Advances in Differential Equations

Local strong solutions of a parabolic system related to the Boussinesq approximation for buoyancy-driven flow with viscous heating

J. I. Díaz, J. M. Rakotoson, and P. G. Schmidt

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Abstract

We propose a modification of the classical Navier-Stokes-Boussinesq system of equations, which governs buoyancy-driven flows of viscous, incompressible fluids. This modification is motivated by unresolved issues regarding the global solvability of the classical system in situations where viscous heating cannot be neglected. A simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. In the present paper, we establish the local-in-time existence and uniqueness of strong solutions for the model problem. The full system of equations and the global-in-time existence of weak solutions will be addressed in forthcoming work.

Article information

Source
Adv. Differential Equations Volume 13, Number 9-10 (2008), 977-1000.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867327

Mathematical Reviews number (MathSciNet)
MR2482584

Zentralblatt MATH identifier
1230.35097

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35Q35: PDEs in connection with fluid mechanics 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]

Citation

Díaz, J. I.; Rakotoson, J. M.; Schmidt, P. G. Local strong solutions of a parabolic system related to the Boussinesq approximation for buoyancy-driven flow with viscous heating. Adv. Differential Equations 13 (2008), no. 9-10, 977--1000. https://projecteuclid.org/euclid.ade/1355867327.


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