Viscosity solutions and uniqueness for systems of inhomogeneous balance laws

This paper is concerned with 
the Cauchy problem 
 
 
$(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad 
u(0,x)=\overline{u}(x),$ 
 
 
for a nonlinear $2\times 2$ hyperbolic system of 
inhomogeneous balance laws 
in one space dimension. As usual, 
we assume that the system is strictly hyperbolic and that each 
characteristic field is either 
 linearly degenerate or genuinely nonlinear. 
 
 
 
Under suitable assumptions on $g$, we prove that there exists 
$T>0$ such that, for every $\overline{u}$ with sufficiently small 
total variation, the Cauchy problem ($*$) 
has a unique "viscosity solution", 
defined for $t\in [0,T]$, 
depending continuously on the initial data.

for a nonlinear n × n hyperbolic system of inhomogeneous balance laws in one space dimension. As usual, we assume that the system is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear (see [12]). Here F : R n → R n is a smooth function, and g: R × R × R n → R n is continuous w.r.t. t, satisfying for every t and every (x, u), (y, v), with k ∈ L 1 (R) and L g > 0 (see [9]). In [8] it was proved the existence of a semigroup S for the system of conservation laws defined on some domain D ⊂ L 1 R; R n containing all integrable functions with sufficiently small total variation, with the following properties. (i) There exists a constant L S such that, for all t ≥ 0 andū,v ∈ D, one has (ii) Ifū ∈ D is piecewise constant, then for t > 0 sufficiently small S tū coincides with the solution of (1.4) which is obtained by piecing together the standard self-similar solutions of the corresponding Riemann problems.
The main result of this paper is concerned with 2 × 2 systems of inhomogeneous balance laws. Theorem 1.1 Let F be a smooth map from a neighborhood of the origin U ⊂ R 2 into R 2 . Assume that the system (1.1) is strictly hyperbolic and every characteristic field is either linearly degenerate or genuinely nonlinear.
In section 6 it will be proved that a viscosity solution is a weak entropic solution of the Cauchy problem (1.1)-(1.2). In section 8 we outline the proof for large initial data, following [7].
From the proof of Theorem 1.1 and in particular from Proposition 4.4 we can make more explicit the dependence of T from the total variation of the initial data. More precisely, the following result holds. .
= P (t, 0)ū yields a "viscosity solution" in the sense of Definition 7.1 to the Cauchy problem (1.1)- (1.2). f) For every δ 1 , δ 2 We remark that the constantδ is related to the domain of definition of the semigroup S for the homogeneous problem (1.4) constructed in [6]. The last property in Theorem 1.2 ensures the possibility of prolonging viscosity solutions as long as their total variation remains bounded byδ/2.
A straigthforward consequence of Theorem 1.1 is the global (in time) existence of viscosity solutions if in (1.3) we replace k(x) with a function k(t, x) ∈ L 1 ([0, +∞[×R) (see for example [9]).
2. Preliminaries. Throughout this paper, the euclidean norm and inner product on R 2 are denoted by | · | and ·, · . U is an open convex subset of R 2 containing the origin, and the map F : U → R 2 is three times continuously differentiable. We denote by PC the family of all piecewise constant functions u ∈ L 1 (R, R 2 ), and T.V.{v; I} will denote the total variation of a function v: R → R 2 over an open interval I. By · and · ∞ we will denote the norms in L 1 and L ∞ respectively.
