ON THE SHIFT DIFFERENTIABILITY OF THE FLOW GENERATED BY A HYPERBOLIC SYSTEM OF CONSERVATION LAWS. Stefano Bianchini S.I.S.S.A. (Trieste) May 1999

Transcription

1 ON THE SHIFT DIFFEENTIABILITY OF THE FLOW GENEATED BY A HYPEBOLIC SYSTEM OF CONSEVATION LAWS Stefano Bianchini S.I.S.S.A. (Trieste) May 999 Abstract. We consier the notion of shift tangent vector introuce in [6] for real value BV functions an introuce in [8] for vector value BV functions. Using a simple ecomposition of u BV in terms of its erivative, we exten the results of [8] to more general shift tangent vectors. This extension allows us to stuy the shift ifferentiability of the flow generate by a hyperbolic system of conservation laws. 99 Mathematics Subject Classification. 35L65. I thank Prof. Alberto Bressan for his useful remarks an suggestions. I also thank Marco Bertola for clarifying explanations on iemannian manifols. Typeset by AMS-TEX

2 . Introuction In this paper we aress th question of ifferentiability of the flow generate by a strictly hyperbolic system of conservation laws. The primary motivation for the introuction of shift ifferentials comes from the theory of hyperbolic conservation laws [5,6,3], in particular the evolution of first orer perturbation of initial ata. Other potential applications are variational problems an optimal control of solutions. It is well known that if S is L contractive semigroup generate by a scalar conservation, then in general the map u S t u, for fixe t, is not ifferentiable in the usual L ifferential structure. However, for any Lipschitz continuous map, one can introuce a new ifferential structure on the space BV, efining a tangent space T u at u BV: given a set Γ of continuous paths γ : [0, ] γ() BV, with γ(0) = u, consier the equivalence relation efine as γ γ if 0 γ (u) γ (u) L = 0. (.) The tangent space T u at u is by efinition the elements of the set Γ/. Every equivalence class can be regare as a first-orer tangent vector at the point u. The stanar choice is to consier a family T u of tangent vectors which can be put in a one-to-one corresponence with L (). More precisely, T u is efine as the family of all equivalence classes of the maps γ v (). = u + v, v L (). (.2) As it is shown in [6], this choice is not aequate for escribing a first-orer variation of the flow map S t : in fact it happens that u S t u oes not map T u = L into T St u = L. For example consier Burgers equation with the family of initial conitions u t + [ u 2 /2 ] = 0, (.3) x u (0, x) = x χ [0,] (x). (.4) By χ I (x) we enote here an in the following the characteristic function of the interval I. Choose for example u. = x χ [0,] (x). In this case u correspons to the path (.) with tangent vector v = x χ [0,] (x) L (). Assuming > 0, the corresponing solution of (.3)-(.4) is u (t, x) = x + t χ [0, +t] (x). (.5) Observe that, for t > 0, the map u (t, ) efine in (.5) is Lipschitz continuous but nowhere ifferentiable because the location x (t) = + t of the shock varies with. Therefore, the it 0 u + u, t > 0, is not well efine as an element of the space T St ū = L (). In [6] it is stuie a ifferent space T u of tangent vectors, u BV(), which can be put into a oneto-one corresponence with L (Du). Here Du enotes the (signe) aon measure corresponing to the istributional erivative of u. The basic iea is the following. In the special case where v is 2

3 continuously ifferentiable with compact support, to v it is associate the equivalence class of the map u, where u is implicitly efine as u (x + v(x)) = u(x), (.6) for all 0 sufficiently small. It is then shown that this corresponence can be uniquely extene to the whole space L (Du). Observe that in (.2) the graph of u is obtaine by lifting the graph of u vertically by v. On the other han, in (.6), the graph of u is shifte horizontally by v. This motivates the term shift-ifferential use in the sequel. A map which is ifferentiable w.r.t. the tangent vectors in T u is sai to be shift ifferentiable. The main result in [6] shows that the flow generate by a single conservation law u S t u is shift ifferentiable almost everywhere w.r.t. t, i.e. outsie a countable set {t k } k N. In [?] it is introuce a ifferent approach to first orer perturbations of a strictly convex scalar conservation law. The main result of [?] is that if the initial atum u 0 satisfies (u 0 ) x C, then the it w =. S t (u 0 + v) S t u 0 (.7) 0 exists in the weak sense of measures, for all v L L +, an the map v w is a linear an boune operator from L + to the space of boune aon measures on. This means that the operator u 0 S t u 0 is Gâteaux ifferentiable in u 0, in some weak sense. It can be shown that if u + v generates a shift tangent vector, then the measure w is equal to adu, where a L (Du) is the shift tangent vector generate by S t (u 0 + v). Of course the it (.) is much stronger than (.7). In [8] the construction of shift ifferentials is extene to the case of vector value function u. If u : n is a BV function, a shift tangent vector is efine in terms of: i) a ecomposition of u into n scalar components; ii) a n-tuple of functions (v,..., v n ) L (Du), etermining the rate at which each component of u is shifte. This is accomplishe by assigning a matrix value function A : M n n, where we enote with M n n the set of iagonalizable n n matrices with real eigenvalues. The eigenvectors of A correspon to the local scalar ecomposition of u, while the eigenvalues of A etermine the shift rate of the corresponent component of u. Denoting with, the scalar prouct in n, the assumption on A are: A) if l i, r i enote the left an right eigenvectors of A, normalize such that r i = an l j, r i = δ j i, then they are Borel measurable an uniformly boune; A2) the eigenvalues λ i belong to L (Du). Given A M n n satisfying A, A2, the main result of [8] is the construction of an equivalent class of paths γ : u L loc which etermine the shift tangent vector A. In this paper we consier a class of paths γ : u L loc, an show how these paths generate a space T u of tangent vectors at u BV. elying on the previous observations an the results of section 4, we will call the elements of T u shift tangent vectors. Before introucing the space T u, in section 2 we stuy a simple case: we consier a constant matrix A an a piecewise constant function u with a single jump. In this case the path u L loc (, n ) is efine consiering the solution of the linear equation u + A u x = 0. (.8) The stuy of this simple case leas to the introuction of a new istance on the space M n n such that (M n n, ) is a complete metric space. 3

