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)
A remark on regularity criterion for the dissipative quasi-geostrophic equations. (English) Zbl 1154.76339
Summary: This paper concerns with a regularity criterion of solutions to the 2D dissipative quasi-geostrophic equations. Based on a logarithmic Sobolev inequality in Besov spaces, the absence of singularities of $\theta $ in $[0,T]$ is derived for $\theta $ a solution on the interval [$0,T$) satisfying the condition $$\nabla^{\perp}\theta \in L^r(0,T; \dot B^0_{p,\infty})\quad \text{for} \frac{2}{p}+\frac{\alpha}{r}=\alpha,\frac{4}{\alpha} \leqslant p \leqslant \infty$$ This is an extension of earlier regularity results in the Serrin’s type space $L^r(0,T;L^p)$.

76D50Stratification effects in viscous fluids
76D03Existence, uniqueness, and regularity theory
35Q35PDEs in connection with fluid mechanics
76U05Rotating fluids
Full Text: DOI
[1] Chae, D.: On the regularity conditions for the dissipative quasi-geostrophic equations. SIAM J. Math. anal. 37, 1649-1656 (2006) · Zbl 1141.76010
[2] Chae, D.; Lee, J.: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Comm. math. Phys. 233, 297-311 (2003) · Zbl 1019.86002
[3] Constantin, P.; Majda, A.; Tabak, E.: Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495-1533 (1994) · Zbl 0809.35057
[4] Constantin, P.; Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. anal. 30, 937-948 (1999) · Zbl 0957.76093
[5] Ju, N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equation in the Sobolev space. Comm. math. Phys. 251, 365-376 (2004) · Zbl 1106.35061
[6] Kato, T.; Ponce, G.: Commutator estimates and the Euler and Navier -- Stokes equations. Comm. pure appl. Math. 41, 891-907 (1988) · Zbl 0671.35066
[7] Kozono, H.; Ogawa, T.; Taniuchi, Y.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242, 251-278 (2002) · Zbl 1055.35087
[8] Majda, A.; Tabak, E.: A two-dimensional model for quasi-geostrophic flow: comparison with the two-dimensional Euler flow. Phys. D 98, 515-522 (1996) · Zbl 0899.76105
[9] Pedlosky, J.: Geophysical fluid dynamics. (1987) · Zbl 0713.76005
[10] Serrin, J.: On the interior regularity of weak solutions of the Navier -- Stokes equations. Arch. ration. Mech. anal. 9, 187-195 (1962) · Zbl 0106.18302
[11] Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. (1993) · Zbl 0821.42001
[12] Triebel, H.: Theory of function spaces. (1983) · Zbl 0546.46028
[13] Wu, J.: The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity 18, 139-154 (2005) · Zbl 1067.35002