Shift-Differentiability of the Flow Generated by a Conservation Law

The paper introduces a notion of "shift-differentials" for maps with values in the space BV. These differentials describe first order variations of a given function $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition.


-Introduction
In the analysis of variational problems related to conservation laws 4], a major source of di culties is the fact that the ow map: u(0; ) 7 !u(t; ) is usually not di erentiable w.r.t. the linear structure of L 1 .As a simple example, consider Burgers' equation u t + u 2 =2 x = 0 (1:1) with the family of initial conditions u (0; x) = x 0;1] (x): (1:2) By I we denote here the characteristic function of the interval I. Assuming > 0, the corresponding solution of (1.1)-(1.2) is u (t; x) = x 1 + t 0; p 1+ t] (x): (1:3) Observe that the map 7 !u (0; ) describes a smooth curve in L 1 , namely a segment.However, for t > 0, the map 7 !u (t; ) is Lipschitz continuous but nowhere di erentiable because the location x (t) = p 1 + (1 + )t of the shock varies with .
To cope with this situation, in 3] a generalized notion of di erential was studied, for maps 7 !u taking values within a class of piecewise Lipschitz functions.The ow generated by a hyperbolic system of conservation laws was proved to be di erentiable in this generalized sense, as long as the solutions remain piecewise Lipschitz continuous.
Aim of the present paper is to introduce a new di erential structure on the space BV of integrable functions with bounded variation, and show that the ow generated by a scalar conservation law is generically di erentiable w.r.t.this structure.We conjecture that a similar result also holds for n n strictly hyperbolic systems.
Let a function u 2 BV be given.In order to de ne a \tangent space" at u, one can follow a procedure which is now standard in di erential geometry.On the family of continuous maps : 0; ] 7 !L 1 , with (0) = u and > 0 (possibly depending on ), consider the equivalence relation 0 i lim !0 1 ( ) ? 0 ( ) L 1 = 0: (1:4) Every equivalence class could be regarded as a \ rst-order tangent vector" at the point u.The set b T u of all these equivalence classes, however, is extremely large, and cannot be adequately described.Therefore, one usually works with a particular subset T u b T u of tangent vectors which admit a suitably nice representative.
The standard choice at this stage is to consider a family T u of tangent vectors which can be put in a one-to-one correspondence with L 1 (IR).More precisely, T u is de ned as the family of all equivalence classes of the maps 7 !v ( ) : = u + v; v 2 L 1 (IR): (1:5) In this paper we study a di erent space T u of tangent vectors, which can be put into a one-to-one correspondence with L 1 (Du).Here Du denotes the (signed) Radon measure corresponding to the distributional derivative of u 2 BV.When v 2 C 1 c L 1 (Du), to v we associate the equivalence class of the map 7 !u , where u is implicitly de ned as u ?
x + v(x) = u(x) (1:6) for all 0 su ciently small.We then show that this correspondence can be uniquely extended to the whole space L 1 (Du).Observe that in (1.5) the graph of u is obtained by lifting the graph of u vertically by v. On the other hand, in (1.6), the graph of u is shifted horizontally by v.
This motivates the term \shift-di erential" used in the sequel.Sections 2 and 3 are concerned with the de nition and the basic properties of shift di erentials.
In particular, we study the relation between the existence of a shift di erential for a map 7 !u and its di erentiability in the usual sense.In Section 4 we consider the semigroup S generated by the scalar conservation law w t + f(w) x = 0: (1:7) Denoting by S t u = w(t; ) the solution of (1.7) with initial condition w(0; x) = u(x), we show that (generically) the ow map u 7 !S t u is shift di erentiable.This result provided the initial motivation for a study of shift di erentials.We believe that our approach can also be useful in the analysis of variational problems in the class of BV functions, in particular those related to conservation laws.A rst example of \shift tangent vectors", in connection with piecewise constant approximate solutions of a 2 2 system of conservation laws, can be found in 2].

-Shift Tangent Vectors
Throughout the following, our basic function space is the normed space X : Here L 1 (IR) refers to the standard Lebesgue measure.The elements of X are thus equivalence classes of functions.For sake of de niteness, we shall always deal with left continuous representatives.By Lip(IR) we denote the space of Lipschitz continuous functions.The Lipschitz constant and the L 1 norm of v are denoted by Lip(v) and kvk 1 , respectively.
Given u 2 X, for every v 2 L 1 (Du) we shall construct a rst-order variation 7 !u of u, consistent with the de nition (1.6) in the smooth case.A preliminary de nition is needed.
