Cauchy problem for a class of quasilinear hyperbolic systems is well posed on a family of Besov spaces

In this paper we consider the two dimensional Cauchy problem for the quasilinear systems{∂tu+a(u)∂xu=0∂tu+au(0,x)=u0(x), with u=(u1,…,uN)u=(u1,…,uN) and a(u)=(ajk(u))j,k=1N real N×NN×N matrix, with entries C∞C∞, such that the eigenvalues of a(0)a(0) are real and distinct, that is, the system is hyperbolic at u=0u=0. We show that a family of Besov spaces, containing the Hölder spaces, near u=0u=0 is continuously preserved by the flow of the above Cauchy Problem.