Self-similar turbulent bursts: existence and analytic dependence

Abstract

We consider the equation of turbulent burst propagation $$ u_t=l(t)\Delta u^m-k\frac{u^m}{l(t)} $$ with $m>1$ and $k \ge 0$. Here $l(t)$ denotes the radius of the support of the turbulent distribution. The equation possesses special self-similar solutions of the form $$ u(x,t)= t^{-\mu/(m-1)}f(\zeta),\ \ \zeta=|x|t^{\mu-1}, $$ which are non-negative and have compact support in the space variables. A most interesting feature of such solutions is that the decay exponent is anomalous. It has been proved that general solutions approach for large $t$ these self-similar profiles (intermediate asymptotics). We prove here the analytic dependence with respect to the relevant parameter $k$, of the anomalous exponent appearing in the special solution. The method used is in itself a complete proof of the existence of the self-similar solution with anomalous exponent, different from that recently given in [11]. This new proof also gives us the analyticity of $f^{m-1}$, as a function of $k$ and $\z$. The main technical contribution is a shooting method from the free boundary, which uses a fixed-point theorem for complex analytic functions, and produces analytic regularity. A major novelty of the paper consists in showing that the family of self-similar solutions can be continued for $k<0$ (representing a reaction term) down to a critical value $-k_*<0$ at which $\mu$ has an asymptote.