Differentiability of solutions to hyperbolic initial-boundary value problems

Abstract:

This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form $(\partial /\partial t - \sum {A_j}\partial /\partial {x_j} - B)u = F$ on $[0,T] \times \Omega ,Mu = g$ on $[0,T] \times \partial \Omega ,u(0,x) = f(x),x \in \Omega$. Assuming that ${\mathcal {L}_2}$ a priori inequalities are known for this equation, it is shown that if $F \in {H^s}([0,T] \times \Omega ),g \in {H^{s + 1/2}}([0,T] \times \partial \Omega ),f \in {H^s}(\Omega )$ satisfy the natural compatibility conditions associated with this equation, then the solution is of class ${C^p}$ from [0, T] to ${H^{s - p}}(\Omega ),0 \leq p \leq s$. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.