De nition 1.Let u 2 X and v 2 L 1 (Du).Consider a family of functions v 2 L 1 (Du) \ Lip(IR), with 2 0; ].We say that v hat-converges to v, and write v ?!v, provided that (2:1) Observe that the last two conditions in (2.1) essentially depend on the parametrization of the family fv g.The next lemma shows that, for every v 2 L 1 (Du), it is always possible to construct a family of functions v hat-converging to v.
Applying the above lemma with r( ) = p , from (2.2) it follows Lip( v ) p and k v k 1 p .Therefore, v ?!v.
We now introduce a convenient notation for certain functions, obtained by shifting horizontally the graph of u.
De nition 2. If u 2 X and Lip( v ) < 1, we denote by v ?u the function implicitly de ned by (v ?u) ?
With the above notation, we can now introduce the main de nition in this paper.
De nition 3. Fix u 2 X and consider a path 7 !u 2 L 1 (IR), de ned on some interval 2 0; ], with u 0 = u.We say that the path u generates the shift tangent vector v 2 L 1 (Du) if, for some functions v ?!v, one has lim !0 1 u ?v ?u L 1 (IR) = 0: (2:4) Roughly speaking, this means that in rst approximation the functions u are obtained by shifting the graph of u horizontally by the amount v.
We will show that the above de nition does not depend on the choice of the approximating functions v .Moreover, if it exists, the vector v 2 L 1 (Du) is uniquely determined by the curve 7 !u .A few preliminary lemmas are needed.Lemma 2. For all u 2 X and 2 Lip(IR) \ L 1 (IR) one has Z IR u(x) 0 (x)dx = ?Z IR (x)Du(x): (2:5) Indeed, when 2 C 1 c (IR), (2.5) follows simply by the de nition of the measure Du.In the general case, (2.5) is proved by an approximation argument, using the fact that u is not only in BV(IR) but also in L 1 (IR).
Remark 2. In the same setting of De nition 3, let (2.4) hold for some family of functions v ?!v.Consider any other family ṽ ?!v.Then by Theorem 1 it follows This shows that the de nition of shift tangent vector does not depend on the choice of v in (2.4).
Corollary 1. Assume that the curve 7 !u ( with u 0 = u) generates the shift tangent vector v 2 L 1 (Du).Then for all ' 2 C 0 c (IR) one has lim !0 1 Z IR u (y) ?u(y) '(y)dy = ?Z IR v(x)'(x)Du(x): (2:16) Proof.Take v hat-converging to v and set ṽ 0 so that ṽ ?u = u for all .Using (2.13) we obtain Denote by the absolutely continuous measure having density (u ?u)= w.r.t.Lebesgue measure.The above corollary states that, if u generates the shift tangent vector v, then the measures converge weakly to the measure having density ?v w.r.t. the measure Du.
On the other hand, according to the standard de nition, a function w 2 L 1 (IR) is the di erential of the curve 7 !u at = 0 if lim !0 u ?u ?w L 1 (IR) = 0: (2:17) The following theorem describes the relations between the shift di erentiability of a curve and its di erentiability in the usual sense.By wdx we denote here the measure having density w w.r.t.Lebesgue measure and by ?vDu the measure having density ?v w.r.t. the measure Du.
Theorem 2. Consider a map 7 !u 2 X, with u 0 = u, generating the shift tangent vector v 2 L 1 (Du).Then this map is di erentiable at = 0 in the usual sense if and only if vDu is absolutely continuous w.r.t.Lebesgue measure.In this case we have vDu = ?wdxwith w de ned by (2:17).
We conclude this section by proving one more lemma, for later use.

-Shift Di erentials
Throughout the following, we consider a locally Lipschitz continuous operator mapping the function space X into itself.
De nition 4. We say that is shift di erentiable at the point u 2 X along the direction v 2 L 1 (Du) if there exists w 2 L 1 (D (u)) such that lim !0 1 w ?(u) ?(v ?u) L 1 (IR) = 0 for some v ?!v, w ?!w.In this case, we call w the shift derivative of at the point u along v and write w = $ r v (u).
Remark 4. The Lipschitz continuity of and Remark 2 imply that, if is shift di erentiable at u along the direction v, then (3.1) holds for all ṽ ; w hat-converging to v and $ r v (u) respectively.
Moreover, Remark 3 shows that, if a shift derivative exists, it is necessarily unique.
De nition 5. We say that is shift-di erentiable at u if there exists a continuous linear map : L 1 (Du) !L 1 (D (u)) such that the following holds.For all v 2 L 1 (Du) there exist v ?!v and w ?!v such that lim !0 1 w ?(u) ?(v ?u) L 1 (IR) = 0: In this case we say that is the shift di erential of at u and write = $ r (u).
