Particle dynamics inside shocks in Hamilton–Jacobi equations 0 1 Kostya Khanin1 and Andrei Sobolevski2,3 0 1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada 2 2 Institute for Information Transmission Problems of the Russian Academy of n Sciences, Moscow, Russia a J 3 Laboratoire J.-V. Poncelet (UMI 2615 CNRS), Moscow, Russia 4 E-mail: [email protected], [email protected] ] h p Abstract. CharacteristiccurvesofaHamilton–Jacobiequationcanbeseenasaction - minimizing trajectories of fluid particles. For nonsmooth “viscosity” solutions, which h give rise to discontinuous velocity fields, this description is usually pursued only up t a to the moment when trajectories hit a shock and cease to minimize the Lagrangian m action. In this paper we show that for any convex Hamiltonian there exists a uniquely [ defined canonical global nonsmooth coalescing flow that extends particle trajectories 1 and determines dynamics inside the shocks. We also provide a variational description v of the corresponding effective velocity field inside shocks, and discuss relation to the 8 9 “dissipative anomaly” in the limit of vanishing viscosity. 4 0 . 1 PACS numbers: 47.10.Df, 47.40.Nm, 02.30.Yy, 02.40.Ft, 02.40.Xx 0 0 1 : v Submitted to: Philosophical Transactions of the Royal Society A i X r a Particle dynamics inside shocks in Hamilton–Jacobi equations 2 1. Introduction The Hamilton–Jacobi equation ∂φ (t,x)+H(t,x,∇φ(t,x)) = 0 (1) ∂t plays an important role in a large variety of mathematical and physical problems. Apart from analytical mechanics, it appears in description of a whole range of extended dissipative systems featuring nonequilibrium turbulent processes, from microscales of condensed matter and statistical physics through mesoscale setting of free-boundary fluid to macroscale cosmological evolution (see, e.g., a non-exhaustive collection of references in [1]). The central issue in a study of nonlinear evolution for (1) is to understand, both from the mathematical and physical points of view, the behaviour of the system after the inevitable formation of singularities. A theory of weak solutions for a general Hamilton–Jacobi equation, employing the regularization by infinitesimal diffusion, exists since the 1970s [2, 3, 4]. In the one- dimensionalsettingthistheoryisessentiallyequivalenttotheearliertheoryofhyperbolic conservation laws in fluid mechanics, developed in the 1950s [5, 6, 7]. In more than one dimension, however, the two theories are no longer parallel. The theory of weak solutions for the Hamilton–Jacobi equation is closely related to calculus of variations, and from this point of view one can say that introduction of diffusion is motivated essentially by stochastic control arguments [8]. In the present paper we adopt a related but somewhat complementary viewpoint, in which the Hamilton–Jacobi equation is considered as a fluid dynamics model, and construct the flow of “fluid particles” inside the shock singularities of a weak solution. A useful example to be borne in mind when thinking about (1) — and arguably the most widely known variant thereof — is the Riemann, or inviscid Burgers, equation ∂u +u·∇u = 0, ∇×u = 0. (2) ∂t It is obtained for the Hamiltonian H(t,x,p) = |p|2/2 by setting u(t,x) = ∇φ(t,x). Note that in (2) it is essential that the velocity field u is curl-free, so this model is indeed equivalent to the Hamilton–Jacobi equation (1). The Riemann equation may in turn be considered as a limit of vanishing viscosity of the Burgers equation ∂uµ +uµ ·∇uµ = µ∇2uµ, ∇×uµ = 0, (3) ∂t so solutions of (2) can be defined as limits of smooth solutions to (3) as the positive parameter µ goes to zero. The Burgers equation is in fact very special: it can be exactly mapped by the Cole– Hopf transformation into the linear heat equation and therefore explicitly integrated [5]. Nonetheless the qualitative behaviour of solutions to