4 In section 3, relying on a simple ecomposition of u in terms of its measure erivative, we efine the matrix value functions A which can generate shift tangent vectors at u. These vector value functions are calle amissible generators. Our efinition is much less restrictive than the one in [8]. It is obvious that two matrix value functions A an à efine the same path if, roughly speaking, their ifference acts on the vector space orthogonal to u: for example, if l n is a left eigenvector for the constant matrix A an Du(x) is orthogonal to l for all x, then the path generate by (.8) is inepenent from the value of the eigenvalue λ corresponing to l. It follows that, ifferently from the scalar case, a path u L loc oes not etermine uniquely the amissible generator, in general. However, we give two criteria to say whether two matrix value functions A an à efine the same shift tangent vector or not. In section 4 we show that our efinition coincies with the one given in [6] for the scalar case. In some sense our construction gives the most natural path, as the path γv(u) consiere in (.2) is the easiest choice in that case. Moreover we show that the matrix value functions consiere in [8] are amissible generators an the efinition of shift tangent vector given there coincies with ours. Finally in section 5 we aress the question of the application to hyperbolic systems of conservation laws. Extening [6], we introuce the shift ifferential of a map Φ : L loc L loc. oughly speaking, Φ is shift ifferentiable at u BV if Φ(u) BV an for all shift tangent vectors A T u, Φ maps a path generating A into a path generating a shift tangent vector B T Φ(u). In general the semigroup S t generate by an hyperbolic system of conservation laws is not efine on the whole L loc, but on an L close subset of BV. Since with our efinition there are shift tangent vectors generate only by paths γ() with Tot.Var(γ()) = +, 0 we nee to restrict the space T u, i.e. we nee to consier a subspace M(u) of T u. The shift ifferential of a map Φ is now efine using M(u) instea of the entire T u. Finally we give three examples of the applications to the Lipschitz continuous semigroup S t generate by a hyperbolic system of conservation laws. The first example consier a simple 2 2 Temple class system, i.e. a system whose rarefaction curves are straight lines. In this case we show that the shift ifferentiability of the map u S t u occurs only if we restrict the space T u to the shift tangent vectors that shift inepenently the two iemann invariants. We show also that this subset M(u) of T u is the biggest set M(u) such that the shift ifferentiability of the map u S t u occurs. The last two examples show that in general it is ifficult to etermine which subspace M(u) of T u shoul be consiere to prove the shift ifferentiability of the map u S t u, an this subspace coul be very small. 2. The iemann problem for linear systems We begin with the most elementary case: a iemann problem for a linear hyperbolic system with constant coefficients. Without any loss of generality, we consier a function u L loc (; n ) with a single jump in 0, i.e. u(x) =. vχ [0,+ ) (x), v n, an a constant matrix A M n n. We recall that with M n n M n n we enote the space of n n iagonalizable matrices with real eigenvalues: if with r i, l i n the left an right eigenvectors corresponing to the eigenvalues λ i, then r r n > 0, r i =, l j, r i = δ j i, i, j =,..., n, (2.) 4

5 where, is the scalar prouct in n, δ j i is the kronecher s elta an is a norm of n. For sake of efiniteness, we shall use (v,..., v n ) = v + + v n, but all the following results are vali for any equivalent norm in n. To efine in this simple case the shift tangent vector A at u, we consier the linear system { ut + Au x = 0 (2.2) u(0, x) = vχ [0,+ ) (x), whose solution is u(t, x) = l i, u(0, x λ i t) r i = i= χ [λi t,+ )(x) l i, v r i, (2.3) where, is the scalar prouct on n. We now introuce a notation for the functions obtaine by (2.3). This notation is the obvious generalization of efinition 2 in [8]. Definition 2.. If u L loc evaluate at time, i.e. i= an A Mn n are as above, we enote by A u the solution of (2.3) A u(x). = u(, x) = χ [λi,+ )(x) l i, v r i. (2.4) i= Note that this efinition coincies with efinition 6 of [8], since the matrix A is constant; in particular each component l i, u r i of u is shifte of the amount λ i. In the following sections we shall nee to estimate A u à u L, where A, à Mn n. If we enote by ũ(t, x) the solution of { ũt + Ãũ x = 0 (2.5) ũ(0, x) = vχ [0,+ ) (x), an explicit computation gives A u à u L = = u(, x) ũ(, x) x u(, x/) ũ(, x/) x = u() ũ() L, where we use the fact that u an ũ are self similar. The last formula, ivie by, can be use to efine a istance on the space of iagonalizable matrices: Definition 2.2. Given two matrices A, à Mn n an a vector v n, consier the two iemann problems efine in (2.2) an (2.5), an enote by u(t, x) an ũ(t, x) their respective solutions. We efine the function (A, Ã; v) as (A, Ã; v) =. u() ũ() L = u(, x) ũ(, x) x. (2.7) (2.6) Moreover we efine the istance (A, Ã) by (A, Ã). = sup (A, Ã; v). (2.8) v: v = In the following theorem we prove that is actually a istance in M n n. 5

