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Decay of solutions to nonlinear parabolic equations: renormalization and rigorous results
Lack of hyperbolicity in the twofluid model for twophase incompressible flow
1.  Department of Mathematics, University of Houston, Houston, Texas 772043008, United States, United States 
2.  Department of Mathematics, The Hebrew University, Jerusalem, Israel 
Much attention has centered on reformulating details of the model to avoid this awkwardness. This paper takes a different approach: a study of the nonhyperbolic operator itself. The objective is to understand the nature of illposedness in nonlinear, as distinct from linearized, models.
We present our initial study of the nonlinear operator that occurs in the twofluid equations for incompressible twophase flow. Our research indicates that one can solve Riemann problems for these nonlinear, nonhyperbolic equations. The solutions involve singular shocks, very low regularity solutions of conservation laws (solutions with singular shocks, however, are not restricted to nonhyperbolic equations). We present evidence, based on asymptotic treatment and numerical solution of regularized equations, that these singular solutions occur in the twofluid model for incompressible twophase flow. The Riemann solutions found using singular shocks have a reasonable physical interpretation.
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