We assume that the system (1.1) is strictly hyperbolic. More precisely, we assume that each matrix A(u) .
It will be convenient to work with a set of Riemann coordinates v = (v 1 , v 2 ). We can assume that the origin in the u-coordinates corresponds to the origin in the v-coordinates, and that the map v → u(v) is a local diffeomorphism. We define Ω . = u −1 (U ). In these new variables, we can consider a family of right and left eigenvectors r i (v), l i (v), i = 1, 2, of the matrix A(u(v)) .
= DF (u(v)), depending smoothly on v, normalized according to Given a function ϕ defined on Ω, its directional derivative at the point v along the vector field r i is denoted by As usual, we assume that, for each i = 1, 2, the i-th characteristic field is either linearly degenerate, so that Now let ε > 0 be given. Following [6], for i = 1, 2, through each point v we construct a parametrized curve σ → ψ ε i (v, σ) which coincides with the rarefaction curve ψ + i through v for σ ≥ − √ ε and with the shock curve ψ − i for σ ≤ −2 √ ε. This is done by choosing a smooth function ϕ: −1], and by defining i.e. the strengths of waves in the approximate solution of the Riemann problem.
Let v: R → R 2 be a piecewise constant function with bounded support. Call x 1 < . . . < x N the points where v has a jump. Let us define, at each point x α , the quantities The total strength of wave in v is then defined as As usual, the sum ranges here over the set A of all couples of approaching waves. We recall that, when x α < x β , the two waves σ iα , σ jβ approach if either i = 2, j = 1, or else i = j, the i-th family is genuinely nonlinear and min{σ iα , σ jβ } < 0. Remark that, since the functions ψ ε i converge in the C 2 -norm to ψ i , then V ε and Q ε converge uniformly to the Glimm's functionals V and Q respectively.
In order to prove the existence of a SRS for (1.4), in [6] it was constructed a family (S ε ) of semigroups generating approximate solutions, in the sense of Definition 2.2. This construction is based on a modified wave-front tracking algorithm (see [2]).
Let v , v ∈ Ω, and assume that, for some for some integers h, k. Let us definê and consider the averaged speed Finally, interpolate between the two previous speeds by setting We are now in a position to give the following , satisfying the following conditions.
whereω is the state with coordinates ((2k , defined according to (2.1). We regard (i)-(ii) as a set of approximate Rankine-Hugoniot conditions. The ε-admissible wavefronts of the second family are defined in an entirely similar way.
is an ε-approximate solution if all of its lines of discontinuity are ε-admissible wavefronts.
For every ε > 0 let us define the domain In [6] it was proved that there existsδ > 0 such that the following proposition holds.
, for every t, s ≥ 0 andv,w ∈ D ε (δ); (iii) for everyv ∈ D ε (δ), the map t → S ε tv is an ε-approximate solution; (iv) the map t → V ε (S ε tv ) + Q ε (S ε tv ) is nonincreasing for everyv ∈ D ε (δ). The SRS S is obtained as the limit of (S ε ) for ε → 0. In order to prove property (ii) above, it was introduced a Riemann-type metric on D ε (δ), by defining a weighted length for a set of admissible paths. The weighted length of an elementary path γ is defined as

