Large-deviations estimates in Burgers turbulence with stable noise initial data
This paper presents two interesting results concerning the tail distribution of the Hopf-Cole solution \(u(x,t)\) of the inviscid Burgers equation \(\partial_t u+\partial_x(u^2/2)=0\), when the initial datum \(u(x,t=0)\) is a (two-sided) Lévy process. The author uses large deviations techniques to obtain his results. First he studies the case when the initial potential \(\psi(\cdot,0)\) (such that \(u(x,t=0)=-\partial_x \psi\)) is a completely skewed to the left stable process with index \(\alpha\in(1,2]\), i.e. its Laplace transform is \(E(\text{exp}\{\lambda\psi(1,0)\})=\text{exp}\{c\lambda^{\alpha}\}\), \(\lambda>0\). In this case it holds: \[ -\log P(u(x,t)>y)\sim c^{-1/(\alpha-1)}(\alpha-1)^2(2\alpha-1)^{-1}\alpha^{-\alpha/(\alpha-1)}ty^{(2\alpha-1)/(\alpha-1)} \] when \(y\rightarrow\infty\). Afterwards the case when the initial potential \(\psi(\cdot,0)\) is a stable Lévy process of index \(\alpha\in(1/2,2)\) which is not completely skewed, is considered. The characteristic function of such a process has the form: \[ E[\text{exp}\{i\lambda\psi(1,0)\}]= \text{exp}\{-c | \lambda| ^{\alpha}(1- i \text{sgn}(\lambda))\beta \tan(\pi \alpha/2)\}. \] The result can be expressed \[ \kappa\leq \liminf_{y\rightarrow\infty}y^{2\alpha-1}P(u(x,t)>y)\leq\limsup_{y\rightarrow\infty}y^{2\alpha-1}P(u(x,t)>y)\leq 2\alpha \kappa, \] where \(\kappa=\frac{c 2^{\alpha-1}(1-\alpha)(1+\beta)}{(2\alpha-1)\Gamma(2-\alpha)\cos(\pi \alpha/2)}t^{\alpha-1}\) for \(\alpha\neq 1\) and \(\kappa=\frac{c}{\pi}\) if \(\alpha=1\).
scientific article; zbMATH DE number 1279478