6 Theorem 2.3. The function : M n n of iagonalizable matrices M n n M n n. The space (M n n efine in (2.8) is a istance on the space, ) is a complete metric space. Proof. It is obvious form (2.7) an (2.8) that (A, Ã) 0 an (A, Ã) = (Ã, A). Suppose now that (A, Ã) = 0. By efinition, the istance (A, Ã) can be efine as the L istance between the two solution of the iemann problem (2.2) an (2.5), respectively: thus to prove A = à is equivalent to say that the matrix A is uniquely etermine by the solution of the iemann problems (2.2), when v varies in n. Assume that v is equal to a right eigenvector of A, namely v = r l, with l n. Then the solution u of (2.2) is a single shock traveling with spee λ l, u(t, x) = r l χ [λl t,+ )(x), an since by hypotheses (A, Ã; r l) = 0, u must be a solution also of the equation (2.5): in fact u an ũ are self similar, an thus if they coincie at t =, they are equal for all t 0. Consequently (2.3) implies that Ãr l = λ l r l. (2.9) Since (2.9) hols for all l {,..., n} an à Mn n, it follows A = Ã. To prove (A, Ã) (A, Â) + (Â, Ã), for all A, Ã,  Mn n, note that for all v n we have (A, Ã; v) = u(t) ũ(t) L u(t) û(t) L + û(t) ũ(t) L = (A, Â; v) + (Â, Ã; v) (A, Â) + (Â, Ã), (2.0) where û is the solution of (2.2) with the matrix Â. Taking the supremum of the left han sie of (2.0), we conclue (A, Ã) (A, Â) + (Â, Ã). This conclues the proof that is a metric. complete metric space. For any v n an A, à Mn n we can write Av Ãv In fact, from (2.2) an (2.5) it follows u(t, x) ũ(t, x)x = t 0 We now prove that this istance makes M n n u(, x) ũ(, x) x = (A, Ã; v). (2.) Au(t, x) x Ãũ(t, x) xxt = t(av Ãv). (2.2) Now let A k M n n be a Cauchy sequence in (M n n, ), an efine u k (t) as the self similar solution of the iemann problem (2.2) with the matrix A k an initial atum vχ [0,+ ) (x). Since u k (t) u l (t) L t (A k, A l ), we have that, for any fixe t, u k (t) u (t) is a Cauchy sequence in L (), converging to a unique it u(t): u k (t, x) u(t, x) x = 0. (2.3) k + Note that (2.) implies that A k is a Cauchy sequence in M n n, an then there exists a matrix A such that k + Ak = A. (2.4) 6 a

7 If we write equation (2.2) with the matrix A k in weak form an we let k +, (2.3) an (2.4) imply for all φ C (, ) that u(t, x)φ t (t, x) + Au(t, x)φ x (t, x)xt + + vφ(0, x)x = 0, + (2.5) so that u is a weak solution of the system { ut + Au x = 0 u(0, x) = vχ [0,+ ) (x), (2.6) This conclues the proof, because it is well known that the system (2.6) has a solution for all v n if an only if A is iagonalizable (see [2]). emark 2.4. It is easy to see that the space M n n [ ] [ 0 0 is not a vector space: for example ] [ ] =, 0 an while the two matrices are in M n n, their sum is not. The same result can be prove for the prouct of two matrices. Note that M n n is star shape, an in fact satisfies also (αa, αã) = α (A, Ã), (2.7) for any α > 0, A, Ã Mn n. While M n n is a manifol, the istance oes not efine a iemannian structure: namely it can be shown that there are no metric tensors generating. A corollary of theorem 2.3 is Corollary 2.5. The istance is stronger that the usual operator norm : for any two matrices A, Ã Mn n A Ã = sup Av Ãv (A, Ã). (2.8) v n Proof. Formula (2.8) follows immeiately from (2.) an the efinition (2.8). The following remark will be important in the following section. emark 2.6. Consier two matrices A, Ã Mn n. It is clear that A Ã = 0 implies (A, Ã) = 0, but in general, fixe a vector v n, we coul have Av = Ãv an (A, Ã; v) > 0. Consier for example the vector v = (, ) an the following matrices [ ] [ ] [ ] [ ] [ ] /5 /5 A =, A = = /5 2/5 With easy computation one can verify that (A, A 2 ; v) = 44/5 while A v = A 2 v. 3. Shift tangent vectors: the vector case In this section we introuce a space of shift tangent vectors for a function u BV(; n ). These are functions in L loc (; n ) whose istributional erivative is a boune vector measure on. We efine f. = Du/ Du, the Lebesgue ecomposition of Du w.r.t. its total variation measure Du. Without any loss of generality, in the following we assume u right continuous. We now give the following efinition: 7

