The Principal Investigator and his collaborators study hyperbolic conservation laws, a class of partial differential equations that covers many of the fundamental equations in applied math, and in particular in fluid dynamics. The main lines of current research are: (A) a systematic approach to the study of one-dimensional systems of conservation laws based on geometric properties of their eigen-frames; (B) employing vanishing viscosity approximations to obtain interaction estimates for conservation laws in several space dimensions; (C) radially symmetric solutions to systems of conservation laws, and in particular solutions to the compressible Euler system.
All of the projects are motivated by the need to provide rigorous statements about the properties of physical models that are used by scientists and engineers. The results assess the range of validity of various physical models and will be of use in delimiting and refining existing models. Particular emphasis is put on understanding solutions of complex, nonlinear equations that are used in simulations of e.g. high speed flight, traffic flow, combustion and detonations.
''(Federal Award ID1009002). The project addresses mathematical issues originating in physical models that describe fluid flow, and in particular gas flow. While the basic models have been known since the 1750s, we still lack answers to basic questions related to existence and qualitative properties of the solutions. A mathematical analysis of these issues is a first step toward a proper understanding of extreme regimes of fluid flow that are great practical interest. Examples of such include supersonic flight, detonations, plasma flow, meteorology etc.These are instances where actual experiments are costly or not possible, while computer simulations may not be reliable. A fundamental challenge is to assess the limits of validity of simplified models, and thus be in a position to provide improvements when necessary. What makes the problems mathematically challenging is their non-linear structure. While near-equilibrium situations have been analyzed with some success. However, for large amplitude flows (far from equilibrium) the non-linear nature of the processes becomes dominant and we currently lack proper answers to even basic questions in such situations. The present project considers a class of models related to gas flow (the Euler equations), as well as systems that, in a precise sense, has a similar structure. Common to all the equations is that they describe phenomena for conserved quantities where information propagate at finite speeds(so-called hyperbolic conservations laws). A major divide in the available theory is between (spatially) one-dimensional vs. multi-dimensional problems. The latter is far more complex and relatively littleis rigorously known in this case. The completed research can be described under three sub-projects: (1) Classification of systems of equations with prescribed geometric structures.In a collaboration with Professor Irina Kogan (North Carolina State University) we have considered the issue of how the underlying geometry characterizes a system of conservation laws. The main part of the activity was to directed toward clarifying how many systems possess certain geometric properties, including the possession of an associated entropy field.During the funding period we published three refereed papers related to geometric structures for systems of hyperbolic conservation laws. (2) Building and analysis of concrete solutions to the Euler equationsOne case considered solutions with repeated shock reflections, while another work resolved all possible pairwise wave interactions of elementary waves. (3) Study of radially symmetric waves. These are the simplest solutions exhibiting multi-dimensional effects.Three works dealing with the construction and stability properties of radially symmetric solutions, both in physical models as well as simplified toy models were considered. In addition a project on the compressible Navier-Stokes equations (roughly, Euler equations with dissipative terms modeling heat conduction and viscosity) was completed and resulted in one peer reviewed paper. Finally, in collaboration with the postdoc Geng Chen (NSF-funded), we initiated a study of regularity properties of the Euler system for isentropic flow. This work points out a basic problem by proving that there is no quantity whose regularity (in terms of variation) always decreases in time when evaluated along solutions. (This work has subsequently been published in a peer reviewed journal.)