a parabolic regularization of (1) for a general convex Hamiltonian ∂φµ +H(t,x,∇φµ) = µ∇2φµ (4) ∂t Particle dynamics inside shocks in Hamilton–Jacobi equations 3 in the limit of vanishing viscosity is similar to that for the Burgers equation. It turns out that as µ → 0 there exists a limit φ(t,x) = limφµ(t,x) which is called the viscosity solution. Remarkably the viscosity solution can be described by a purely variational construction which does not use the viscous regularization at all. Below we briefly recall the main ideas of this variational approach. Assume that the Hamiltonian function H(t,x,p) is smooth and strictly convex in the momentum variable p, i.e., is such that for all (t,x) the graph of H(t,x,p) as a function of p lies above any tangent plane and contains no straight segments. This implies that the velocity v = ∇ H(t,x,p) is a one-to-one function of p. In addition, p the Lagrangian function L(t,x,v) = max [p·v −H(t,x,p)] p under the above hypotheses is smooth and strictly convex in v, although it may not be everywhere finite: e.g., the relativistic Hamiltonian H(t,x,p) = 1+|p|2 corresponds to the Lagrangian L(t,x,v) that is defined for |v| ≤ 1 as − 1p−|v|2 and should be considered as taking value +∞ elsewhere. This will not happpen if in addition one assumes that the Hamiltonian H grows superlinearly in |p|. The relation between the Lagrangian and the Hamiltonian is symmetric: they are Legendre conjugate to one another. This relation can also be expressed in the form of the Young inequality: L(t,x,v)+H(t,x,p) ≥ p·v. This inequality holds for all v and p and turns into equality whenever v = ∇ H(t,x,p) p or p = ∇ L(t,x,v). The two functions ∇ H(t,x,p) and ∇ L(t,x,v) are inverse to v p v each other; we will call them the Legendre transforms at (t,x) of p and of v. (Usually the term “Legendre transform” refers to the relation between the conjugate functions H and L; here we follow the usage adopted by A. Fathi in his works on weak KAM theory [9], which is more convenient in the present context.) Note that if H(t,x,p) = |p|2/2, then L(t,x,v) = |v|2/2 and the Legendre transform reduces to the identity v ≡ p, blurring the distinction between velocities and momenta. This is another very special feature of the Burgers equation. Now assume that φ(t,x) is a strong solution of the inviscid equation (1), i.e., a smooth function that satisfies the equation in the classical sense. For an arbitrary trajectory γ(t) the full time derivative of φ along γ is given by dφ(t,γ) ∂φ = +∇φ·γ˙ = ∇φ·γ˙ −H(t,γ,∇φ) ≤ L(t,γ,γ˙), (5) dt ∂t where at the last step the Young inequality is used. This implies a bound for the mechanical action corresponding to the trajectory γ: t2 φ(t ,γ(t )) ≤ φ(t ,γ(t ))+ L(s,γ(s),γ˙(s))ds. (6) 2 2 1 1 Z t1 Equality in (5) is only achieved if γ˙ is the Legendre transform of ∇φ at (t,γ): γ˙(t) = ∇ H(t,γ,∇φ(t,γ)). (7) p Particle dynamics inside shocks in Hamilton–Jacobi equations 4 This represents one of Hamilton’s canonical equations, with momentum given for the trajectory γ by p (t) := ∇φ(t,γ(t)). The other canonical equation, p˙ = −∇ H, follows γ x from (1) and (7) because ∂∇φ p˙ (t) = +(∇⊗∇φ)·γ˙ = −∇ H(t,γ,∇φ)−∇ H ·(∇⊗∇φ)+(∇⊗∇φ)·γ˙ γ x p ∂t Therefore the bound (6) is achieved for trajectories satisfying Hamilton’s canonical equations. This is a manifestation of the variational principle of the least action: Hamiltonian trajectories (γ(t),p (t)) are (locally) action minimizing. In particular, γ if the initial condition φ(t = 0,y) = φ (y), (8) 0 is a fixed smooth function, the identity t φ(t,x) = φ (γ(0))+ L(s,γ(s),γ˙(s))ds 0 Z 0 holds for a minimizer γ such that γ(t) = x. The least action principle can be used to construct the viscosity solution corresponding to the initial data (8): t φ(t,x) = min φ (γ(0))+ L(s,γ(s),γ˙(s))ds . (9) γ:γ(t)=x 0 Z (cid:16) (cid:17) 0 This is the celebrated Lax–Oleinik formula (see, e.g., [10] or [9]), which reduces a PDE problem (1), (8) to the variational problem (9) where minimization is extended to all smooth curves γ such that γ(t) = x. The function φ defined by (9) is smooth at those points (t,x) where the minimizing trajectory is unique. In this case, the minimizer can be embedded in a smooth family of minimizing trajectories whose endpoints at time 0 and t are continuously distributed aboutγ(0)andγ(t) = x. Apieceofinitialdataφ getscontinuouslydeformedaccording 0 to (5) along this bundle of trajectories into a piece of smooth solution φ to (1) defined in a neighbourhood of x at time t. Of course the Hamilton–Jacobi equation is satisfied in a strong sense at all points where φ is differentiable. However, the crucial feature of (9) is that generally there will be points (t,x) with several minimizers γ , which start at different locations γ (0) but bring the same value i i of action to x = γ (t). Just as above, each of these Hamiltonian trajectories will be i responsible for a separate smooth “piece” of solution. Thus for locations x′ close to x the function φ will be represented as a pointwise minimum of smooth pieces φ : i φ(t,x′) = min φ (t,x′). i i As all γ have the same terminal value of action, all these pieces intersect at (t,x): i φ (t,x) = φ (t,x) = ... = φ(t,x). Thus the neighbourhood of x at time t is 1 2 partitioned into domains where φ coincides with each of the smooth functions φ and i satisfies the Hamilton–Jacobi equation (1) strongly. These domains are separated by surfaces of various dimensions where two, or possibly three or more, pieces φ i intersect and hence φ is not differentiable. Such surfaces are called shocks. Note that Particle dynamics inside shocks in Hamilton–Jacobi equations 5 a function φ defined by the Lax–Oleinik formula is continuous everywhere, including the shocks; it is its gradient that suffers a discontinuity. In general, there are infinitely many continuous functions that match the initial condition (8) and are differentiable and satisfy the Hamilton–Jacobi equation (1) apart from some shock surfaces, just as φ. A standard one-dimensional example of such nonuniqueness is provided by φ (t,x) = min(α|x| − α2t/2,0), which for any α > 0 satisfies the initial condition α φ (0,x) = 0 and the equation ∂φ /∂t + |∂φ /∂x|2/2 = 0 apart from the shock rays α α α x = ±αt/2, x = 0. Whatdistinguishesthefunctionφdefinedby(9)fromallthese“weak solutions,” and grants it with important physical meaning, is that φ appears in the limit of vanishing viscosity for the regularized equation (4) with the initial condition (8) (see, e.g., [3]). For a smooth Hamiltonian it can be proved that once shocks are created they never disappear, although they can merge with one another. Another important physical feature of viscosity solutions is that minimizers can only merge with shocks but never leave them: all minimizers coming to some (t,x) in (9) originate at t = 0. It is easy to see that this is not so in the above example of φ , where minimizers emerge from x = 0 α at all times t > 0. Moreover, in a solution φ given by (9) a minimizer that has come to a shock cannot be continued any longer as a minimizing trajectory. Indeed, wherever it comes, there will be other trajectories originated at t = 0 that will bring smaller values of action to the same location. Hence for the purpose of the least action description (9) Hamiltonian trajectories are discontinued as soon as they are absorbed by shocks. The set of trajectories which survive as minimizers until time t > 0 is decreasing with t, but at all times it is sufficiently large to cover the whole space of final positions. Let us now adopt an alternative viewpoint and consider the Hamilton–Jacobi equation as a fluid dynamics model, assuming that Hamiltonian trajectories (7) are described by material “particles” transported by the velocity