Long time existence of a class of contact discontinuities for second order hyperbolic balance laws

The study of long time existence of classical smooth solutions for second order quasilinear wave equations has received much attention (see e.g. [1] and the references given there). Results were also obtained for continuous semilinear waves with gradient jumps on a characteristic hypersurface in [2] and for C 1 quasilinear waves with second order derivatives jumps on a characteristic hypersurface in [3] , [4] . In this paper we show how the methods and results of [2] can be extended to a class of continuous weak solutions, with gradient jumps on a characteristic hypersurface, for some second order quasilinear balance laws.


Introduction
The study of long time existence of classical smooth solutions for second order quasilinear wave equations has received much attention (see e.g. [1]and the references given there).Results were also obtained for continuous semilinear waves with gradient jumps on a characteristic hypersurface in [2] and for C 1 quasilinear waves with second order derivatives jumps on a characteristic hypersurface in [3] , [4] .In this paper we show how the methods and results of [2] can be extended to a class of continuous weak solutions, with gradient jumps on a characteristic hypersurface, for some second order quasilinear balance laws.

Statement of the results
Let Ω ⊂ R N be a bounded open set lying locally on one side of its boundary ∂Ω, where ∂Ω is a C ∞ manifold of dimension (N − 1).We shall consider the balance law if t > 0, x ∈ R N , where x = (x 1 , . . ., x N ) is the space variable, t (sometimes called where We shall assume that and also that the following "null condition" holds : if p = tr (p 0 , . . ., p N ) satisfies |p| ≤ R and p 2 0 − 1≤j≤N p 2 j = 0, then 0≤i,j≤N f ij (p)p i p j = 0. (2.3) We shall consider weak solutions to (2.1) which satisfy the initial condition The initial data will have to satisfy appropriate compatibility conditions.To describe those conditions, let ψ ∈ C ∞ (R N , R) be such that ψ < 0 in Ω, ψ > 0 in R N \ Ω, dψ = 0 at each point of ∂Ω, and let ϕ be the (at least local near is tangent to Σ.We shall restrict ourselves to solutions to (2.1), (2.4) which satisfy the conditions lim for all m ∈ N and all a ∈ ∂Ω.Of course (2.5) can be expressed in terms of z 0 , z 1 only.Put Σ(t) = {(s, x) ∈ Σ, s = t}, D(t) = 0<s<t ({s} × Ω(s)), where {s} × Ω(s) is the bounded connected component of ({s} Σ(s).We have the following local existence result.
The solution z described in Thm 2.1 is a contact discontinuity.To obtain long time existence results, we assume that Ω is convex and the total curvature of ∂Ω in the normal direction is nonvanishing(so Then Σ is global in t > 0 (cf. [2]).We also introduce the following smallness assumptions.We assume that z 0 , z 1 depend on a small parameter > 0, and that for some ε 0 > 0 and all α ∈ N N , one can find Denote by T ε the supremum of all T > 0 such that Thm 2.1 holds.Then we have the following long time existence result.
To prove Thm 2.1, it is enough to show that (3.
But this last relation is easily seen to hold if we make use of the relations 3), and of (3.5).Taking (3.7) into account, we finally conclude that ) be such that z = u and such that z vanishes at some point of ∂Ω.It is easily seen that z satisfies (2.1) in D(T ), (2.4) on {0} × Ω and (3.1).

Proof of Theorem 2.2
When each F i is identically constant, Thm 2.2 has been proved in [2] .We are going to use the same method as in [2] in order to prove estimates which will enable us to obtain Thm 2.2 by a continuation method.If h(t, x) is a function of t, x and . We have the following energy estimate (where X, Λ ij are as before).
Proposition 4.1 One can find δ, C > 0 such that the following holds.
where dσ is the canonical hypersurface measure on S(T ).
Proof of Prop 4.1.One writes Lw • ∂ t w as the sum of a divergence and a quadratic form in w , and integrates over D(T ).This is done as in the proof of Prop 5.1 of [2] (see also Prop 3.4 of [4] ).We may omit the details.
Denote by Γ 1 , . . ., Γ n the vector fields where where z (s, x) = {∂ α z(s, x), |α| = 2}, Using the calculus properties of the derivatives Γ α (cf. [6]), we find that ,Ω(s) ≤ r; (4.3) [λ] means sup{ν ∈ Z, ν ≤ λ} and r is small.To estimate J 1 + J 2 in (4.2), we may use the following result.ds ≤ ε 0 hold, then Admitting Prop 4.2 for a moment, and using it to estimate J 1 + J 2 , we obtain from (4.2), (4.3), if ε is small : ,Ω(s) ≤ r for 0 ≤ s ≤ t, and k ≥ 1.Now, as proved in Prop B.1 of [2] , we have the following variation on an inequality of [7] one can find C > 0 such that for all U ∈ C ∞ (D(T )) and all (t, x) ∈ D(T ) : ],Ω(s) and applying the Gronwall inequality to (4.4), we deduce that ,Ω(s) ≤ r.So finally we obtain that ds ≤ εk,r with εk,r small, and sup ds ≤ εk,r , and ε ≤ εk,r (with εk,r small), as a simple argument using (4.5) and (4.6) shows.Since (4.6) holds, it follows from the results of [5] (and from well known results for the classical Cauchy problem) that we may continue z up to t = T + η, for some η > 0 (as a solution to (2.1) in D(T + η) satisfying (3.1) on S(T + η)), provided that ε and ε T 0 (1 + s) − N −1 2 ds are small.A standard reasoning then gives the lower bounds for T ε stated in Theorem 2.2, and Theorem 2.2 is proved.So it remains to prove Prop 4.2.To prove Prop 4.2 we may proceed as in the proof of Prop 4.2 of [2] (in which f ij ≡ 0 for all i, j).Denote by M 1 , . . ., M l the vector fields Λ ij , 1 ≤ i < j ≤ N .One first proves by induction that one can find ε 0 > 0, and C βkα > 0 for any β ∈ N l , k ∈ N, α ∈ N N +1 , with ε 0 and C βkα independent of T , such that on S(T ), if 0 < ε ≤ ε 0 , β ∈ N l , k ∈ N, α ∈ N N +1 .Then estimates involving Γ α can be deduced (see [2] ).In [2] , the jump of ∂ t z across S(T ) satisfies a differential equation along the integral curves of X; in the present situation, it satisfies a first order quasilinear partial differential equation on S(T ).Indeed, put again u j = ∂ j z, 0 ≤ j ≤ N .In D(T ) we have f jk (u)∂ j u k + f (u), (4.8) .Estimates of H are given in [2] .From (4.10) it is not hard to deduce that (4.7) is true if β = k = α = 0.The general case of (4.7) follows by obvious adaptations of the reasonings of [2] .This completes the proof of Prop 4.2.Hence the proof of Theorem 2.2 is also complete.