In other words is shift di erentiable at u if it is shift di erentiable along each direction v 2 L 1 (Du) and the map v 7 !$ r v (u) is linear.
Remark 5.By Theorem 1, the shift di erential of is a linear continuous map whose norm is bounded by the local Lipschitz constant of .
The next theorem shows that, in order to prove the shift di erentiability of at a point u, it su ces to check that it holds along a dense set of directions.
Theorem 3. Let : X 7 !X be locally Lipschitz continuous.If is shift di erentible at u 2 X along all the directions v 2 Y , where Y is a dence subspace of L 1 (Du), and if the map : Y !L 1 (D (u)) that associates v to the shift derivative $ r v (u) is linear, then is shift di erentiable in u and $ r (u) = e where e is the continuous extension of to the space L 1 (Du).
Proof.Theorem 1 and the local Lipschitz continuity of imply that is continuous w.r.t. the norm of L 1 (Du), hence we can extend by continuity to the whole space L 1 (Du).We call e this extension.Now consider any v 2 L 1 (Du) and choose a sequence v k 2 Y converging to v in L 1 (Du).

-Application to Conservation Laws
Throughout the following, by S : X 0; +1) !X we denote the semigroup generated by the scalar conservation law (1.7).We assume that f 2 C 2 (IR) with f 00 (s) c > 0 for all s 2 IR.By de nition, (t; x) 7 !(S t u)(x) is the unique entropic solution of the Cauchy problem w t + f(w) x = 0 w(0; ) = u 2 X: It is well known that S is a contractive semigroup in the L 1 (IR) norm 5,7,9].An explicit representation of the function w(t; ) = S t u is provided by the theory of generalized characteristics 6] and by the integral formula of Lax 8].Fix any time t > 0. For u 2 X, x 2 IR, consider the right and left limits != lim y!x (S t u)(y), and de ne (u; x) : = x ?f 0 (! )t: Otherwise stated, the points (u; x) are the intersections with the x-axis of the maximal and the minimal backward characteristic from the point (t; x), respectively.Here we do not explicitly indicate the dependence on t, because the time t > 0 will be kept xed throughout the sequel.The maps x 7 !(u; x) are nondecreasing 8].If x 0 < x 00 one has ?(u; x 0 ) + (u; x 0 ) ? (u; x 00 ) + (u; x 00 ): If S t u is continuous at a point x, then in (4.2) !+ = !?and we simply write (u; x) : = + (u; x) = ?(u; x).For u 2 X and y 2 IR, consider the function F y u ( ; x) = t f 0 ( ) ? f( ) + Z x?f 0 ( )t y u( )d : (4:4) For any y; y 0 , it is clear that the functions F y u and F y 0 u di er only by a constant.Therefore, for a xed x, the set of values where F y u is minimized does not depend on y.This set of values will be For simplicity, we shall write F u ( ; x) instead of F ?1 u ( ; x).The following results were proved in 8].
a) At time t, the left and right limits of the solution of (4.1) at a point x are given by (S t u)(x?) = max S u (x); (S t u)(x+) = min S u (x): b) S t u is continuous at x if and only if the set S u (x) contains a single point.
We now state the main theorem of the paper.
Theorem 4. Assume that, at a given time t > 0, the entropic solution of (4:1) does not contain interacting shocks, nor centers of a compression wave.In other words, assume that for every x 2 IR the set S u (x) contains at most two points.Then the map S t is shift di erentiable at the point u.
The proof relies on a sequence of preliminary lemmas.First, we show that the multifunction (u; x) 7 !S u (x) is upper semicontinuous 1].In the following, the distance of the point x to the compact set K is written d(x; K).
Lemma 5. Let 7 !u 2 X (u 0 = u) be a continuous map de ned on some interval 0; 0 ], with ku k 1 R for some R > 0 and all .Let A be an open set, and K be a compact set such that S u (x) A 8x 2 K: ( Then there exists 2 (0; 0 ] and > 0 such that S u (x) A whenever 2 0; ]; d(x; K) : (4:7) Proof.If the conclusion does not hold, then there exists sequences x j , j and !j 2 S u j (x j ) such that d(x j ; K) !0, j !0 but !j = 2 A for all j .Since the !j are uniformly bounded and A is S u (x) = f! ?; !+ g consisting of exactly two points.Then, for every " > 0 there exists > 0 such that ?S t u (x) 2 ?!??"; !?+ " ?!+ ?"; !+ + " 8x 2 x ?; x + ] (4:11) for all small enough.

