Critical thresholds in Eulerian dynamics

In this thesis, we study the critical regularity phenomena in Eulerian dynamics, u_t+u\cdot \nabla u=F(u,D u,\cdots), here $F$ represents a general force acting on the flow and by regularity we seek to obtain a large set of sub-critical initial data. We analyze three prototype models, ranging from the one-dimensional Euler-Poisson equations to two-dimensional system of Burgers equations to three-dimensional, four-dimensional and even higher-dimensional restricted Euler systems. We begin with the one-dimensional Euler-Poisson equations, where $F$ is the Poisson forcing term together with the usual γ-law pressure. We prove that global regularity of the Euler-Poisson equations with γ\geq 1 depends on whether or not the initial configuration crosses an intrinsic critical threshold. Next, we discuss multi-dimensional examples. The first multi-dimensional example that we focus our attention on is the two dimensional pressureless flow, where F=ε \Delta u. Our analysis shows that there is a uniform \textsl{BV} bound of the solutions u^ε. Moreover, if the initial velocity gradient \nabla u₀ does not have negative eigenvalues, then its vanishing viscosity limit is the smooth solution of the corresponding equations of the inviscid fluid flow. The second multi-dimensional example we discuss here is the restricted Euler dynamics, where \nabla F= \displaystyle\frac{1}{n} \mathrm {tr} (\nabla u)²I_{n\times n}\,. Our analysis shows that for the three-dimensional case, the finite-time breakdown of the restricted Euler system is generic, and for the four-dimensional case, there is a surprising global existence for sub-critical initial data. Further analysis extends the above result to the general $n$-dimensional ($n>4$) restricted Euler system.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2007

Author dc:creator

• Wei, Dongming

Advisor dc:contributor.advisor

• Tadmor, Eitan

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/1903/7229

OAI identifier oai:identifier

oai:drum.lib.umd.edu:1903/7229