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 -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 -dimensional space, where . 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 -dimensional incompressible Navier-Stokes equations and the -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 -norm of the global solutions.
TUESDAY, November 28, 2006