Generalized solution of a mixed problem for linear hyperbolic system

In the first part of this article, we will prove an existence-uniqueness result for generalized solutions of a mixed problem for linear hyperbolic system in the Colombeau algebra. In the second part, we apply this result to a wave propagation problem in a discontinuous environment.


Introduction
In 1982, Colombeau introduced an algebra G of generalized functions to deal with the multiplication problem for distributions,see Colombeau [1,2]. This algebra G is a differential algebra which contains the space D of distributions. Furthermore, nonlinear operations more general than the multiplication make sense in the algebra G. Therefore the algebra G is a very convenient one to find and study solutions of nonlinear differential equations with singular data and coefficients. Consider the mixed problem for the linear hyperbolic system in two variables v ik (t) U k (0, t) + H i (t) i = 1, . . . , r t ≥ 0 + Compatibility conditions (1) where Λ, F and V are (n × n) matrices whose terms are discontinuous functions. The matrix Λ is real and diagonal such that In the case where Λ ∈ L ∞ R 2 + and F ∈ W −1,∞ loc R 2 + , multiplicative products of distributions appear in system (1), and so there is no general way of giving a meaning to system (1) in the sense of distribution. This hyperbolic system even when it is in the form of a system of conservation laws does not admit any solutions distributions in general see [3]. Our approach is to study (1) in Colombeau's algebra [1,2], and under some hypotheses on Λ, F , ν and H, the system (1) admits an unique solution in G R 2 + . This result completes work already made in the global case by M. Oberguggenberger [4]. The second part of this article, we will apply this result to the wave propagation problem in a discontinuous environment. the following system c R and c L are real constants, u 0 and v 0 are continuous almost everywhere. For this problem one can find a classical solution on {0 ≤ x < x 0 : t ≥ 0} and {x > x 0 : t ≥ 0}, so imposing a transmission condition in x = x 0 : the continuity of u and v, one will have a classical solution on {x ≥ 0 , t ≥ 0}. Further if (u 0 , v 0 ) are generalized functions, one can show that the problem (2) has a unique solution (U, V ) ∈ G R 2 + × G R 2 + , without having us need of the passage conditions, in the same way one shows that this solution admits an associated distribution that is equal to the classical solution by adjusting.

Existence and uniqueness
We recall some definitions from the theory of generalized functions which we need in the sequel. We define the algebra G (R m ) as follows Obviously E [R m ] with point wise multiplication is an algebra but C ∞ (R m ) is not a subalgebra. Then given ϕ ∈ A 1 (R m ) and ε ∈ ]0 , 1[, we define a function ϕ ε by An element of E [R m ] is called "moderate" if for every compact subset K of R m and every differential operator D = ∂ k1 x1 , . . . , ∂ km xm there is N ∈ N such that the following holds if for every compact subset K of R m and every differential operator D, there is N ∈ N such that : In what follows, the elements of G R 2 will be written with capital letters and their representatives in E M R 2 with small letters. Furthermore we use the following simplified notations : In our work we need a subset of E M R 2 + that contains elements u satisfying the following properties : Definition 1 A generalized function U ∈ G R 2 + admitting a representative u with the property (a) (respectively (b)) is called globally bounded (respectively locally logarithmic growth). Theorem 1 Let F , Λ and A be n × n matrices with coefficients in G R 2 + , suppose that: there exists r as :

