Motivated by the quasi-geostrophic model, the authors study the equation:
where and they prove the following theorems:
(1) Let be a function in . For , set . If (and ) satisfies for every the level set energy inequalities
where is a constant.
(2) Let , for . Assume that is bounded in and ; then is in .
From these two theorems, the regularity of solutions to the quasi-geostrophic equation follows.
(3) Let be a solution to an equation
with , a singular integral operator. Assume also that verifies the level set energy inequalities. Then, for every , there exists such that is bounded in .