Isolated singularities for solutions of the nonlinear stationary Navier-Stokes equations

Author:
Victor L. Shapiro

Journal:
Trans. Amer. Math. Soc. **187** (1974), 335-363

MSC:
Primary 35Q10

MathSciNet review:
0380158

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The notion for (u, *p*) to be a distribution solution of the nonlinear stationary Navier-Stokes equations in an open set is defined, and a theorem concerning the removability of isolated singularities for distribution solutions in the punctured open ball is established. This result is then applied to the classical situation to obtain a new theorem for the removability of isolated singularities. In particular, in two dimensions this gives a better than expected result when compared with the theory of removable isolated singularities for harmonic functions.

**[1]**A. P. Calderón and A. Zygmund,*Singular integrals and periodic functions*, Studia Math.**14**(1954), 249–271 (1955). MR**0069310****[2]**Günter Hellwig,*Partielle Differentialgleichungen*:*Eine Einführung*, Mathematische Leitfäden, Teubner Verlagsgesellschaft, Stuttgart, I960; English transl., Blaisdell, Waltham, Mass., 1964. MR**22**#5794; MR**30**#3286.**[3]**O. A. Ladyženskaja,*Mathematical problems in the dynamics of a viscous incompressible flow*, Fizmatgiz, Moscow, 1961; English rev. ed., Gordon and Breach, New York, 1969. MR**27**#5034a; MR**40**#7610.**[4]**Victor L. Shapiro,*Topics in Fourier and geometric analysis*, Mem. Amer. Math. Soc. No.**39**(1961), 100. MR**0147826****[5]**Victor L. Shapiro,*Fourier series in several variables*, Bull. Amer. Math. Soc.**70**(1964), 48–93. MR**0158222**, 10.1090/S0002-9904-1964-11026-0**[6]**Victor L. Shapiro,*Characteristic planes and pointwise solutions of the heat equation.*, Indiana Univ. Math. J.**20**(1970/1971), 115–133. MR**0264248****[7]**Victor L. Shapiro,*Removable sets for pointwise solutions of the generalized Cauchy-Riemann equations*, Ann. of Math. (2)**92**(1970), 82–101. MR**0437898****[8]**Lipman Bers, Fritz John, and Martin Schechter,*Partial differential equations*, American Mathematical Society, Providence, R.I., 1979. With supplements by Lars Gȧrding and A. N. Milgram; With a preface by A. S. Householder; Reprint of the 1964 original; Lectures in Applied Mathematics, 3A. MR**598466****[9]**A.-P. Calderón and A. Zygmund,*Local properties of solutions of elliptic partial differential equations*, Studia Math.**20**(1961), 171–225. MR**0136849****[10]**R. H. Dyer and D. E. Edmunds,*Removable singularities of solutions of the Navier-Stokes equations.*, J. London Math. Soc. (2)**2**(1970), 535–538. MR**0265788**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35Q10

Retrieve articles in all journals with MSC: 35Q10

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1974-0380158-2

Keywords:
Stationary Navier-Stokes equations,
isolated singularities,
nonlinear,
distribution solution

Article copyright:
© Copyright 1974
American Mathematical Society