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Seminar Abstracts

Linghai Zhang, Lehigh University

"Exact limits of global solutions of some dissipative partial differential equations"

Abstract

 

Motivated by many very interesting results, we will establish exact limits for the $L2$-norm multiplied by the sharp rate of decay of the global solutions, as time approaches positive infinity, of the Cauchy problems for an abstract dissipative partial differential equation in $n$-dimensional space, where $n\geq1$. The model includes the one-dimensional cubic Korteweg-de Vries-Burgers equation, the one-dimensional cubic Benjamin-Ono-Burgers equation, the two-dimensional nonlocal quasi-geostrophic equation, the $n$-dimensional incompressible Navier-Stokes equations and the $n$-dimensional magnetohydrodynamics equations as particular examples. The main ideas in the analysis are Fourier transform, time-dependent decomposition of frequency space and lower limit and upper limit estimates.

For certain other model equations (for example, the fluid dynamics equations in geophysics, the quasi-geostrophic equations with fractional-order derivative, the Cahn-Hilliard equation, the non-degenerate system of filtration type, and the Kuramoto-Sivashinsky equation), which are not covered by the aforementioned abstract differential equation, we can apply the same idea as above to obtain the exact limit of the $L2$-norm of the global solutions.

TUESDAY, November 28, 2006
Time: 1:30 P.M.
Location: PPB 300