Space-time stationary solutions for the Burgers equation (Q2862637)

The authors study the stationary solutions for the Burgers equation which, in one space dimension, reduces to NEWLINE\[NEWLINE \partial_t u(x,t)+\frac 12\partial_x u^2(x,t)=-\partial_x F(x,t),NEWLINE\]NEWLINE where \(u(x,t)\) denotes the velocity and \(-\partial_x F(x,t)\) the acceleration. The authors consider entropy solutions, which are globally well defined and unique for a large class of initial velocity and forcing terms, since solutions of this equation develop discontinuities in finite time.NEWLINENEWLINEThey study the long-term behavior when the forcing \(-\partial_x F(x,t)\) is a space-time stationary random process. Here, the key ingredient is the Lax-Oleinik variational principle. Also, few results have been available in the noncompact setting and in this work, forcing with homogeneous probability distribution in space and time are considered.