Definition 2.4 Let ]a, b[ be an open interval. An elementary path is a map
for some constant K > 0. In order to simplify notations, we dropped the dependence on ε.
If γ is a pseudopolygonal, we define its weighted length γ ε as the sum of the weighted lenghts of its elementary paths.
On the domain D ε (δ) we can define the Riemann-type metric The property (ii) in Proposition 2.3 was proved in [6] showing that the distance d ε is uniformly equivalent to the usual L 1 distance, and that S ε Since V ε → V and Q ε → Q uniformly, ifv ∈D(δ) thenv ∈ D ε (δ) for every ε > 0 small enough. For every t ≥ 0 and everyv ∈D it can be shown that there exists S tv . = lim ε→0+ S ε tv . Moreover S is a semigroup satisfying properties (i) and (ii) of Proposition 2.3, and property (iv) with V ε and Q ε replaced by V and Q. By property (ii), we can extend S ε and S respectively to [0, +∞[×cl D ε (δ) and [0, +∞[×clD(δ), where "cl" denotes the closure in the L 1 topology.
3. Construction of approximate solutions. In this section we will construct a sequence (P ν ) of operators, which will converge uniformly to an operator P satisfying the properties listed in Theorem 1.1.
For every fixed ν ∈ N, let us define an approximating function g ν of g, piecewise constant w.r.t. x, in the following way. Let It is easily seen that every g j is Lipschitz continuous w.r.t. v, with the same Lipschitz constant L g of g.
In order to define the approximate evolution operators for the nonhomogeneous problem, the idea is to evolve from τ = kε ν for a time interval of length ε ν with the homogeneous approximate semigroup, and then to make the correction v → v + ε ν g ν (τ + ε ν , ·, v), which gives the contribution of the nonhomogeneous term.
Let us considerv ∈ D εν (δ). For every t ∈ [0, ε ν [ and m ∈ N let us define and then, by induction on Clearly, if n ≥ k, and t ≥ nε ν then we have In the following Lemma we shall prove some basic properties of g ν .
Lemma 3.1 For every fixed t and everyv ∈ PC with bounded total variation, the composed map x → g ν (t, x,v(x)) has bounded total variation and These estimates gives concluding the proof.
4. Basic estimates. In the following, in order to simplify notations, we will omit the dependence on ε. Since the maps ψ ε i converge to ψ i in the C 2 norm, all the estimates will hold uniformly in ε. Letg h > 0, and let σ i , σ i , i = 1, 2, be respectively the strength of the waves generated by the Riemann problems with data ( =g(x ± , v ± ). With the notations of Section 2, we have that Since E i (w, w) = 0 for every w, it follows where O(1) denotes the Landau order symbol, i.e. a quantity which remains uniformly bounded (in absolute value) by some constant C, depending only on the system (1.1). Finally, from the Lipschitz continuity ofg w.r.t. v, we obtain Ifg is continuous at x, the last term at the right hand side of (4.1) vanishes and then Let γ: ]a, b[→ L 1 be an elementary path as in (2.2), and let us assume thatg is piecewise constant w.r.t. x, continuous at every point is an elementary path. Let us denote by σ iα the strength of the waves of the Riemann problems associated to ω α .
Let us denote byR iα , P iα and Q the quantities corresponding respectively toR iα , P iα and Q for γ .
Here A A denotes the symmetric difference between the two sets A and A . From (4.2) we obtain In order to estimate |Q − Q |, observe that In the same way one obtains .
Then, with the above notations, one has, for every Proof. If sgn σ iα = sgn σ iα , by Lemma 4.1 and the estimate If sgn σ iα = sgn σ iα , the same estimate follows by the definition of B and (4.7) Proof. If γ: Using (4.2) and (4.6), we can estimate the length of the difference of two elementary pathsγ .
Hence there exists a constant C > 0 such that and then, by definition of d ε , (4.7) holds.
Let us denote by T the set of all times t ∈ [0, T ] such that t = k2 −n for some k, n ∈ N, and let us define the set Let {t j ,v j } j be a dense subset of [0, T ] ×D(δ/2). By a diagonal argument, we can extract a subsequence of {P ν }, again denoted by {P ν }, such that for every τ ∈ T and every j such that t j ≥ τ , there exists P (t j , τ )v j . = lim ν→+∞ P ν (t j , τ )v j . If t j , t i ≥ τ , passing to the limit for ν → +∞ in (4.10) with (t,v) = (t j ,v j ) and (s,w) = (t i ,v i ) we obtain By continuity, P can be extended to a Lipschitz function defined on I ×D satisfying, Since the domain I ×D is compact in the product topology of R × R × L 1 (R, R 2 ), we have that P ν converges uniformly to P in this topology.
It is easy to see that P is a local evolution operator, that is 2) Indeed, if t, s, τ ∈ T , by the definition of P ν we have that for every ν large enough and for everyv ∈D(δ/2), P ν (t, τ )P ν (τ, s)v = P ν (t, s)v. Passing to the limit for ν → +∞, we obtain P (t, τ )P (τ, s)v = P (t, s)v. Since T is dense in [0, T ], by continuity we have that (5.2) holds.
Following the proof of Corollary 3 of [4], we have: 6. Weak entropic solutions. In this section we will prove that the trajectories of P are weak entropic solutions of (1.1).

