zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. (English) Zbl 1204.35063

Motivated by the quasi-geostrophic model, the authors study the equation:

t θ+v·θ=-Λθ,x N ,divv=0,

where Λθ=(-Δ) 1/2 θ and they prove the following theorems:

(1) Let θ(t,x) be a function in L (0,T;L 2 ( N ))L 2 (0,T;H 1/2 ( N )). For λ>0, set θ λ :=(θ-λ) + . If θ (and -θ) satisfies for every λ>0 the level set energy inequalities

N θ λ 2 (t 2 ,x)dx+2 t 1 t 2 N |Λ 1/2 θ λ | 2 dxdt N θ λ 2 (t 1 ,x)dx,0<t 1 <t 2 ,


sup x N |θ(T,x)|C * θ 0 L 2 T N/2 ,

where C * >0 is a constant.

(2) Let Q r =[-r,0]×[-r,r] N , for r>0. Assume that θ(t,x) is bounded in [-1,0]× N and v| Q 1 L (-1,0;BMO); then θ is C α in Q 1/2 .

From these two theorems, the regularity of solutions to the quasi-geostrophic equation follows.

(3) Let θ be a solution to an equation

t θ+u·θ=-Λθ,x N ,divu=0,

with u j =R ¯ j [θ], R ¯ j a singular integral operator. Assume also that θ verifies the level set energy inequalities. Then, for every t 0 >0, there exists α such that θ is bounded in C α ([t 0 ,[× N ).

35B65Smoothness and regularity of solutions of PDE
35B45A priori estimates for solutions of PDE
42B37Harmonic analysis and PDE
86A99Miscellaneous topics in geophysics
76D03Existence, uniqueness, and regularity theory