field u(t,x), which is the Legendre transform of the momenta field p(t,x) = ∇φ(t,x). From this new perspective it is no longer natural to accept that particles annihilate once they reach a shock. Can therefore something be said about the dynamics of those particles that got into the shock, notwithstandingthefactthattheirtrajectoriesceasetominimizetheaction? The problem here comes from the discontinuous nature of the velocity field u, which makes it impossible to construct classical solutions to the transport equation γ˙(t) = u(t,γ). In dimension 1 the answer to the question above is readily available. Shocks at each fixed t are isolated points in the x space and as soon as a trajectory merges with one of them, it continues to move with the shock at all later times. This description is related to C. Dafermos’ theory of generalized characteristics [11] which, in fact, can be extended to a much more general situation of nonconvex Hamiltonians and systems of conservation laws. However, in several space dimensions shocks become extended surfaces and already for equation (1) with a strictly convex Hamiltonian dynamics of trajectories inside shocks is by no means trivial. The main goal of the present paper is to describe a natural and canonical construction of such dynamics. Particle dynamics inside shocks in Hamilton–Jacobi equations 6 First results in this direction were obtained by I. Bogaevsky [12, 13] for the Burgers equation (3). Bogaevsky suggested to consider the transport problem for a smooth velocity field uµ: γ˙µ(t) = uµ(t,γµ), γµ(0) = y. Since uµ for µ > 0 is a smooth vector field, this equation defines a family of particle trajectories forming a smooth flow. The next step is to take the limit of this flow as µ ↓ 0. Bogaevskyprovedthatthislimitexistsasanon-differentiablecontinuousflow, for which the forward derivative γ˙(t+0) = lim [γ(t+τ)−γ(t)]/τ is defined everywhere. τ↓0 If γ(t) is located outside shocks, this derivative coincides with u(t,γ(t)). Otherwise there are several limit values of velocity u , and Bogaevsky discovered an interesting i explicit representation for γ˙(t+0): it coincides with the center of the smallest ball that contains all u . It should be remarked that uniqueness of a limit flow in the case of i a quadratic Hamiltonian was earlier observed by P. Cannarsa and C. Sinestrari in the contextofpropagationofsingularitiesfortheeikonalequation anddifferentialinclusions [14, Lemma 5.6.2]. The original approach in [12, 13] is based on the specific properties of the Burgers equation and cannot be applied in the case of general convex Hamiltonians. In particular, the method uses the identity of velocities and momenta, which does not hold in the general setting. At the same time the common wisdom says that all Hamilton– Jacobi equations with convex Hamiltonians must have similar properties. In this work we consider the above strategy, consisting in the parabolic regularization of the Hamilton–Jacobi equation and investigation of the vanishing viscosity limit for the corresponding regularized flow, in the case of general convex Hamiltonians. We show that such a limit exists, and derive an explicit representation for the forward velocity of the limit flow that extends the above result for the Burgers equation. Yet the mechanism powering these results in the general case is completely different. It is based on the fundamental uniqueness of a possible limit behaviour of γµ, which we discuss in detail below. The paper is organized as follows. In Section 2 we study the local structure of a viscosity solution near a singularity. We also introduce the notion of admissible velocity at a singularity and show that it can be determined uniquely. Moreover, the unique admissible velocity provides a solution to a certain convex minimization problem, which generalizes Bogaevsky’s construction of the center of the smallest