8 Definition 3.. Consier a matrix value function A : M n n. Given u BV(; n ), we say that A is a amissible generator of a shift tangent vector at u if (A(y), 0; f(y)) Du (y) < +, (3.) where (A(y), 0; f(y)) is efine in (2.7). We enote the class of amissible generators for u as Am(u). For all u BV, it is easy to prove the existence of the following it u( ). = u(x). x Since our efinition of shift tangent vector at u epens only on the erivative Du of u, we assume u( ) = 0. Given u BV, we can obviously write u(x) = u( ) + f(y)χ (,x] (y) Du (y) = u( ) + f(y)χ [y,+ ) (x) Du (y). (3.2) If A(y) is an amissible generator for u, then we consier for any y the solution w(t, x; y) of the iemann problem { wt + A(y)w x = 0 (3.3) w(0, x; y) = f(y)χ [y,+ ) (x), an we efine the path A u L loc as A u(x) =. w(, x; y) Du (y) = l i (y), f(y) χ [y,+ ) (x λ i (y))r i (y) Du (y) i= = l i (y), f(y) χ [y+λi (y),+ )(x)r i (y) Du (y). i= (3.4) If A is constant, (3.4) coincies with (2.4). With these notations we can rewrite conition (3.) as w(, x; y) w(0, x; y) xy < +. (3. ) We first show that the function A u is well efine as an element of L loc for all 0. Lemma 3.2. Given a function u L loc BV(; n ), the function A u efine in (3.4) is a path in L loc (; n ) such that Proof. Since we have 2 A u u L w(, x; y) w(0, x; y) x Du (y) = (A, 0; f(y)) Du (y). (3.5) 8 (A, 0; f(y)) Du (y) +,

9 the function w(, x; y) w(0, x; y) is in L ( Du x), an then (3.4) is in L loc, thus efine a.e.. Finally, if we change the orer of integration, we can write A u u L = A ( ) u(x) u(x) x = w(, x; y) w(0, x; y) Du (y) x The conclusion follows. w(, x; y) w(0, x; y) x Du (y) = 2 (A(y), 0; f(y)) Du (y). emark 3.3. In general the function A u nees not to be in BV(; n ) if n 2. In fact, consier the following example ( + ) u = (u, u 2 ) = 2 i χ [i,+ )(x), 0 BV(; 2 ), i= an the matrix value function efine as [ ] 2 i 0 A(i) = 2 2 i = [ ] [ 0 2 i 0 2 i 0 2 i The solution of each iemann problem (3.3) is (0, 0) x < i 2 i w(, x; i) = (, 2 i ) i 2 i x < i + 2 i (, 0) x i + 2 i ] [ ] 0 2 i. If we set (v, v 2 ) = v + v 2, an easy computation shows that (A(i), 0; f(i)) = 2 i + 2 < 3, an then A is an amissible generator at u. The weak erivative of (3.7) has measure norm + 2 i+, an thus, for all 0 <, A u is not in BV(; 2 ). The scalar case is a particular situation, since for all φ Cc () we have A u(x)φ (x)x = w(, x; y)φ (x)x Du (y) 2 = f(y)χ [y+λi (y),+ )(x)φ (x) = 2 = f(y)φ(y + λ i (y)) Du (y) φ C 0Tot.Var(u). This implies that A u BV(). In general we can prove the following theorem: + f(y) φ (x)x Du (y) y+λ i (y) Theorem 3.4. Assume that the left eigenvectors l i (y) of A(y) Am(u), satisfying (2.), are uniformly boune by a constant M. Then the function A u is in BV for all 0. Proof. We recall that by (2.) the right eigenvectors are normalize. If φ is a Cc () function an we enote with φ its erivative, we have A u(x)φ (x)x = w(, x; y)φ (x)x Du (y) 2 n ( ) + = l i (y), f(y) r i (y) φ(x)x Du (y) y+λ i (y) = i= n i= l i (y), f(y) r i (y)φ(y + λ i ) Du (y) 2nM φ C 0Tot.Var(u). 9 (3.6) (3.7)

10 This conclues the proof, since the above formula is a efinition of the space BV. emark 3.5. Definition 3. is the largest class of amissible generators at u BV. One can restrict the class of amissible generators, for example assuming the matrix value function A Am(u) to be uniformly iagonalizable in the sense of theorem 3.4, or using the istance in (3.) instea of (A, 0; f(y)). However all the following results are inepenent of the set of amissible generators, an then we will use efinition (3.). Before giving the efinition of shift tangent vector, we prove the following theorem. Theorem 3.6. If A an à are two amissible generators at u, then inf 0 sup 0 A u à u L A u à u L A(y)f(y) Ã(y)f(y) Du (y), (A(y), Ã(y); f(y)) Du (y). (3.8) emark 3.7. Note that by formula (2.) we have Note also that (A(y), Ã(y); f(y)) Du (y) A(y)f(y) Ã(y)f(y) Du (y) (A(y), 0; f(y)) Du (y) + if A, à are amissible. To prove the theorem we nee a preinary lemma. (A(y), Ã(y); f(y)) Du (y). (Ã(y), 0; f(y)) Du (y) < +, Lemma 3.8. The function (A u u)/ converges to A(y)f(y) Du (y) in the weak sense of measures. Proof. If φ is a C 0 c () function, we have A u(x) u(x) φ(x)x = = 2 n i= ( ) w(, x; y) w(0, x; y) φ(x)x Du (y) ( ) l i y (y), f(y) r i (y) φ(x)x Du (y). y+λ i (y) (3.9) Since φ is uniformly continuous in, we have that for every y. Note that (2.7) yiels y φ(x)x = λ i (y)φ(y), 0 y+λ i (y) ( l i y (y), f(y) r i (y) φ(x)x) i= y+λ i (y) φ C0(A(y), 0; f(y)), 0

