Higher-Order for the Multidimensional Generalized BBM-Burgers Equation: Existence and Convergence Results

We study the global existence of solutions for the multidimensional generalized BBM-Burgers equations of the form u(t) + Sigma(d)(j=1) f(j)(u)(xj) = delta Sigma(d)(j=1) u(xjxjt) + Sigma(d)(j=1)(Sigma(N)(n=1)(-1)(n+1)gamma(n)partial derivative(2n)(xj)u), for (x, t) is an element of R(d) x R(+), with initial data u(x, 0) = u(0)(x), x is an element of R(d), as alpha > 0, gamma(n) > 0, n = 1,..., N approach zero, and f is a sufficiently smooth function. We also deal with the convergence of solutions of this Cauchy problem, and the proofs are based instead on DiPerna's uniqueness theory for entropy measure-valued solutions.