Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations
The authors study the asymptotics of solutions of the initial value problem for a class of Burgers-type equations, and for a related differential-difference analogue, in terms of shifted wave-trains solutions. These equations are \(f_t+\varphi(f)f_x=\varepsilon f_{xx}\) and \({dF\over dt}+\varphi(F){F(x,t)- F(x-\varepsilon,t)\over\varepsilon}=0\), respectively; the initial conditions in each case are \(f(x,0)=\begin{cases} 0, & x<x^-,\\ f^0(x), & x^-\leq x\leq x^+, \\ 1, & x>x^+,\end{cases}\) where \(0<x^-<x^+<\infty\), \(0\leq f^0(x)\leq 1\). In this paper the authors prove their results for the second of these equations; proofs for the first are left to a further paper. Further improvements (for both equations) are to be given in a joint work with A. E. Tumanov. The problem studied is one that goes back to Gelfand, and of which particular cases have been studied over the years by various authors, for example, for \(\varphi(f)\) linear (Burgers equation) or monotonic, as well as for other special cases. Analogous results have been given for the second equation by G.~M.~Henkin and V.~M.~Polterovich. Here results are obtained for a large class of functions \(\varphi\). The equations considered have a variety of physical applications, including to the displacement of oil by water in a porous medium, and to economics.
scientific article; zbMATH DE number 2164953