ball. In Section 3 we demonstrate that for any nonsmooth viscosity solution there exists a unique continuous nonsmooth flow of trajectories tangent to the field of admissible velocities. Section 4 contains concluding remarks. Acknowledgments This work is supported by the NSERC Discover Grant and the Russian Foundation for Basic Research, project RFBR-CNRS–07–01–92217; the second author is supported in Particle dynamics inside shocks in Hamilton–Jacobi equations 7 part by the French Agence Nationale de la Recherche, project ANR–07–BLAN–0235 OTARIE. We acknowledge the hospitality of Observatoire de la Cˆote d’Azur (Nice, France), where part of this work has been performed, as well as support of the French Ministry for National Education. We are sincerely grateful to J´er´emie Bec, Patrick Bernard, Ilya Bogaevsky, Yann Brenier, Philippe Choquard, and Boris Khesin for numerous valuable discussions. It is our special pleasure to express gratitude to Uriel Frisch, who made crucial contributions to the current revival of interest in the “Burgers turbulence” and related fields, and who is turning 70 in 2010. 2. Local structure of viscosity solutions and admissible momenta Let φ be a viscosity solution to the Hamilton–Jacobi equation (1) with initial data (8). If there is a single minimizer coming to (t,x), then φ is differentiable at this point and ∂φ φ(t+τ,x+ξ) = φ(t,x)+ τ +∇φ·ξ +··· ∂t = φ(t,x)−H(t,x,∇φ)τ +∇φ·ξ +···, where dots ··· stand for higher-orger terms. If φ is not differentiable at (t,x), this means that there are several minimizers γ such that γ (t) = x, each bringing to (t,x) i i a different piece φ of solution. Then the Lax–Oleinik formula implies that i φ(t+τ,x+ξ) = min φ (t+τ,x+ξ) i i = φ(t,x)+min (−H τ +p ·ξ)+···, i i i where p := ∇φ (t,x) and H := H(t,x,p ). i i i i In the latter case neither of the expressions −H τ + p · ξ provides an adequate i i linear approximation to the difference φ(t + τ,x + ξ) − φ(t,x), but they all majorize this difference up to a remainder that is linear or higher-order depending on τ and ξ. Evidently, so does too the linear form −Hτ +p·ξ for any convex combination p = λ p , H = λ H i i i i Xi Xi with λ ≥ 0, λ = 1. In convex analysis these convex combinations are called i i i supergradientsPof φ at (t,x) and the whole collecton of them, which is a convex polytope with vertices (−H ,p ), is called the superdifferential of φ [14, 15]. We use i i R. T. Rockafellar’s notation ∂φ(t,x) for the superdifferential [15]. To avoid a possible misunderstanding it should be noted that, although uniqueness of minimizer coming to (t,x) implies differentiability of φ at t and earlier times, it does not imply its differentiability at any t+τ > t. The following example shows how this may happen. The function defined for τ ≥ 0 by 4 4 φ(t+τ,x+ξ) = φ(t,x)− τ3 −2|ξ|τ − (|ξ|+τ2)3/2 3 3 is a viscosity solution of the equation ∂φ/∂t+|∂φ/∂x|2/2 = 0 that satisfies the smooth initial condition φ(t,x+ξ)−φ(t,x) = −4|ξ|3/2. For τ > 0 a shock appears at ξ = 0, but 3 Particle dynamics inside shocks in Hamilton–Jacobi equations 8 differentiability at τ = 0 is recovered because ∂φ(t+τ,x) = {−8τ2}×[−4τ,4τ] shrinks to (0,0) as τ ↓ 0. Such points (t,x) are called preshocks [1]. The particular case of preshocks is an instance of a general fact: replacing gradients with superdifferentials allows to recover continuous differentiability, but in a weaker sense. Namely, suppose (t ,x ) converges to (t,x) and the sequence (−H ,p ) ∈ n n n n ∂φ(t ,x ) has a limit point (−H,p). By definition of superdifferential, n n φ(t +τ,x +ξ)−φ(t ,x ) ≤ −H τ +p ·ξ +···; (10) n n n n n n passing here to the limit and using continuity of φ, we see that (−H,p) ∈ ∂φ(t,x). Therefore the superdifferential ∂φ(tx) contains all the limit points of superdifferentials ∂φ(t ,x ) as (t ,x ) converges to (t,x). n n n n Suppose (t,x) is a point of shock where