Abstract
It is shown that if [0,\(\hat T\)) is the maximal interval of existence of a smooth solutionu of the incompressible Euler equations in a bounded, simply connected domain Ω\( \subseteq\) R 3, then\(\int_0^{\hat T} {\left| {\omega ( \cdot ,t)} \right|_{L^\infty (\Omega )} } dt = \infty\), where ω=∇×u is the vorticity. Crucial to this result is a special estimate proven in Ω of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity.
Similar content being viewed by others
References
Adams, R.: Sobolev Spaces. New York: Academic Press 1975
Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math.17, 623–727 (1959)
Agmon, S., Douglis, A., Nirenberg, L.: Estimates over the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math.17, 35–92 (1964)
Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of Smooth Solutions for the 3-D Euler Equations. Commun. Math. Phys.94, 61–66 (1984)
Ferrari, A.: On the Blow-up of Solutions of the 3-D Euler Equations in a Bounded Domain. Ph.D. thesis, Duke University, 1992
Kato, T., Lai, C.Y.: Nonlinear Evolution Equations and the Euler Flow. J. Funct. Anal.56, 15–28 (1984)
Kato, T., Ponce, G.: Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue SpacesL p s (R 2). Revista Matematicas Iberoamericana2, 73–88 (1986)
Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math.34, 481–524 (1981)
Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. In: Applied Math. Sciences Series, Vol.53, Berlin, Heidelberg, New York: Springer 1983
Majda, A.: Vorticity and the Mathematical Theory of Incompressible Fluid Flow. Lecture Notes, Princeton University, 1986–1987
Solonnikov, V.A.: On Green's Matrices for Elliptic Boundary Value Problems I. Trudy Mat. Inst. Steklov110, 123–170 (1970)
Solonnikov, V.A.: On Green's Matrices for Elliptic Boundary Value Problems II. Trudy Mat. Inst. Steklov116, 187–226 (1971)
Taylor, M.: Pseudodifferential Operators and Nonlinear PDE. Boston, MA: Birkhäuser 1991
Temam, R.: On the Euler Equations of Incompressible Perfect Fluids. J. Funct. Anal.20, 32–43 (1975)
Wahl, W. von: Vorlesung über das Außenraumproblem für die instationären Gleichungen von Navier-Stokes. Rudolph-Lipschitz-Vorlesung, Sonderforschungsbereich 256. Nichtlineare Partielle Differentialgleichungen, Bonn, 1989
Wahl, W. von: Estimating ∇u by ∇·u and ∇×u. To appear in Math. Methods in the Appl. Sciences
Yanagisawa, T.: A Continuation Principle for the Euler Equations for Incompressible Fluids in a Bounded Domain. Preprint
Zajaczkowski, M.: Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations in a Bounded Domain. Bull. Polish Acad. of Sciences Mathematics37, 169–181 (1989)
Author information
Authors and Affiliations
Additional information
Communicated by J.L. Lebowitz
Rights and permissions
About this article
Cite this article
Ferrari, A.B. On the blow-up of solutions of the 3-D Euler equations in a bounded domain. Commun.Math. Phys. 155, 277–294 (1993). https://doi.org/10.1007/BF02097394
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02097394