagation of singularities for a hyperbolic system with characteristics of variable multiplicity

1986 ◽  
Vol 101 ◽  
pp. 111-130 ◽  
Author(s):  
Chisato Iwasaki ◽  
Yoshinori Morimoto
Keyword(s):  
Cauchy Problem ◽  
Wave Front ◽  
Hyperbolic System ◽  
Wave Front Set ◽  
Variable Multiplicity ◽  
The Cauchy Problem ◽  
Propagation Of Singularities ◽  
Admissible Trajectory

In this paper we consider the Cauchy problem for a hyperbolic system with characteristics of variable multiplicity and construct a certain solution whose wave front set propagates precisely along the so-called “broken null bicharacteristic flow”, in other words, along the admissible trajectory if we use the terminology of [6].

scholarly journals PROPAGATION OF SINGULARITIES IN THE SOLUTIONS TO THE BOLTZMANN EQUATION NEAR EQUILIBRIUM

2008 ◽  
Vol 18 (07) ◽  
pp. 1093-1114 ◽  
Author(s):  
RENJUN DUAN ◽  
MENG-RONG LI ◽  
TONG YANG
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Initial Data ◽  
Cross Section ◽  
The Other ◽  
Spatially Inhomogeneous ◽  
The Cauchy Problem ◽  
Spatially Inhomogeneous Boltzmann Equation ◽  
Propagation Of Singularities ◽  
Derivatives Of

This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of Boudin–Desvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

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