k smooth branches φ meet. It follows from i the above discussion that for a particle moving from a shock point (t,x) all possible values of the velocity v must be such that the corresponding momentum p(v) belongs to the convex hull of the available momenta p = ∇φ (t,x), 1 ≤ i ≤ k. However, i i one can say even more. For small positive τ not all the branches φ will contribute to i the solution φ at a point (t+τ,x+vτ), but only those of them that correspond to a minimum in min (−H +p ·v). Denote the corresponding set of indices i i i I(v) := {1 ≤ j ≤ k:−H +p ·v = min(−H +p ·v)}. (11) j j i i i The set I(v) has the following physical meaning: if particle moves away from a shock with a given velocity v then only φ and p for j ∈ I(v) are relevant. Geometrically j j one can say that the convex hull of p , j ∈ I(v), is the p-projection of the face of j the superdifferential ∂φ(t,x) that looks toward an infinitesimal observer who has just left (t,x) with velocity v. This implies that any possible velocity v must satisfy the following condition. Admissibility Condition. A velocity v∗ is admissible if and only if the corresponding momentum belongs to the convex hull of the momenta p ,j ∈ I(v∗). j Namely, p∗(v∗) = λ p , λ ≥ 0, λ = 1. (12) j j j j X X j∈I(v∗) j∈I(v∗) To make the admissibility argument rigorous it is necessary to have some control over the remainder term in (10). A natural function class that contains viscosity solutions of Hamilton–Jacobi equations and in which such control is possible is formed by semiconcave functions [14]. We refer a reader interested in careful proofs of this and other convex analytic results used in this paper to monographs [14, 15]. Remarkably the Admissibility Condition determines the velocity v∗ uniquely. Lemma 1 (Uniqueness) Let φ be a viscosity solution to the Cauchy problem (1), (8). Then at any (t,x) there exists a unique admissible velocity v∗. Moreover this admissible velocity v∗ is the unique point of the global minimum for the following function Lˆ(v) := L(t,x,v)−min (−H +p ·v). (13) i i i Particle dynamics inside shocks in Hamilton–Jacobi equations 9 Proof. Recall that L(t,x,v) is a strictly convex function of v because of assumptions formulated in the Introduction. Rewriting L (v) := L(t,x,v)+H −p ·v, Lˆ(v) = max L (v), (14) i i i i i we see that Lˆ(v) is a poitwise maximum of a finite number of strictly convex functions and therefore is strictly convex itself. Furthermore, because the Hamiltonian H(t,x,p) is assumed to be finite for all p, its conjugate Lagrangian L(t,x,v) grows faster than any linear function as |v| increases, and all its level sets are bounded. Thus Lˆ(v) attains its minimum at a unique value of velocity v∗. Let us show that this point of minimum v∗ satisfies the admissibility condition. Indeed, ∇ L (v∗) = ∇ L(t,x,v∗)−p = p∗ −p . v i v i i Suppose that p∗ does not belong to the convex hull of p , j ∈ I(v∗). Then there exists a j vector h such that (p∗−p )·h < 0 for all j ∈ I(v∗). It follows that L (v∗+ǫh) < L (v∗) j j j for all j ∈ I(v∗) if ǫ > 0 is sufficiently small. Hence, Lˆ(v∗ + ǫh) < Lˆ(v∗) for sufficiently small ǫ, which contradicts our assumption that v∗ is a point of minimum. This contradiction proves that v∗ is admissible. To prove uniqueness we show that if vˆ is admissible then it is a unique point of ˆ ˆ global minimum for the function L. Using the strict convexity of L, we obtain L (vˆ+h) = L(t,x,vˆ+h)+H −p ·(vˆ+h) j j j > L (vˆ)+∇ L(t,x,vˆ)·h−p ·h = L (vˆ)+(pˆ−p )·h, j v j j j where pˆ is the Legendre transform of vˆ. Since vˆ is admissible, pˆ = λ p , where all j j j λ ≥ 0and λ = 1. Hence, λ (pˆ−p )·h = [( λ )pˆ− λ p P]·h = [pˆ−pˆ]·h = 0. j j j j j j j j j j j It follows thPat (pˆ−p )·h ≥ 0Pfor at least one j ∈PI(vˆ). ThuPs Lˆ(vˆ+h) > Lˆ(vˆ), which j implies that vˆ is a unique point of global minimum for Lˆ. This observation concludes the proof. The