scholarly journals On the existence of weak solutions for the initial-boundary value problem in the Jeffreys model of motion of a viscoelastic medium

2004 ◽  
Vol 2004 (10) ◽  
pp. 815-829 ◽  
Author(s):  
D. A. Vorotnikov ◽  
V. G. Zvyagin
Keyword(s):  
Boundary Value Problem ◽  
Weak Solutions ◽  
Viscoelastic Medium ◽  
Dimensional Space ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Constitutive Law ◽  
Existence Of Weak Solutions ◽  
Initial Boundary Value

This paper deals with the initial-boundary value problem for the system of motion equations of an incompressible viscoelastic medium with Jeffreys constitutive law in an arbitrary domain of two-dimensional or three-dimensional space. The existence of weak solutions of this problem is obtained.

GLOBAL WEAK SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM FOR SCREW PINCHES IN PLASMA PHYSICS

2009 ◽  
Vol 19 (06) ◽  
pp. 833-875 ◽  
Author(s):  
JIANWEN ZHANG ◽  
SONG JIANG ◽  
FENG XIE
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Plasma Physics ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Field Equations ◽  
Initial Boundary Value

This paper is concerned with an initial-boundary value problem for screw pinches arisen from plasma physics. We prove the global existence of weak solutions to this physically very important problem. The main difficulties in the proof lie in the presence of 1/x-singularity in the equations at the origin and the additional nonlinear terms induced by the magnetic field. Solutions will be obtained as the limit of the approximate solutions in annular regions between two cylinders. Under certain growth assumption on the heat conductivity, we first derive a number of regularities of the approximate physical quantities in the fluid region, as well as a lot of uniform integrability in the entire spacetime domain. By virtue of these estimates we then argue in a similar manner as that in Ref. 20 to take the limit and show that the limiting functions are indeed a weak solution which satisfies the mass, momentum and magnetic field equations in the entire spacetime domain in the sense of distributions, but satisfies the energy equation only in the compact subsets of the fluid region. The analysis in this paper allows the possibility that energy is absorbed into the origin, i.e. the total energy be possibly lost in the limit as the inner radius goes to zero.

Weak solutions to the initial-boundary value problem for the shearing of non-homogeneous thermoviscoplastic materials

1989 ◽  
Vol 113 (3-4) ◽  
pp. 257-265 ◽  
Author(s):  
Nicolas Charalambakis ◽  
François Murat
Keyword(s):  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Temperature Dependent ◽  
Initial Boundary Value

SynopsisWe prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.

scholarly journals Formulation and analysis of a parabolic transmission problem on disjoint intervals

2012 ◽  
Vol 91 (105) ◽  
pp. 111-123 ◽  
Author(s):  
Bosko Jovanovic ◽  
Lubin Vulkov
Keyword(s):  
Boundary Value Problem ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Input Data ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
One Dimensional ◽  
Initial Boundary Value

We investigate an initial-boundary-value problem for one dimensional parabolic equations in disjoint intervals. Under some natural assumptions on the input data we proved the well-posedness of the problem. Nonnegativity and energy stability of its weak solutions are also studied.

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