Self-similar turbulent bursts existence and analytic dependence. (English) Zbl 0823.35084

Summary: We consider the equation of turbulent burst propagation \[ u_ t= l(t) \Delta u^ m-k (u^ m/l(t)) \] with \(m>1\) and \(k\geq 0\). 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 nonnegative 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 [S. P. Hastings and L. A. Peletier, Eur. J. Appl. Math. 3, No. 4, 319-341 (1992; Zbl 0769.76025)]. This new proof also gives us the analyticity of \(f^{m-1}\), as a function of \(k\) and \(\zeta\). 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.


35K55 Nonlinear parabolic equations
76F05 Isotropic turbulence; homogeneous turbulence
35B40 Asymptotic behavior of solutions to PDEs


Zbl 0769.76025