This paper is concerned with the elliptic system (0.1) ∆v = φ, ∆φ = |∇v|^2 posed in a bounded domain Ω ⊂ R^N, N ∈ 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 Ω. Assuming that Ω 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 Ω 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) v_t− ∆v = θ, θ_t− ∆θ = |∇v|^2, where v and θ represent the fluid velocity and temperature, respectively. The system (0.1), with φ = −θ, is the stationary version of (0.2).
Large solutions for a system of elliptic equations arising from fluid dynamics
LAZZO, Monica;
2005-01-01
Abstract
This paper is concerned with the elliptic system (0.1) ∆v = φ, ∆φ = |∇v|^2 posed in a bounded domain Ω ⊂ R^N, N ∈ 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 Ω. Assuming that Ω 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 Ω 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) v_t− ∆v = θ, θ_t− ∆θ = |∇v|^2, where v and θ represent the fluid velocity and temperature, respectively. The system (0.1), with φ = −θ, is the stationary version of (0.2).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
