Differential and Integral Equations

Self-similar turbulent bursts: existence and analytic dependence

Fernando Quirós Gracián and Juan L. Vázquez

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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.

Article information

Source
Differential Integral Equations, Volume 8, Number 7 (1995), 1677-1708.

Dates
First available in Project Euclid: 12 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1368397752

Mathematical Reviews number (MathSciNet)
MR1347975

Zentralblatt MATH identifier
0823.35084

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 76F05: Isotropic turbulence; homogeneous turbulence 76R50: Diffusion [See also 60J60]

Citation

Quirós Gracián, Fernando; Vázquez, Juan L. Self-similar turbulent bursts: existence and analytic dependence. Differential Integral Equations 8 (1995), no. 7, 1677--1708. https://projecteuclid.org/euclid.die/1368397752


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