-Proof of Theorem 4.
Let u 2 X and t > 0 be given, according to the assumptions of Theorem 4. Because of Theorem 3, it su ces to prove the existence of a dense subspace Y L 1 (Du) and a linear operator : Y !L 1 ?D(S t u) such that, for every path !u (u 0 = u) which generates a shift tangent vector v 2 Y , the corresponding path !S t u generates the shift tangent vector v.A careful choice of the space Y will considerably simplify our computations.?S t u (x) ?u (x) dx "; (5:9) where 7 !u and 7 !u are any paths generating the shift tangent vectors v and v, respectively.
One easily checks that ṽ 2 Y , ?ṽ (x) = ( v)(x) if x = 2 x; x + 0 ] and x 2 B 2 (ṽ).If w ?! ṽ we can thus apply Lemma 10 and Theorem 1 and obtain the existence of some 2 (0; 0 ) such that lim sup !0 ": with w linear on each of the intervals x?M ; x] and x; x+M ].If ;M is de ned as in (4.17), calling J the interval enclosed by the points x and x + v 0 , we now have S t (ṽ ?u)(x) ?w ?S t u(x) We can now complete the proof of Theorem 4. Let v 2 Y and v ?!v as in (5.6).Since the supports of the functions v are uniformly bounded, we can choose a constant R large enough so that S t (v ?u) = S t u outside the interval ?R; R].Moreover since v also has compact support, it is not restrictive to assume that w ?S t u = S t u outside ?R; R].Theorem 4 will thus be proved if we show that L : = lim sup !0 1 Z R ?R ?S t (v ?u) (y) ??w ? (S t u) (y) dy = 0: (5:14) Let " > 0 be given.By Lemmas 10, 11 and 12, for each x 2 ?R; R] there exists x > 0 such that lim sup !0 1 Z x+ x x? x ?S t (v ?u) (y) ??w ?(S t u) (y) dy 0 if x 2 B 1 B 2 (v), " if x 2 B 3 (v).
Covering the compact interval ?R; R] with nitely many neighborhoods, say x i ?i ; x i + i ], i = 1; : : :; N, we obtain 1 Z R ?R ?S t (ṽ ?u) (x) ?(w ?S t u)(x) dx N X i=1 1 Z x i + i x i ?i ?S t (ṽ ?u) (x) ?(w ?S t u)(x) dx: (5:15) Taking the lim sup of (5.15) as !0, one nds L X x2B 3 (v) " n"; where n is the number of points where v has a jump.Observing that n is independent of " while " > 0 is arbitrary, this yields (5.14).
Corollary 2. Let u 2 X be given.Then the ow map S t generated by the conservation law (1:6) is shift di erentiable at u for all except countably many times t > 0.
Proof.It su ces to show that, for any > 0, the set of times t > at which the assumptions of Theorem 4 do not hold is at most countable.For each rational numbers p, call x p ( ), the generalized characteristic for the solution of (4.1), with x p ( ) = p.It is well known that this characteristic is uniquely de ned for t .For p; q rationals, call t pq the rst time at which the characteristics through p and q meet, i.e.
t pq : = min t; x p (t) = x q (t) : Set t pq : = 1 if the two characteristics remain disjoint for all t > .Now suppose that the assumptions in Theorem 4 do not hold at some time t > .Then, for some x 2 IR, there exist three distinct characteristics through the point (t; x), of the form x(s) = x + i (s ?t) i = 1; 2; 3; s 2 0; t]; with 1 > 2 > 3 .Choose rational numbers p; q such that x ? 1 (t ? ) < p < x ? 2 (t ? ) < q < x ? 3 (t ?): We then have t = t p;q .Since the set ft pq g is countable, the corollary is proved.
hold.Then the curve 7 !ugenerates a shift tangent vector v 2 L 1 (Du) if and only if the measure wdx is absolutely continuous w.r.t. the measure Du, and the equality wdx = ?vDuholds.Proof.1. Assume that the curve 7 !u generates the shift tangent vector v and that (2.17) Indeed, (2.21) is clear if w is continuous with compact support.The general case where w 2 L 1 (IR) is proved by a standard approximation argument.
Since the two intervals around ! ?; !+ are disjoint, and since S t u cannot have upward jumps, for any 2 0; ] the corresponding solution S t u will have a downward jump at some point x , with x0 = x, jx ?xj , and such that If x 2 B 3 (v), then for every " > 0 there exists > 0 such that o : Lemma 11.