Critical Threshold Phenomena in Nonlinear Balance Laws

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

We plan to study the so-called critical threshold phenomena associated with different nonlinear balance laws, in which the persistence of global features of the solutions hinges on a delicate balance between nonlinear convection and a variety of forcing mechanisms. Thus, for example, solutions for nonlinear hyperbolic conservation laws will develop generic singularities in finite time, whereas the existence of balancing forces in other time-dependent problems, e.g., the 2-D incompressible Euler equations, retain global smoothness for all time. This project proposes to analyze those borderline cases, where the persistence of features such as smoothness and finite-time breakdown vs. time decay etc., does not fall into any one particular category. Instead, global features depend on crossing critical threshold associated with the intrinsic configurations of our problems. The presence of various forcing mechanisms in nonlinear convection-dominated PDEs changes the physical situation, and is responsible for the complexity of the underlying problem. Our proposed research falls into four major sub-categories, all tied to the central issue of critical threshold phenomena in nonlinear balance laws: (i) The question of global smoothness/finite-time breakdown for Euler-Poisson equations. We also plan to study the critical threshold phenomena for solutions of Euler-Poisson equations, its relation to semi-classical limits of nonlinear Schrodinger equations and to augment these studies by high-resolution numerical simulations; (ii) Lack of L2-concentrations (-- and hence global existence) of weak solution for Euler equations, depending on the initial configuration in appropriate borderline regularity spaces; (iii) Questions of global vs. local existence for restricted Euler and Navier-Stokes dynamics; and finally, (iv) The issue of crossing the critical sub-characteristic threshold condition in hyperbolic relaxation problems, where there is a balance between different orders of nonlinear convective waves. The breakdown of waves on the beach is a familiar phenomenon. This breakdown phenomenon depends on whether the waves accumulate sufficient strength, height etc. and in general, they depend on whether the initial configuration crosses intrinsic critical thresholds which distinguish between finite time breakdown and long term persistence of the smooth wave patterns. The goal of this project is to study a variety of critical threshold phenomena in problems governed by time-dependent problems. While many such problems develop finite-time singularities and other problems retain global smooth solutions, we focus on borderline cases, where intrinsic features of the solutions like smoothness vs. generic finite time breakdown, boundedness, time decay, etc, hinge on the delicate balance between the nonlinear convection and the (possibly nonlinear) forcing terms. In particular, the persistence of such global features does not fall to any particular category, but instead, these features depend on crossing critical threshold associated with the general configurations of our problems, very much like the conditional breakdown of waves on the beach. These are precisely the kind of problems that are of great research interest in various applications. The critical threshold phenomenon in nonlinear balance laws is not well understood, and the available techniques to study such phenomena need to be further investigated. In this context, there are many issues to be clarified, inter connections to be analyzed -- even in simplified settings, and general understanding of the critical threshold phenomena in realistic situations is sought in terms of both analytical and numerical studies. Professors E. Tadmor and H. Liu will continue their ongoing cooperative and individual research on the critical threshold phenomena in the context of Euler-Poisson equations, incompressible Euler equations, long time existence and finite time breakdown of restricted Euler and Navier-Stokes dynamics, and hyperbolic relaxation systems.