zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
On the regularity of the pressure of weak solutions of Navier-Stokes equations. (English) Zbl 0574.35070
The aim of this paper is to prove the regularity property pL 5/3 for the pressure of weak solutions of Navier-Stokes equations in a bounded or an exterior domain. This result was known only for the whole space. The method to prove this property uses a new potential theoretical estimate of the linearized equation with different integration exponents in space and time. Using this regularity property of the pressure, it is possible to prove the existence of a weak solution of Navier-Stokes equations which is smooth for large |x| in an exterior domain. This result has been proved by L. Caffarelli, R. Kohn and L.Nirenberg [Commun. Pure Appl. Math. 35, 771-831 (1982; Zbl 0509.35067)] for the whole space.

MSC:
35Q30Stokes and Navier-Stokes equations
35D10Regularity of generalized solutions of PDE (MSC2000)
35D05Existence of generalized solutions of PDE (MSC2000)
76D05Navier-Stokes equations (fluid dynamics)
References:
[1]M. E. Bogovski, Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Soviet Math. Dokl.20, 1094-1098 (1979).
[2]L. Caefarelli, R. Kohn andL. Nirenberg, Partial Regularity of Suitable Weak Solutions of the Navier-Stokes Equations. Comm. Pure Appl. Math.35, 771-831 (1982). · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[3]W.Erig, Die Gleichungen von Stokes und die Bogovski-Formel. Diplomarbeit, Universit?t-Gesamthochschule Paderborn 1982.
[4]D. Fujiwara andH. Morimoto, AnL r -Theorem of the Helmholtz-decomposition of vector fields. J. Fac. Sci. Univ. Tokyo24, 685-699 (1977).
[5]Y. Giga, The Stokes operator inL r spaces. Proc. Japan Acad.57, 85-89 (1981). · Zbl 0471.35069 · doi:10.3792/pjaa.57.85
[6]Y. Giga, Analyticity of the semigroup generated by the Stokes operator inL r spaces. Math. Z.178, 297-329 (1981). · Zbl 0461.47019 · doi:10.1007/BF01214869
[7]T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J.12, 115-140 (1982).
[8]H. Sohr, Zur Regularit?tstheorie der instation?ren Gleichungen von Navier-Stokes. Math. Z.184, 359-375 (1983). · Zbl 0506.35084 · doi:10.1007/BF01163510
[9]H. Sohr, Optimale lokale Existenzs?tze f?r die Gleichungen von Navier-Stokes. Math. Ann.267, 107-123 (1984). · Zbl 0552.35059 · doi:10.1007/BF01458474
[10]H. Sohr andW. von Wahl, A New Proof of Leray’s Structure Theorem and the Smoothness of Weak Solutions of Navier-Stokes Equations for Large |x|. Bayreuth. Math. Sehr.20, 153-204 (1985).
[11]V. A. Solonnikov, Estimates for Solutions of Nonstationary Navier-Stokes Equations. J. Soviet Math.8, 467-529 (1977). · Zbl 0404.35081 · doi:10.1007/BF01084616
[12]W. von Wahl, Regularit?tsfragen f?r die instation?ren Navier-Stokesschen Gleichungen in h?heren Dimensionen. J. Math. Soc. Japan32, 263-283 (1980). · Zbl 0456.35073 · doi:10.2969/jmsj/03220263
[13]W. von Wahl, The Equationu? + A(t)u=fin a Hilbert Space andL p -Estimates for Parabolic Equations. J. London Math. Soc.25, 483-497 (1982). · Zbl 0493.35050 · doi:10.1112/jlms/s2-25.3.483
[14]W. von Wahl, ?ber das Verhalten f?rt?0 der L?sungen nichtlinearer parabolischer Gleichungen, insbesondere der Gleichungen von Navier-Stokes. Bayreuth. Math. Schr.16, 151-277 (1984).
[15]W. von Wahl, Klassische L?sbarkeit im Gro?en f?r nichtlineare parabolische Systeme und das Verhalten der L?sungen f?rt??. Nachr. Akad. Wiss. G?ttingen, 5. Akademie der Wissenschaften, G?ttingen 1981, 131-177 (1981).