We study the initial value problem for the quasi-geostrophic type equations
where , ( ) is a fixed parameter and is divergence free and determined from through the Riesz transform , with a permutation of . The initial data is taken in the Sobolev space with negative indices. We prove local well-posedness when
We also prove that the solution is global if is sufficiently small.
Submitted November 26, 1996. Published June 12, 1998.
Math Subject Classification: 35K22, 35Q35, 76U05.
Key Words: Quasi-geostrophic equations, Weak data, Well-posedness.
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School of Mathematics, Institute for Advanced Study
Princeton, NJ 08540. USA