11 an then we can use Lebesgue s ominate convergence theorem in (3.9) when 0: A u(x) u(x) 0 φ(x)x = = l i (y), f(y) r i (y)φ(y) Du (y) i= A(y)f(y)φ(y) Du (y). (3.0) Proof of Theorem 3.6. Since A(y)f(y) Du (y) is the weak it of the sequence (A u u)/, the linearity of the integrals gives A u(x) à u(x) 0 for all φ C c (), an a stanar argument yiels ( φ(x)x = A(y)f(y) Ã(y)f(y) ) φ(y) Du (y), A(y)f(y) Ã(y)f(y) Du (y) inf 0 The other inequality follows from a argument similar to (3.6): since A u(x) à u(x) x = 2 u (x) ũ (x) x. ( ) w(, x; y) w(, x; y) Du (y) x w(, x; y) w(, x; y) x Du (y) = (A(y), Ã(y); f(y)) Du (y), it follows sup 0 A u à u L (A(y), Ã(y); f(y)) Du (y). In the following example, we prove that the its in (3.8) may in general be ifferent. Example 3.9. Consier the vector function u = (u, u 2 ) efine as u (x) = u 2 (x) = 0, 0 x < 0 n 2 i min{c c i i, } 3 i + n 3 n < x < c i 3 i n, c i = 0,, 2 i= i= x > i= an the fixe matrix A efine as [ ] 0 A = = 0 [ ] [ ] [ ] The function u is essentially Vitali s function multiplie by the unitary vector (, 0). We now stuy the integral I() = u (x) u(x) x,

12 where u is the solution the system (2.2) evaluate at time, namely u (x) = ( ) 2 u (x ) + ( ) 2 u (x + ). Note that by the efinition of u, we have I(3 n ) = I(3 ). An explicit computation of I(3 ) gives +3 I(3 ) = 3 u 3 (x) u(x) x With the same construction we can compute Note that I(2/3 n ) = I(2/9) = 9 2 ( u 3 2 (x) x = ) = ( ) = A(y)f(y) Du (y) = Du(y) = 35 24, (A(y), 0; f(y)) Du (y) = 2Du(y) = 2 3 2, as theorem 3.6 requires. We now introuce the main efinitions of this paper. Definition 3.0. Fix u BV(; n ) an consier the partition of Am(u) efine by the following equivalence relation : if A, à Am(u), then A à if sup 0 A u(x) à u(x) x = 0. (3.) We efine the shift tangent vectors A at u as the elements A of the set Am(u)/. emark 3.. In general, given u BV, it is ifficult to give a precise characterization of the equivalence class A. For example, if u Cc (; n ) an A, à Mn n are two constant matrices, the application of Lebesgue ominate convergence theorem shows that A u(x) 0 à u = Au x (y) Ãu x(y) y, an thus in this case A à if Au x(y) = Ãu x(y) for all y. emark 2.6 shows instea that if u has a jump, in general A an à belong to ifferent equivalence classes. However, using (3.8), we can say that if (A(y), Ã(y); f(y)) Du (y) = 0, then A Ã. Conversely if A(y)f(y) Ã(y)f(y) Du (y) > 0, then A u an à u generate two ifferent shift tangent vectors. 2

13 Definition 3.2. Consier a path u L (; n ), efine in some interval [0, ]. We say that the path u generates the shift tangent vector A Am(u)/ if for some amissible generator A A we have 0 u A u L = 0. (3.2) emark 3.3. It is easy to prove using (3.) that the above efinition oes not epen on the choice of the representative A A. Moreover, if it exists, the shift tangent vector A, not the amissible generator, is uniquely etermine by the curve u. 4. Equivalence of the other efinitions In this section we show that efinition 3.2 coincies with the efinition of shift tangent vector given in [6] for the scalar case an in [8] in the vector case. If u is a function in BV(; ), then instea of a matrix value function A Am(u) we have a function a L (Du): in fact in this case conition (3.) reuces to a(y) Du (y) < +, (3. ) since (a, 0; v) = a v. Moreover it is easy to verify that the set Am(u)/ coincies with L (Du), because in the scalar case the its (3.8) coincie an then a ã if an only if a = ã almost everywhere w.r.t. the measure Du. We recall that in [6] the shift tangent vector a L (Du) is efine by the equivalence class w.r.t. the L norm of the path a u, where (a u)(x + a (x)) = u(x), (4.) an a is a Lipschitz continuous function such that a (y) a(y) Du (y) = 0, 0 sup Lip(a ) <, 0 a = 0. (4.2) 0 In [6] it is shown that this class is uniquely etermine by the function a L (Du), so that the efinition is consistent. The following proposition shows that our efinition of shift tangent vectors coincies with the one above. Proposition 4.. Given a L (Du), let a be a Lipschitz continuous function such that (4.2) hol, an let a u be the L loc function efine in (4.). Then 0 a u a u L = 0, (4.3) where a u is efine the path in L loc efine in (3.4). Proof. It is simple to verify that the function a u can be written as a u = + f(y)χ [y,+ ) (x a (y)) Du (y), 3

14 an, using (4.2), with easy computation we obtain a u a u x f(y)χ [y,+ ) (x a (y)) f(y)χ [y,+ ) (x a(y))) Du (y)x 2 = a (y) a(y) Du (y) = a a L (Du), an then (4.3) is verifie. We now consier the vector case. Given u BV(; n ), we recall that in [8] a shift tangent vector is etermine by a matrix value function A : M n n having the r i, l i as right an left eigenvectors, an the λ i as eigenvalues such that the functions r i, l i : n are Borel measurable an uniformly boune an λ i L ( Du ). In this case the path A u is efine by consiering a matrix value function A : M n n such that i) for each given (0, ], its right an left eigenfunctions ri, li, : n an its eigenvalues λ i : are Lipschitz continuous an boune. Moreover, they remain constant outsie a compact interval K ; ii) for each i =,..., n one has { ri (x) r i (x) + l i,(x) l i (x) + λ i (x) λ i (x) } µ u (x) = 0, (4.4) 0 max 0 j ( 0 /4 Lip(ri ) + Lip(l i, ) + Lip(λ i ) + λ ) i = 0, (4.5) ( { λ j } ri (x) r i (x) + l i, (x) l i (x) ) Du (x) = 0, (4.6) ri, sup l i, <, (4.7) i, Next, for any integer k Z, let Pk be the points P k. = k 2 3/4, an let Ik an J k be the open intervals centere at Pk an with lengths 2 3/4 an 3/4, respectively. Finally set A u(x). = l i, (Pk ), u(x λ i (Pk )) ri (Pk ), for x Ik. (4.8) i= The last result of this section shows that, following efinition 3.4, A u generates the shift tangent vector A etermine by A A. Proposition 4.2. If A is efine as above, then A is amissible. Moreover, if we consier the path A u efine in (3.4), we have 0 A u A u L = 0. (4.9) It follows that A u generates A, where A is the shift tangent vector etermine by A A. Proof. By the assumptions on A, there exists a constant M such that sup i l i M. (4.0) 4

