A Degenerate Pseudoparabolic Regularization of a Nonlinear Forward-Backward Heat Equation Arising in the Theory of Heat and Mass Exchange in Stably Stratified Turbulent Shear Flow

Abstract

The authors analyze an initial-boundary value problem for the equation \[u_t = \varphi (u_x )_x + \tau \psi (u_x )xt,\] where $\tau $ is a positive parameter, $\varphi ,\psi :{\bf R} \to {\bf R}$, $\varphi $ is nonmonotone, $\psi $ is strictly increasing and uniformly bounded in ${\bf R}$, and $|\varphi '(p)| = O(\psi '(p))$ as $p \to \pm \infty $. The equation arises as a (new) model for turbulent heat or mass transfer in stably stratified shear flows, in which case $u_x $ is nonnegative, $\varphi (p) > 0$ for $p > 0$, and $\varphi (0) = \varphi ( + \infty ) = 0$. Well-posedness is proved and, in the model case, the qualitative behavior of solutions is studied. In particular it is shown that smooth solutions may become discontinuous in finite time, and that such solutions converge to a piecewise constant spatial profile as $t \to \infty $. This behavior is in agreement with experimental observations and numerical computations.