Advances in Differential Equations

Blow-up of a class of solutions with free boundaries for the Navier-Stokes equations

V. A. Galaktionov and J. L. Vazquez

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We study the blow-up behaviour of a class of solutions of the incompressible Navier-Stokes equations in $\mathbf{R}^2$ which are described by a parabolic free-boundary problem with a nonlocal term. We prove that the solutions blow up in finite time globally in space and calculate the corresponding rates and profiles. The asymptotic profile and the free boundary are obtained as a nonviscous limit, solution of a nonlocal first-order equation of the Hamilton-Jacobi type. This means that the final highly nonstationary stage of such flows is essentially a convection-type singularity. The viscosity term, though asymptotically negligible, provides the eventual regularity of the limit profiles. Other types of unstable blow-up and global patterns with a non-generic behaviour are discussed.

Article information

Adv. Differential Equations, Volume 4, Number 3 (1999), 297-321.

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35B40: Asymptotic behavior of solutions 35R35: Free boundary problems 76D05: Navier-Stokes equations [See also 35Q30]


Galaktionov, V. A.; Vazquez, J. L. Blow-up of a class of solutions with free boundaries for the Navier-Stokes equations. Adv. Differential Equations 4 (1999), no. 3, 297--321.

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