15 The matrix value function A : M n n is then uniformly strictly hyperbolic on, an for every y the istance (A, 0) is boune by (A(y), 0) = sup (A(y), 0; v) = sup v n, v n, v = v = λ i (y) sup l i (y), v M i= v n, v = λ i (y) l i (y), v r i (y) i= λ i (y). i= (4.) Since λ i belongs to L ( Du ), (4.3) implies that A is amissible. Now we prove (4.9). For any y we evaluate the L istance between the solution of the iemann problem (2.2) with matrix A(y) an the one with matrix A (y): if v n has norm v =, we have (A(y), A (y); v) = χ [λi (y)t,+ )(x) l i (y), v r i (y) i= χ [λ i (y)t,+ )(x) l i, (y), v ri (y) i= λ i (y) λ i (y) l i (y), v i= M i= i= L ( ) + χ [λ i (y)t,+ )(x) l i (y), v r i (y) l i, (y), v ri (y) λ i (y) λ i (y) + max{ λ j } j L ( M ri (y) r i (y) + l i, (y) l i (y) ). We note in fact that the integran function is ifferent from 0 only for x [ max j i= { λ j, max{ λ j ]. j (4.2) It follows from (4.2) an (4.4) (4.7) that (A(y), A (y)) Du (y) = 0. (4.3) 0 If M is a boun also for l i, in (4.7), a similar argument gives (A (Pk ), A (y); v) M y Pk n Lip(λ i ) i= + y Pk max{ λ j } j ( ) MLip(ri ) + Lip(l i, ), i= (4.4) If y is in Jk, (4.4) an (4.4) (4.7) yiel (A (Pk ), A (y)) Du (y) = 0. (4.5) 0 k J k 5

16 The conclusion follows easily from (4.3) an (4.5), noting that A u A u L = k I k A u(x) A u(x) x (A(y), A (y)) Du (y) + k J k (A (y), A(y)) Du (y). 5. Application to systems of conservation laws In this section we stuy the shift ifferentiability of the flow generate by a hyperbolic system of conservation laws. Following [6], we consier an operator Φ : L loc L loc. Definition 5.. We say that Φ is shift ifferentiable at u BV(; n ) along the shift tangent vector A Am(u)/ if Φ(u) BV(; n ) an there exists B Am(Φ(u))/ such that 0 B Φ(u) Φ(A u) L = 0, (5.) for some A A, B B. Moreover if there exists a map Λ : Am(u)/ Am(Φ(u))/ such that for all A Am(u)/ the it (5.2) hols with B = ΛA, then Φ is shift ifferentiable at u. In other wors Φ is shift ifferentiable at u if it is shift ifferentiable along each irection A Am(u)/. Note that Am(u)/ is not a vector space, except in the scalar case. emark 5.2. This efinition points out a major ifficulty when one is ealing with the semigroup S generate by a system of conservation laws: in fact the omain D of efinition of S is an L close subset of BV, while in general the efinition of shift tangent vector uses a path not in BV. Moreover using the same example given in remark 3.3, one can show that there exists shift tangent vectors such that if u is any generating path, then 0 Tot.Var(u ) = +. The above remark motivates the following efinition: Definition 5.3. Let Φ : L loc Dom(Φ) BV such that (5.) hols for u, v Dom(Φ). Given a subset M(u) Am(u)/, we say that Φ is M(u) shift ifferentiable at u if there exists a map Λ : Am(u)/ M(u) Am(Φ(u))/ such that for all A M(u) the it (5.) hols with B = ΛA. In the rest of this section we consier the application of these efinitions to the flow generate by a hyperbolic system of conservation laws { vt + f(v) x = 0 v(0, x) = u(x) (5.2) where v n an f : n n smooth. We recall that, uner various assumption of f (see [,4,0]), (5.2) generates a unique Lipschitz continuous semigroup S t : [0, + ) D D such that D contains all the functions u with sufficiently small total variation with values in some compact set K; 6

