Initial data identification in conservation laws and Hamilton–Jacobi equations

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Abstract

In the scalar 1D case, conservation laws and Hamilton–Jacobi equations are deeply related. For both, in the case of a uniformly convex flux/Hamiltonian, we characterize those profiles that can be attained as solutions at a given positive time corresponding to at least one initial datum. Then, for each of the two equations, we precisely identify all those initial data yielding a solution that coincide with a given profile at that positive time. Various topological and geometrical properties of the set of these initial data are then proved.

Résumé

Dans le cadre scalaire 1d, les lois de conservations scalaires et les équations d'Hamilton-Jacobi sont fortement reliées. Pour chacune de ces classes et dans le cas d'un flux/hamiltonien convexe on caractérise les profils pouvant tre atteints au temps T depuis au moins une donnée initiale via les semigroupes de solutions. Dans un second temps, on caractérise précisément toutes les données initiales produisant une solution coincidant avec le profil donné au temps T. On déduit finalement de cette caractérisation un ensemble de propriétés topologiques/géométriques de l'ensemble des données initiales correspondant un profil donné.

Introduction

Under suitable conditions on the flux f:RR and on the initial datum, solutions to a scalar conservation law in 1 space dimension, namely totu+xf(u)=0, are known to be obtained through u=xU from solutions to the Hamilton–Jacobi equationtU+f(xU)=0. A peculiar feature of these equations is their irreversibility. In particular, in the case of (1.1), inexorable shock formations cause an unavoidable loss of information, so that different initial data may well evolve into the same profile. Usual identification techniques, often based on linearizations or fixed point arguments, have no chances to be effective when dealing with (1.1) or (1.2).

Below, we provide a full characterization of the set of the initial data for (1.1), respectively (1.2), that evolve into a given profile. Geometric and topological properties of this set are also obtained. To this aim, a refinement of the results in [1], see also [2], [3], on the relation between (1.1) and (1.2) had to be obtained and is here presented (see Lemma 4.3).

For any suitable initial datum uo, we denote by (t,x)StCLuo(x) the weak entropy solution to (1.1). Symmetrically, we denote by (t,x)StHJUo(x) the viscosity solution to (1.2). Below, we consider the case of a uniformly convex C2 flux/Hamiltonian f and we obtain complete characterizations of both setsITCL(w):={uoL(R;R):STCLuo=w}and ITHJ(W):={UoW1,(R;R):STHJUo=W} and we use the notation IT whenever we refer to both sets in (1.3).

First, we identify those profiles such that the corresponding set IT is non empty. Here, we consider in detail the case of the conservation law (1.1). A key role is played by the decay of rarefaction waves, a phenomenon typically described through Oleinik decay estimatesfor a.e. x1,x2R with x1<x2f(w(x2))f(w(x1))x2x1T, see (2.4), which are known to be a necessary condition for a profile w to be attained as solution to (1.1) at time T since the classical work [4]. This result was recently improved in [5], extended to systems of conservation laws in [6] and of balance laws in [7], see also the reference texts [8, Chapter 6, Ex. 5] and [9, Theorem 11.2.1]. For related problems dealing with the reachable set of (1.1), also in the case of the initial – boundary value problem, we refer to [10], [11], [12] and [13]. Below, we prove that Oleinik decay estimates are also sufficient to ensure that IT is not empty. This proof is constructive, in the sense that an initial datum in IT is explicitly constructed, see Theorem 3.1.

Once IT is ensured to be non empty, in its characterization as well as in establishing its properties a key role is played by two sets, say Xi and Xii, whose precise definitions are in (2.5), that partition R (up to a negligible set), see Fig. 1. For x varying in the former one, Xi, the value attained at x by any initial datum in IT is essentially uniquely determined: Xi evolves in the subset of R where the assigned profile is regular. On the contrary, for x varying in the latter one, Xii, the value attained at x by any initial datum in IT is subject to rather loose constraints: Xii evolves in the singular part of the assigned profile. Here, singular points are understood as points of jumps for the solutions to (1.1) and angular points for the solutions to (1.2).

