Burgers equation with self-similar Gaussian initial data: Tail probabilities
The author studies the asymptotic behavior of the equation \[ \partial_tu+ u\partial_x u=\mu\partial_{xx}u,\quad u(x,0)= u_0(x)= ds(x)/dx, \] where \(s(x)\) is the fractional Brownian motion \(w_\gamma(x)\), \(\gamma\in (0,2)\). Following the terminolog of gas dynamics, a solution \(u\) and magnitude of discontinuity of \(u\) are called velocity and shock, respectively. The distribution of shocks has the following asymptotic behavior as \(x\to 0\) or \(x\to \infty\): \(0< c_1< F(x) x^\theta< c_2< \infty\), \(x< \varepsilon\), \(-\infty< c_-< \ln\overline F(x) x^{2\theta- 2}< c_+<0\), \(x> \varepsilon^{-1}\), where \(\overline F= 1-F\) and \(\theta= \gamma/2-1\). Estimates of such type are derived both for the distribution of velocity and shock.
scientific article; zbMATH DE number 1279574