Definition 2 the system (1) satisfies the compatibility conditions in
. . , n) are globally bounded, ∂ x Λ i and F i are locally logarithmic growth, so for an initial data U 0 in G (R + ), V i an element in G (R + ) globally bounded and H i in G (R + ), then the problem 1 has an unique solution in G R 2 + .
Proof : The proof of the theorem is an adaptation to the demonstration of the theorem 1.2 in [4], therefore one is going to give the big lines rightly. Let λ a representative of Λ in G R + + such that with λ i satisfies the property (a) and ∂ x λ i satisfies the property (b).
Let f and a are any representatives of F and A in G R 2 + with f satisfies (b). v, h and u 0 are any representatives of V , H and U 0 in G (R + ) with v satisfies (a). so Let's consider the following problem if we denote γ ε i the corresponding characteristic curve to λ ε i then the problem I ε admits an unique solution u ε , u ε i ∈ C ∞ R 2 + given by where t 0 is such that the curve γ i cuts the axis (0t) at a point P i (0, t 0 ). u ε i is C ∞ function, so it remains to show therefore that u ε i is moderate growth. from assumptions, we have Let K 0 be a compact in R + , we draw the straight line with a slope −M , the determination domain K T of the solution u ε i does not depend on ε.
Proof : for i = 1, . . . , r, and from the integral equation that verified by u ε i we have and the proof is completed by applying the Gronwall's lemma to the function For i = r + 1, . . . , n it is the same way with t 0 = 0, v = 0, h = 0. the next of the proof of theorem 1, we have ∃N 1 ∈ N such that : ∀φ ∈ A N1 (R + ) therefore according to the lemma, we have for the other derivatives, differentiating the system (I ε ) for example with regard to x, one gets a system similar to the first. And because ∂ x Λ is locally logarithmic growth one gets the same estimation as before, . . . , then one has u ε i ∈ E M (R 2 + ) i = 1, . . . , n either the existence of the solution for the problem (1) is in G(R 2 + ).

Uniqueness
Let U , V two solutions in G(R 2 + ) of the problem (I ε ), with the same initial data and the same boundary values. One must show that so u ε is a representative of U and G(R 2 + ) and if v ε is a representative of V in G(R 2 + ) then u ε − v ε ∈ N (R 2 + ) see [2]. indeed : u ε − v ε verifies the same problem that previously and therefore the demonstration is the same. Then one has Remark 1 To get the solution in the case where Λ ∈ L ∞ (R 2 + ), F ∈ W −1,∞ (R 2 + ), one uses the following result. see [4, proposition 2] Proposition 1 a) Let ω ∈ W −1,∞ loc (R 2 + ) then there exist U ∈ G(R 2 ) such that: U is associated to ω and U is locally logarithmic growth. b) let ω ∈ L ∞ (R 2 ) then there exist U ∈ G(R 2 ) such that: U is associated to ω and U is globally bounded, and ∂ α U is locally logarithmic growth. α = (α 1 , α 2 ) such that |α| = α 1 + α 2 = 1 Remark 2 For g ∈ L ∞ (R + ) one can find G ∈ G(R + ) such that: and there exist a representative g ε of G such that g ε is nil at the neighborhood of 0 for all ε.

Application
Consider the problem ( 2 ) For the initials data u 0 , v 0 continuous almost everywhere, and nil at neighborhood of 0. the problem (2) admits a classic solution for 0 < x < x 0 : t ≥ 0 and x > x 0 : t ≥ 0 and while imposing a passage condition on the x 0 (continuity of u and v at the point x 0 ) then one will have a solution on u(x, t) = u 0 (γ 1 (x, t, 0)) on (I) v(0, t) on (II) so one designates by Γ the characteristic curve comes from of (0, 0) the part (I) designates the set of (x, t) ∈ R 2 + below Γ. and the part (II) the set the points (x, t) over Γ (see the figure (2)). γ 1 the connected curve characteristic corresponding to c. γ 2 the connected curve characteristic corresponding to −c.  Proof c ∈ L ∞ (R + ), from the proposition (1) there exists C ∈ G(R + ) such that C ≈ c c is globally bounded and ∂ x C is locally logarithmic growth. And so, from the theorem 1, there exists an unique solution U , V in G(R 2 + ) of the problem (2).
To show that U ≈ u we suppose that (x, t) belongs to the region limited by the broken characteristic curve Γ comes from the origin and the axis (ox) which we note (region I).
If (x, t) is over of this curve, the demonstration is identical but with reflection (region II) and for (x, t) ∈ Γ (the characteristic curve comes from the origin) this set is negligible.