17 there exists a constant L such that S t u S s w L L( t s + u w L ); each trajectory S t u is a weak entropic solution of the Cauchy problem (5.2) with initial atum u; if u is a piecewise constant function, then, for small t, S t u coincies with the function obtaine piecing together the solutions of the corresponing iemann problems. We are intereste in the shift ifferentiability of the map u S t u, for fixe t. The first example shows the application of the above efinitions to the semigroup S t generate by a simple 2 2 Temple class system. We recall that in the case of Temple class systems the omain D of the semigroup can be extene to vector value functions with arbitrary large total variation (see [2,3]). Example 5.4. Consier the following Temple class system: ( ) u u t + = 0 ( u + v ) x v v t + = 0 u + v If we choose the two iemann coorinates w = u + v an w 2 = v/u, the system becomes { (w ) t = 0 (w 2 ) t + w (w 2 ) x = 0 We assume that the initial ata w,0, w 2,0 BV(; ) assume values in some square [a, b] [a, b] 2, with a, b > 0. It is well known that the omain [a, b] [a, b] is invariant for the semigroup S t generate by (5.3) ([2,3]). If we consier the characteristics lines of the secon equation, i.e. the integral lines of the ODE { ẋ = x w,0 (x) = t = w,0 (z)z, (5.5) x(0) = y y(t,x) then the solution of (5.4) can be obtaine by the metho of characteristics: The semigroup S t is then x (5.3) (5.4) w (t, x) = w,0 (x), w 2 (t, x) = w 2,0 (y(t, x)). (5.6) S t : BV(; [a, b] [a, b]) [0, + ) BV(; [a, b] [a, b]) (w,0, w 2,0 ) (w (t, x), w 2 (t, x)). = (w,0 (x), w 2,0 (y(t, x))) (5.7) The class of shift tangent vectors M(w,0, w 2,0 ) Am(w,0, w 2,0 )/ that we consier are those generate by shifting inepenently the two components w an w 2 : more precisely, M(w,0, w 2,0 ) is the class of shift tangent vectors generate by the set of amissible generators { A M 2 2 : A = [ λ 0 0 λ 2 ], λ L (Dw ), λ 2 L (Dw 2 ) }. (5.8) It is clear that in this case M(w,0, w 2,0 ) is homeomorphic to the space L (Dw,0 ) L (Dw 2,0 ), an then it is a vector space. The following result is essentially theorem 3 of [6]. 7

18 Theorem 5.5. Suppose that u S t u is Y shift ifferentiable at u, where Y. = Then u S t u is M shift ifferentiable. { [ ] } A M 2 2 λ 0 : A =, λ 0 λ i Cc (), i =, 2. (5.9) 2 Proof. The proof is an easy extension of the proof of theorem 3 in [6]. If is less than min{/lip(λ ), /Lip(λ 2 )}, then λ i w i coincies with the function implicitly efine by (λ i w i )(x + λ i (x)) = w i (x). If we write (5.5) for the shifte case, we have { ẋ = λ w,0 (x) x(0) = y = t = x (t,y) y λ w,0 (z)z, (5.5 ) an then the solution is We can write S t (λ w,0, λ 2 w 2,0 )(x (t, y)) = (λ w,0 (x (t, y)), λ 2 w 2,0 (y)). (5.0) S t w 2 (x) = w 2,0 (y(t, x)) = λ 2 w 2,0 (y(t, x) + λ 2 (y(t, x))) = S t (λ 2 w 2,0 ) (x ( t, y(t, x) + λ 2 (y(t, x)) )) ( = S t (λ 2 w 2,0 ) x + x( t, y(t, x) + λ 2 (y(t, x)) ) ) x. (5.) Now we show that x (t, y(x) + λ 2 (y(t, x))) x x w,0 (x) y(t,x) λ (z)dw,0 (z) + w,0(y(t, x)) λ 2 (y(t, x)), (5.2) w,0 (x) if Dw,0 is continuous in x, y(t, x). Inee (5.6 ) is ifferentiable at = 0 if an only if w,0 is continuous in x, y(t, x), an its erivative coincies with (5.2). It is clear that the convergence in (5.2) is in L (Dw 2 (t)) if an only if the atomic part of Dw an Dw 2 are isjoint at t = 0 an t. In fact the left han sie of (5.2) is uniformly boune for all > 0, an (5.2) hols outsie the countable set of jumps of w,0. Using Lebesgue s ominate convergence theorem the conclusion follows. The map Λ : M(w,0, w 2,0 ) Am(S t u) is then { v (x) v 2 (x) x w (x) y(t,x) v (x) λ (z)dw (z) + w (y(t, x)) λ 2 (y(t, x)) w (x) (5.3) Using theorem 5.5 we can say that, given w = (w, w 2 ) such that the atomic parts of Dw an Dw 2 are isjoint, the map w S t w efine in (5.8) is M(w, w 2 ) shift ifferentiable for all t such 8

19 that the atomic parts of D(S t w) an D(S t w) 2 are isjoint. In this simple case it is easy to prove that Λ is a boune linear operator from M(w,0, w 2,0 ) to M(w (t), w 2 (t)). Note that one cannot expect the map w S t w to be M(w) shift ifferentiable for a bigger set M(w). Consier in fact figure 5.a. In this case the initial conition u is one single shock along the first iemann invariant. The solution S t u is then a single travelling shock. In this case Am(u) is all M n n. Consier a matrix A M n n, an assume that the function A u has two jumps for > 0: this implies that the A is not in the class M(w) consiere in (5.8). The solution S t (A u) is in figure 5.b: in fact it is easy to prove that the new jumps are splitte in waves of the two families. It is then clear that for all t > 0 the map u S t u is not shift ifferentiable along A: in fact no matrix A M n n can generate the wave patterns of figure 5.b if t > 0. t t λ λ 2 x Figure 5.a Figure 5.2b The following examples show that in general the map u S t u is not M(u) shift ifferentiable even if M(u) contains very simple tangent vectors. The conclusion is that, given a hyperbolic system (5.2), it is very ifficult to etermine the class M(u) in which the shift ifferentiability of the map u S t u occurs. Moreover the set M(u) coul be extremely small: in example 5.6, M(u) is only biimensional. Example 5.6. Consier the wave fronts configuration of figure 5.2a: three shocks of ifferent families interact at the same point, but only two survive to the interaction. It is clear that, by stanar Gm s interaction estimates, one can construct this configuration if the vanishing shock has size of the orer of the prouct of the sizes of the other two. For simplicity we assume that the two surviving shocks o not change in position or size: this can be achieve consiering a systems in which two equations are inepenent of the others, for example the 3 3 system v t ( u x ) = 0 u t + = 0 v x z t + ( 3 + z)z x + v x = 0 Consier the configuration of figure 2.b, in which the vanishing shock has been shifte of an amount : the amissible generator A is then 0 x x A(y) = I x > x where I is the ientity matrix, an x is a point between the vanishing shock an the other two. The shaowe region represents the centere rarefaction wave generate by the interaction of the two surviving shocks. 9 x

