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 ( ∂ / ∂ t − ∑ A j ∂ / ∂ x j − B ) u = F (\partial /\partial t - \sum {A_j}\partial /\partial {x_j} - B)u = F on [ 0 , T ] × Ω , M u = g [0,T] \times \Omega ,Mu = g on [ 0 , T ] × ∂ Ω , u ( 0 , x ) = f ( x ) , x ∈ Ω [0,T] \times \partial \Omega ,u(0,x) = f(x),x \in \Omega . Assuming that L 2 {\mathcal {L}_2} a priori inequalities are known for this equation, it is shown that if F ∈ H s ( [ 0 , T ] × Ω ) , g ∈ H s + 1 / 2 ( [ 0 , T ] × ∂ Ω ) , f ∈ H s ( Ω ) 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 {C^p} from [0, T] to H s − p ( Ω ) , 0 ≤ p ≤ s {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.