Initial data identification in conservation laws and Hamilton–Jacobi equations
Introduction
Under suitable conditions on the flux and on the initial datum, solutions to a scalar conservation law in 1 space dimension, namely to are known to be obtained through from solutions to the Hamilton–Jacobi equation 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 , we denote by the weak entropy solution to (1.1). Symmetrically, we denote by the viscosity solution to (1.2). Below, we consider the case of a uniformly convex flux/Hamiltonian f and we obtain complete characterizations of both sets and we use the notation whenever we refer to both sets in (1.3).
First, we identify those profiles such that the corresponding set 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 estimates 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 is not empty. This proof is constructive, in the sense that an initial datum in is explicitly constructed, see Theorem 3.1.
Once 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 and , whose precise definitions are in (2.5), that partition (up to a negligible set), see Fig. 1. For x varying in the former one, , the value attained at x by any initial datum in is essentially uniquely determined: evolves in the subset of where the assigned profile is regular. On the contrary, for x varying in the latter one, , the value attained at x by any initial datum in is subject to rather loose constraints: 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 and . 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 containing at most a unique element. Roughly, they correspond to the continuous profiles in the case of (1.1) and to the ones in the case of (1.2). A further consequence, for instance, is that a discontinuity present at 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 on each of the different connected components of and are entirely independent from each other. The complete characterization of then results from the gluing of the characterizations obtained on each of the connected components of and .
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 are then proved. First, we re-obtain its convexity, which was already stated in [14]. Then, the unique extremal point of 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 is non empty, then also 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 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 .
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 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 , refer to [21] or [9, § 1.7] for more details. As usual, we also use functions u in , respectively in , meaning that the restriction of u to any bounded real interval I is in , respectively in . If , then we set and we convene
Construction of a remarkable element of
We now prove that Oleinik Condition (2.4) characterizes those profiles w such that . 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 that will play a key role in the sequel. Clearly, this technique is effective
Characterizations of and
In view of [25, Theorem 1.1], for any initial datum and for all but countably many times T, the map leads to a function in , see also [21, Theorem 1.2] and [9, Theorem 11.3.5]. Therefore, we restrict our analysis below to functions so that .
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 such that . Then, a map is in if and
Geometric properties of
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 .
Proposition 5.1 Let (2.1) hold and T be positive. Fix so that . Then, with respect to the topology, for any , the set is closed; has empty interior.
By (T1), is an set in the topology.
Proposition 5.2 Let (2.1) hold and T be positive. Fix such that and
Proofs related to § 2
The Lebesgue measure in is denoted by . Given , we define the set of its Lebesgue points as the set of those such that . By [26, Corollary 2, Chapter 1, § 7]), .
Below, we often use the decomposition of a BV function u into its absolutely continuous part , its jump part , which is a possibly infinite sum of Heaviside functions, and its Cantor part . Whenever , we say that . Recall that if
Proofs related to § 3
A tool used below is the following classical representation formula for the solutions to (1.1).
Proposition 7.1 Let (2.1) hold and . The solution to in the sense of Definition 2.1 is the map[29, Theorem 2.1]
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 ). Let be the precise representative (5.2) of the initial datum to (1.1) such that the corresponding solution u satisfies . We now prove that conditions (i) and (ii) hold. Since , we write below , with and being a sum of countably many Heaviside functions centered at the points of jump in w. Note that p, and are all weakly increasing. Using the notation (2.3), introduce the set
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 has empty interior, fix in and use the characterization of provided by Theorem 4.2. Note that by Proposition 2.5, there exists an such that either (I) or (II) in Theorem 4.2 holds. Let (I) hold at a given . Define the sequence of initial data Clearly, strongly in as
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).
References (33)
- et al.
A note on front tracking and equivalence between viscosity solutions of Hamilton-Jacobi equations and entropy solutions of scalar conservation laws
Nonlinear Anal. Ser. A: Theory Methods
(2002) - et al.
Sharp decay estimates for hyperbolic balance laws
J. Differ. Equ.
(2009) - et al.
Numerical schemes for conservation laws via Hamilton-Jacobi equations
Math. Comput.
(1995) The Cauchy problem in the large for non-linear equations and for certain first-order quasilinear systems with several variables
Dokl. Akad. Nauk SSSR
(1964)Discontinuous solutions of non-linear differential equations
Usp. Mat. Nauk
(1957)An extension of Oleinik's inequality for general 1D scalar conservation laws
J. Hyperbolic Differ. Equ.
(2008)- et al.
Decay of positive waves in nonlinear systems of conservation laws
Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4)
(1998) Hyperbolic Systems of Conservation Laws
(2000)Hyperbolic Conservation Laws in Continuum Physics
(2016)- et al.
Lower compactness estimates for scalar balance laws
Commun. Pure Appl. Math.
(2012)