Technically, in the above procedure, Dafermos' generalized characteristics play an essential role. They are a usual tool in proving Oleinik estimates and are here used to identify the two sets Xi and Xii. On the other hand, their use is considerably eased by the strict convexity of the flux.

As a byproduct of this characterization, we precisely identify the profiles yielding an IT containing at most a unique element. Roughly, they correspond to the continuous profiles in the case of (1.1) and to the C1 ones in the case of (1.2). A further consequence, for instance, is that a discontinuity present at t=0+ in a solution to (1.1) may not disappear at any finite time.

Moreover, coherently with the finite propagation speed typical of (1.1) and (1.2), the values attained at x by any initial datum in IT on each of the different connected components of Xi and Xii are entirely independent from each other. The complete characterization of IT then results from the gluing of the characterizations obtained on each of the connected components of Xi and Xii.

Instrumental in these proofs is the ability to go back and forth between solutions to (1.1) and solutions to (1.2). To this aim, we needed to complete the results in [1] that deal with the connection from (1.2) to (1.1). Indeed, Proposition 2.3 details how to pass from solutions to (1.1) to solutions to (1.2).

On the basis of the obtained characterizations, several properties of ITCL(w) are then proved. First, we re-obtain its convexity, which was already stated in [14]. Then, the unique extremal point of ITCL(w) is fully characterized and we prove that, remarkably, this set is a cone admitting no finite dimensional extremal faces, see Proposition 5.2.

The characterization below directly shows that as soon as ITCL is non empty, then also ITCLBV(R;R) is non empty, meaning that any profile reached by an initial datum with unbounded total variation can also be reached by a (different) initial datum in BV. Moreover, we prove that ITCL always contains one sided Lipschitz continuous functions but more regular initial data may also be available. The initial datum constructed “prolonging backwards all shocks” yields a solution whose interaction potential [8, Formula (10.10)] is constant on ]0,T[.

Issues related to parameter identification or inverse problems based on (1.1) or (1.2) provide further motivations to the natural theoretical questions outlined above. In particular, we defer to the related paper [14] that motivates the present problem through applications to the study of sonic booms in (1.1), also providing several illustrative examples and visualizations. In the case of (1.2), U is typically the value function associated to a time reversed control problem, f being related to the dynamics and to the running cost, with Uo playing the role of the terminal cost. Here, the present result amounts to characterizing the terminal cost corresponding to given initial cost, see [15, Section 10.3] for further connections to optimal control problems. The present analytic results might also help in numerical investigations such as those in [16], [17], [18], [19].

Sections 2 to 5 collect the analytic results, while all proofs are deferred to sections 6 to 9.

Section snippets

Notation and preliminary results

Throughout, T is fixed and strictly positive. Below, we mostly refer to [20, § 3.2] for results about BV functions. In Section 6 we briefly recall the definition and the main properties of SBV(R;R), refer to [21] or [9, § 1.7] for more details. As usual, we also use functions u in BVloc(R;R), respectively in SBVloc(R;R), meaning that the restriction u|I of u to any bounded real interval I is in BV(I;R), respectively in SBV(I;R). If uBVloc(R;R), then we set u(x±)=limξx±u(ξ) and we convene

Construction of a remarkable element of ItCL(w)

We now prove that Oleinik Condition (2.4) characterizes those profiles w such that ITCL(w). Indeed, if a profile w satisfies Oleinik condition (2.4), then the conservation law (1.1) can be integrated backwards in time, taking w as final datum at time T and yielding a BV initial profile at time 0.

Technically, we reverse the space variable, rather than reversing time, and we explicitly construct an element of ITCL(w) that will play a key role in the sequel. Clearly, this technique is effective

Characterizations of ITCL(w) and ITHJ(W)

In view of [25, Theorem 1.1], for any initial datum uo and for all but countably many times T, the map w=STuo leads to a function pw in SBVloc(R;R), see also [21, Theorem 1.2] and [9, Theorem 11.3.5]. Therefore, we restrict our analysis below to functions wSBVloc(R;R) so that pwSBVloc(R;R).

