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Seminar Abstracts

Linghai Zhang, Lehigh University

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



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