Limites singulares para equações do tipo Rosenau-KdV-RLW e Benney-Lin : existência e convergência de soluções

Abstract

We consider the approximations @tu + @xf(u) + b1@3xu + b2@t@2xu + 2c@t@4 xu = @2xu (4) and @tu + @xf(u) + @2xu + 2b@3 xu + 3c@4xu + 5d@5xu = @2xu (5)

of the Rosenau-KdV-RLW and Benney-Lin equations and supplementing them with an initial condition u(0; x) = u ; ;0(x) (6) we establish the global existence of solutions u ; for the problems (4){(6) and (5){(6). Moreover, we study the limiting behaviour of the sequence u ; when the parameters and are kept in balance and tend to zero, and we prove that the limit function consists of the unique entropy solution of the conservation law @tu + @xf(u) = 0: The tools used will be the Compensated Compactness Theory developed by Tartar-Murat [22, 23, 27, 28] and DiPerna's theory [10, 11] on Entropy Measure-Valued Solutions together with a number of uniform estimates on the sequence u ; obtained during the text.