Proof. We begin by showing that u(t) is a weak solution of (1.1), that is
. Without loss of generality we can assume thatū is piecewise constant. Let us define, for every ν ∈ N, the map u ν (t) .
Following the proof of Lemma 27 in [6], it is easy to see that Recalling that g ν converges uniformly to g on compact subsets of [0, T ] × R × R 2 , and that u ν → u in L 1 loc , it is easy to see that the sequence of functions {ϕ ν }, defined by converges for almost every t ∈ [0, T ] to x, u(t, x)) dx .
By the Lebesgue dominated convergence theorem we have that Hence, passing to the limit in (6.2) for ν → +∞, we obtain (6.1).
It remains to show that u(t) is an entropic solution. Let η be a smooth convex entropy with flux q. We have to show that With the same notations of the first part of the proof, we have that where in the last inequality we used the convexity of η. Reasoning as in (6.2), we can pass to the limit for ν → +∞ obtaining (6.3).

Viscosity solutions and uniqueness.
We give now the definition of viscosity solution for a system of balance laws. For the equivalent definition in the homogeneous case, see [4]. We will prove the equivalence between viscosity solutions and trajectories of P . This in turn will imply that P is independent of the subsequence (P ν ) constructed in section 5, and then it is the unique operator satisfying the properties listed in Theorem 1.1.
Consider now the Riemann problem associated to the initial condition u(0, x) = u − for x < 0, u(0, x) = u + for x > 0, where u ± . = lim x→ξ± u(τ, x), and let ω(t, x) be the standard self-similar solution. Let us define We can now give the following: Theorem 7.2 Letū ∈ D. Then u(t) .
As in the first part of the proof, there exists a constant C 1 > 0 such that Reasoning as in (7.11), we have that where J k is defined in (7.9). Moreover, Indeed, by the choice ofλ, the integrand is zero in the region |x − ξ| ≤λ(k + 1)ε ν , and, by property (iv) in Proposition 2.3, By Corollary 5.1 and the definition of P ν , recalling (7.19) and (7.20), one has From this last estimate we obtain Recalling that P ν converges uniformly to P , passing to the limit for ν → +∞, and by the arbitrariety of ε , we have that (7.4) holds.
We pass now to prove the opposite implication. Proof. Clearly, it suffices to show that u(T ) = P (T, 0)u(0) restricted to any interval of the form J . = [−R +λT, R −λT ], with R >λT arbitrarily large. Let R and δ 0 positive be given. We will prove that the inequality holds for t = T . Clearly, (7.22) is satisfied when t = 0. Let τ be the supremum of the times for which (7.22) holds. By continuity we have that (7.22) holds for t = τ . If τ < T , we show that (7.22) holds for τ +ε for someε > 0, reaching a contradiction. Applying Theorem 7.2 to the map ε → P (τ + ε, τ )u(τ ), it follows that, for every fixed ξ and for every ρ, ε > 0 sufficiently small, Let us define I j . =]ξ j − ρ j , ξ j + ρ j [, I j .
= I j \ {ξ j } and η, with ρ j , ε j > 0 such that and such that the estimates (7.3), (7.23) hold when ξ = ξ j , ρ = ρ j , ε ∈]0, ε j ]. Since the number of ξ j is finite, the function η can be chosen independent of ξ j . By possibly shrinking the values ρ j , we can assume that the I j are pairwise disjoint.
We now have that, given any semigroup P obtained by approximations as in Section 5, a continuous function u: [0, T ] → D is a viscosity solution if and only if u(t) = P (t, 0)u(0). Recalling (5.1) and (5.2), Theorem 1.1 is proved.
8. The case of large data. In [7] the authors consider the problem (1.4)-(1.2) whereū has bounded but possibly large total variation.
As a first result, they prove the existence of a Lipschitz semigroup whose domain contains all functions which are sufficiently close (in the norm of total variation) to a given Riemann data. It is assumed that the corresponding Riemann problem satisfies a linearized stability condition.
The approach is essentially the same of [6], but in this case three different metrics must be used, depending on the solution of the Riemann problem. Indeed this solution may contain two shock waves, or two rarefaction waves, or a shock and a rarefaction wave.
Another result obtained in [7] concerns the existence of a local solution for an arbitrary initial dataū with bounded variation, satisfying suitable stability and non-resonance conditions at every jump point. The uniqueness of this solution is proved within the class of viscosity solutions to (1.4)-(1.2) satisfying the additional condition (C) below.
We remark that the estimates given in sections 3 and 4 of the present paper are based on the Lipschitz continuity of the weigths R iα with respect to the wave strengths. It is not difficult to check that these estimates continue to hold for more general metrics of the form on the domain D considered in [7], containing all suitably small BV perturbations of a Riemann data. In (8.1), the coefficients c iα may depend on the relative position of x α with respect to the (0, 1 or 2) large shocks present in the solution, while d jβ iα may also depend on the relative position of x α and x β . The local existence and uniqueness results stated in Theorems 2 and 3 of [7] thus admit a straightforward extension to the nonhomogeneous case, proved by the same arguments used here in Theorem 1.1.