admissibility property, first formulated above in a somewhat unmanageable combinatorial form (12), turns out to be the optimality condition in a convex minimization problem given by (13), i.e., a much simpler object. In particular, if φ is differentiable at (t,x), then Lˆ(v) = L(t,x,v)+H(t,x,∇φ)−∇φ·v and the minimum in (13) is achieved at the Legendre transform of ∇φ. We thus recover Hamilton’s equation (7). The following reformulation will clarify the connection between admissibility and Bogaevsky’s original construction for the Burgers equation. Let v = ∇ H(t,x,p ) i p i be the velocity corresponding to the limit momentum p and observe that p = i i ∇ L(t,x,v ). The Young inequality implies that H = H(t,x,p ) = p ·v −L(t,x,v ) v i i i i i i and therefore (14) assumes the form Lˆ(v) = max [L(t,x,v)−L(t,x,v )−∇ L(t,x,v )·(v −v )]. i i v i i The quantity in square brackets is known as the Bregman divergence between vectors v and v , a specific measure of their separation with respect to the convex function L i Particle dynamics inside shocks in Hamilton–Jacobi equations 10 [16]. When L(t,x,v) = |v|2/2, the Bregman divergence reduces to (half) the squared distance between the two vectors; hence the admissible velocity v∗ exactly conicides with the centre of smallest ball containing all v , and Bogaevsky’s result is recovered. i ˆ Finally, wediscussthephysicalmeaningofthefunctionL. Consideraninfinitesimal movement from (t,x) with velocity v. Obviously φ(t,x)+L(t,x,v)dt−φ(t+dt,x+ vdt) ≥ 0. It is easy to see that in the linear approximation in dt φ(t,x)+L(t,x,v)dt−φ(t+dt,x+vdt) = Lˆ(v)dt. Hence the unique admissible velocity v∗ minimizes the rate of the difference in action between the true minimizers and trajectories of particles on shocks. In other words, the trajectory on a shock cannot be a minimizer but it does its best to keep its surplus action growing as slowly as possible. 3. The vanishing viscosity limit In the preceding section we constructed a canonical vector field v∗ = ∇ H(t,x,p∗) p correspondingtoagivenviscositysolutionφoftheCauchyproblem(1),(8). Thebasisof thisconstruction, theadmissibilitycondition, appearsasanaturalconsistencycondition between velocities and supergradients. This condition together with the variational principle (13) guarantees uniqueness of the admissible pair (v∗,p∗). The vector field v∗(t,x) can be decomposed into a union of smooth tangent vector fields defined on connected pieces of smooth shock surfaces of various dimensions as well asonthedomainwhereφisdifferentiable, sodynamicsinthelatterdomainorinsideany piece of a smooth shock surface is given locally by a smooth flow. But globally the field v∗ is discontinuous, and it is not immediately clear if there exists an overall continuous flow of trajectories γ that is compatible with v∗ in the sense that γ˙(t+0) = v∗(t,γ). Even less obvious is the uniqueness of such a flow. In order to answer these questions affirmatively we employ the vanishing viscosity limit for the parabolic regularization ∂φµ +H(t,x,∇φµ) = µ∇2φµ, µ > 0, (15) ∂t of the Hamilton–Jacobi equation (1). For sufficiently smooth initial data φµ(t = 0,y) = φ (y)equation(15)hasagloballydefinedstrongsolution, whichislocallyLipschitzwith 0 a constant independent of µ. Moreover, φµ converges as µ ↓ 0 to the unique viscosity solution φ corresponding to the same initial data. Proof of these facts may be found, e.g., [3], where they are established for φ ∈ C2,α. 0 Consider now the differential equation γ˙µ(t) = ∇ H(t,γµ,∇φµ(t,γµ)), γµ(0) = y. (16) p For µ > 0 this equation has a unique solution γµ, which continuously depends on the y initial location y. Fix a point (t,x) with t > 0 and pick for all sufficiently small µ > 0 trajectories γµ such that γµ(t) = x. The uniform Lipschitz property of solutions φµ implies that the curves γµ are uniformly bounded and equicontinuous on some interval