Singularities and Error Bounds for Hyperbolic Equations

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The Euler equations, introduced in 1755, provide a basic mathematical description of the motion of fluids. While these equations are now extensively used in physics and engineering, some fundamental theoretical issues have remained unsolved. An outstanding open question is whether this model is deterministic. In other words, knowing the present configuration of a fluid, under which conditions can we uniquely predict its future behavior? Recent numerical experiments, performed by the principal investigator and collaborators, have identified certain initial states of the fluid which lead to multiple solutions. The present project will investigate the basic mechanism for which Euler's equations may fail to determine a unique solution. Specific examples will be studied, containing one or more spiraling vortices, to understand whether a similar loss of uniqueness occurs for compressible as well as incompressible fluid flow. The analysis will be carried out by a combination of theoretical and computational techniques. Rigorous estimates will be derived on the difference between a numerically computed approximation and the corresponding exact solution. The project will provide a training ground for various graduate students and young researchers.This project will investigate singularities of solutions to nonlinear wave equations. In particular, the analysis will focus on a class of initial value problems for the Euler equations modeling a two-dimensional, inviscid, compressible, or incompressible fluid flow. Based on recent numerical simulations conducted by the PI and collaborators, initial data having an algebraic singularity at the origin are expected to provide the simplest examples of Cauchy problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. This analysis will be carried out by a combination of theoretical and computational techniques. In a neighborhood of a spiraling vortex singularity, the solution will be studied by a suitable transformation of variables. On a domain where the solution is smooth, rigorous a posteriori error bounds for the numerical approximations will be derived. A related project will seek a posteriori error bounds for discrete numerical schemes, such as the Lax-Friedrichs and the Godunov scheme, in the computation of entropy-weak solutions to one-dimensional hyperbolic systems of conservation laws.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.