Publication: Large solutions for a system of elliptic equations arising from fluid dynamics
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2005
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Society for Industrial and Applied Mathematics
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This paper is concerned with the elliptic system (0.1) Delta upsilon=phi, Delta phi=vertical bar del upsilon vertical bar(2) posed in a bounded domain Omega subset of R-N, N is an element of N. Specifically, we are interested in the existence and uniqueness or multiplicity of "large solutions," that is, classical solutions of (0.1) that approach infinity at the boundary of Omega. Assuming that Omega is a ball, we prove that the system (0.1) has a unique radially symmetric and nonnegative large solution with v(0) = 0 (obviously, v is determined only up to an additive constant). Moreover, if the space dimension N is sufficiently small, there exists exactly one additional radially symmetric large solution with v(0) = 0 (which, of course, fails to be nonnegative). We also study the asymptotic behavior of these solutions near the boundary of Omega and determine the exact blow-up rates; those are the same for all radial large solutions and independent of the space dimension. Our investigation is motivated by a problem in fluid dynamics. Under certain assumptions, the unidirectional flow of a viscous, heat-conducting fluid is governed by a pair of parabolic equations of the form (0.2) upsilon(t) -Delta upsilon=theta, theta t-Delta theta=vertical bar del upsilon vertical bar(2), where v and theta represent the fluid velocity and temperature, respectively. The system (0.1), with phi = -theta, is the stationary version of (0.2).
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