Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (1): 41-74.doi: 10.1016/S0252-9602(12)60004-6

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GLOBAL EXISTENCE, UNIQUENESS, AND STABILITY FOR A NONLINEAR HYPERBOLIC-PARABOLIC PROBLEM IN PULSE COMBUSTION

Olga Terlyga1, Hamid Bellout2, Frederick Bloom2*   

  1. 1.Fermi National Laboratory, Batavia, IL 60510, USA 2.Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA
  • Received:2011-07-12 Online:2012-01-20 Published:2012-01-20
  • Contact: Frederick Bloom,bloom@math.niu.edu E-mail:terlyga@fnal.gov; bellout@math.niu.edu; bloom@math.niu.edu
  • Supported by:

    The research reported in this paper is based, in large measure, on the Ph.D. dissertation of the first author at Northern Illinois University.

Abstract:

A global existence theorem is established for an initial-boundary value prob-lem, with time-dependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonlinear mixed hyperbolic-parabolic sys-tem. Using results previously established for the associated linear problem, a fixed point argument is employed to prove local existence for a regularized version of the nonlinear problem with artificial viscosity. Appropriate a-priori estimates are then derived which imply that the local existence result can be extended to a global existence theorem for the regularized problem. Finally, a di?erent set of a priori estimates is generated which allows for takingthelimit as theartificial viscosity parameter converges tozero; the corresponding solution of the regularized problem is then proven to converge to the unique solution of the
initial-boundary value problem for the original, nonlinear, hyperbolic-parabolic system.

Key words: pulse combustion, hyperbolic-parabolic system, global existence, regularity

CLC Number: 

  • 35M33
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