20 t t x Figure 2.a Figure 2.b Assume without any loss of generality that the interaction occurs at t =. Since we are ealing with systems in conservation form, we have for all t 0 that S t (A u)(x) S t u(x)x = A u(x) u(x)x. (5.4) From the picture it is clear that the path S t (A u) oes not generate any shift tangent vector, if t > : in fact, since the two surviving shocks o not move, the shift tangent vector must be 0, but using the conservation property (5.4) one has S t (A u)(x) S t u(x) x S t (A u)(x) S t u(x)x = σ, (5.5) where σ is the size of the shifte shock. With the same analysis, it is clear that the u S t u is shift ifferentiable along A Am(u) if an only if the three shocks are shifte in such a way that they will meet at the same point. If we enote with s i the spee of the i th shock, the shift rates of the three shocks are λ i = ξ s i ξ 2, (ξ, ξ 2 ) 2, i =, 2, 3. The vector (ξ, ξ 2 ) gives the irection in which the interaction point is shifte. In example 5.6 the instability w.r.t. the shift ifferentiability can be relate to the structural instability of the point of interaction of the three shocks (see [7]). We recall that the solution S t u of (5.4) is sai to be structurally stable at the point (τ, ξ) + if, on the half plane t < 0, the function ũ(t, x) =. u(τ + ηt, ξ + ηx) (5.6) η 0 + satisfies one of the following conitions: ũ is a constant function; ũ contains one incoming shock an no other wave; ũ contains two incoming shocks an no other wave. In [7] it is shown that it (5.6) exists an it is a self similar weak solution of u t + f(u) x = 0. In the case consiere in example 5.6, the self similar solution contains 3 incoming shocks, an thus it is structurally unstable. In the next example we show that the same results can be prove for solution structurally stable in each point of the half plane x

21 Example 5.7. Consier the (triangular) system { ut + ( 3 + u)u x = v x v t + vv x = 0 In this case the secon equation is ecouple. This implies that the 2 waves move with a spee inepenent on the value of u: then it is possible to construct the wave pattern of figure 5.3a, for initial ata chosen carefully an small enough: a Lipschitz continuous solution of the first equation, a shock an a centere rarefaction wave of the secon equation interact in such a way that only the shock will survive. t t x Figure 5.3a Figure 5.3b In figure 5.3b it is represente the solution corresponing to A u, where A is the same as in example 5.6. Using the same analysis of example 5.6, one can show that the path S t (A u) oes not generate any shift tangent vector: in fact also in this case the surviving shock oes not move, an then the shift tangent vector shoul be zero. A computation similar to (5.5) gives S t (A u)(x) S t u(x) x Du (x). Note that in this case the solution is structurally stable in the whole plane +. eferences [] F. Ancona, A. Marson, Well-poseness of general n n hyperbolic systems of conservation laws, in preparation. [2] P. Baiti, A. Bressan, The semigroup generate by a Temple class system with large ata, Diff. Integ. Equat. 0 (997), [3] S. Bianchini, The semigroup generate by a Temple class system with non convex flux function, preprint S.I.S.S.A., ef. 07/98/M. [4] F. Bouchut, F. James, Differentiability with respect to initial ata for a scalar conservation law, in Hyperbolic problems: theory, numerics, applications, Fey, Jeltsch eitors, Birkhäuser, ISNM 29 (999), 3 8. [5] A. Bressan,, The unique it of the Gm scheme, Arch. ational Mech. Anal. 30 (995), [6] A. Bressan,. M. Colombo, The semigroup generate by 2 2 conservation laws, Arch. ational Mech. Anal. 33 (995), 75. [7] A. Bressan, G. Guerra, Shift ifferentiability of the flow generate by a conservation law, Discr. Cont. Dynam. Syst 3 (997), [8] A. Bressan, P. G. LeFloch, Structural stability an regularity of entropy solution to hyperbolic systems of conservation laws, In. Univ. Math J. (to appear). [9] A. Bressan, M. Lewicka, Shift ifferentials of maps in BV spaces, in Nonlinear theory of generalize functions, Grossel, Hörmann, Kunzinger, Oberguggenberger eitors, Chapman an Hall/CC, esearch Notes in Mathematics 40 (997), [0] A. Bressan, A. Marson, A variational calculus for shock solutions of system of conservation laws, Comm. Part. Diff. Eq. 20 (995), x

22 [] A. Bressan, T. P. Liu, T. Yang, L stability estimates for n n conservation laws, Arch. ational Mech. Anal. (to appear). [2] L.C. Evans,.F. Gariepy Measure theory an fine properties of functions (992), CC press, Boca aton. [3] D. Serre, Systèmes e lois e conservation, Dierot Eiteur, Paris, 996. [4] J. Smoller Shock waves an reaction iffusion equations (982), Springer Verlag, New York. Stefano Bianchini, S.I.S.S.A. (I.S.A.S.), Via Beirut 2/4, 3403 Trieste, Italy aress: bianchin@sissa.it 22