We proceed with our main result, in the version referring to the Conservation Law (1.1)

Theorem 4.1

Let (2.1) hold and T be positive. Fix wSBVloc(R;R) such that ITCL(w). Then, a map uoL(R;R) is in ITCL(w) if and

Geometric properties of ITCL(w)

On the basis of the characterization provided by Theorem 4.1 and Theorem 4.2, we obtain the following information on topological and geometrical properties of the set ITCL(w).

Proposition 5.1

Let (2.1) hold and T be positive. Fix wSBVloc(R;R) so that ITCL(w). Then, with respect to the Lloc1 topology,

  • (T1)

    for any M>0, the set ITCL(w){uL(R;R):|u|M} is closed;

  • (T2)

    ITCL(w) has empty interior.

By (T1), ITCL(w) is an Fσ set in the Lloc1 topology.

Proposition 5.2

Let (2.1) hold and T be positive. Fix wL(R;R) such that ITCL(w) and wSBV

Proofs related to § 2

The Lebesgue measure in R is denoted by L. Given uLloc(R;R), we define the set Leb(u) of its Lebesgue points as the set of those xR such that limr01rxx+r|u(ξ)u(x)|dξ=0. By [26, Corollary 2, Chapter 1, § 7]), L(RLeb(u))=0.

Below, we often use the decomposition u=uac+uj+uc of a BV function u into its absolutely continuous part uac, its jump part uj, which is a possibly infinite sum of Heaviside functions, and its Cantor part uc. Whenever uc=0, we say that uSBV(R;R). Recall that if uBV(R;R

Proofs related to § 3

A tool used below is the following classical representation formula for the solutions to (1.1).

Proposition 7.1

[29, Theorem 2.1]

Let (2.1) hold and uoL1(R;R). The solution to{tu+xf(u)=0u(0,x)=uo(x) in the sense of Definition 2.1 is the mapu(t,x)=g(xy(t,x)t)wherey(t,x)minimizesys(t,x,y),s(t,x,y)=tf(xyt)+0yuo(ξ)dξ,f(λ)=λg(λ)f(g(λ)),g(λ)=(f)1(λ).

Formula (7.2) is a direct consequence of the classical Lax–Hopf formula, see [29, Theorem 2.1], [30], [31] or [9, § 11.4], adapted to the present assumption (2.1) on f. Here we

Proofs related to § 4

Proof of Theorem 4.1, (necessity)

(Throughout this proof, we set p=pw). Let uo be the precise representative (5.2) of the initial datum to (1.1) such that the corresponding solution u satisfies u(T)=w. We now prove that conditions (i) and (ii) hold.

Since pSBV(R;R), we write below p=pac+ps, with pacAC(R;R) and ps being a sum of countably many Heaviside functions centered at the points of jump in w. Note that p, pac and ps are all weakly increasing.

Using the notation (2.3), introduce the setE:={xR:p is differentiable at x and p

Proofs related to § 5

Proof of Proposition 5.1

We prove the different parts separately.

Proof of (T1): The proof directly follows from [8, Theorem 6.2, Chapter 6].

Proof of (T2): To prove that ITCL(w) has empty interior, fix uo in ITCL(w) and use the characterization of ITCL(w) provided by Theorem 4.2. Note that by Proposition 2.5, there exists an x¯R such that either (I) or (II) in Theorem 4.2 holds.

Let (I) hold at a given x¯R. Define the sequence of initial datauon(x):=uo(x)+χ]pw(x¯)1/n,pw(x¯)+1/n[(x). Clearly, uonuo strongly in L1 as n

Acknowledgement

Both authors thank Enrique Zuazua for having suggested the problem and for useful discussions during the “VII Partial Differential Equations, Optimal Design and Numerics” that took place in Benasque. The second author was supported by the ANR project “Finite4SOS” (ANR 15-CE23-0007).

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