Singularities of solutions of Hamilton-Jacobi equations

This is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton-Jacobi equations in the past decades. We also review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations.


INTRODUCTION
This is a survey paper concerning the progress made for the singularities of the solutions to Hamilton-Jacobi equations in the past decades. We begin with a quote from the paper [KS16] by Khanin and Sobolevski: The evolutionary Hamilton-Jacobi equation (HJ) appears in diverse mathematical models ranging from analytical mechanics to combinatorics, condensed matter, turbulence, and cosmology ⋯. In many of these applications the objects of interest are described by singularities of solutions, which inevitably appear for generic initial data after a finite time due to the nonlinearity of (HJ). Therefore one of the central issues both for theory and applications is to understand the behavior of the system after singularities form.
The notion of viscosity solutions, introduced in the seminal papers [CL83,CEL84], provides the right class of generalized solutions to study existence, uniqueness, and stability issues for problem (HJ). An overview of the main features of this theory can be found in the monographs [BCD97] for first order equations and [FS06] for second order equations.
It is well known that Hamilton-Jacobi equations have no global smooth solutions in general, because solutions may develop singularities due to crossing or focusing of characteristics. The persistence of singularities, i.e, once a singularity is created, it will propagate forward in time up to +∞, affords an evidence of irreversibility for equation (HJ), while the compactness after the evolution of the associated Lax-Oleinik semi-group gives another one ( [ACN16b,ACN16a]).
The expected maximal regularity for solutions of (HJ) is the local semiconcavity of ( , ⋅) for > 0. Indeed, semiconcave functions were used to study well-posedness for (HJ) before the theory of viscosity solution was developed ([Dou61, Kru75,Kry87]). Nowadays, the notion of semiconcavity has been widely used in many mathematical fields, such as [Hru78, CF91, FM00, Rif00,Rif02] in control theory and sensitivity analysis, [Roc82,CM06] in nonsmooth and variational analysis, [Pet07] in metric geometry. Good references on semiconcave functions include the monographs [CS04,Vil09].
To our knowledge, the first paper dealing with the singularities of viscosity solutions of (HJ) is the paper by the first author and Soner ( [CS87]). Thanks to the discovery of semiconcave functions in the study of viscosity solutions of (HJ) ( [CS89]), some propagation results for general semiconcave functions were obtained in [ACS93]. The propagation of singularities of semiconcave functions along Lipschitz arcs was firstly studied in [AC99] and then extend to solutions of Hamilton-Jacobi equations ( [AC02]).
It is the first time in [AC02] the authors introduced the important notion of generalized characteristics for Hamilton-Jacobi equation (HJ), which is a keystone for the further progress later. In one-dimensional case, the idea of generalized characteristics also comes from earlier work by Dafermos [Daf77] on Burgers equation. The readers can also refer to Arnold's book [Arn90] on the shock wave singularities and perestroikas of Maxwell sets and the references therein.
For any (Lipschitz and semiconcave) weak KAM solution of (1.2), an intrinsic method was developed in the paper [CC17]. Using the positive Lax-Oleinik semi-group {̆ } >0 , one can obtain an intrinsic singular characteristic propagating singularities from any singular initial point, or general cut point of ( [CCF17]).
Although singular characteristics satisfy (1.1), the convex hull in the differential inclusion (1.1) is an obvious obstacle to establish uniqueness and stability. The only well-understood system with such well-posedness properties is the system with Hamiltonians quadratic in the momentum variable. A typical example is the Hamiltonian = 1 2 | | 2 , where differential inclusion (1.1) becomes the generalized gradient systeṁ ( ) ∈ + ( ( )), Inspired by earlier works [Bog02,Bog06,Str13], Khanin and Sobolevski essentially proved the existence of singular characteristics satisfying (1.1) without convex hull under some extra conditions on the initial data ( [KS16]). These kinds of singular characteristics are called strict singular characteristics (or broken characteristics in [Str13]). The fact that singular characteristics satisfy more restrictive dynamics than (1.1) might help to obtain some kind of uniqueness result. Indeed, in the recent work [CC20], we solved such a well-posedness problem in ℝ 2 for non-critical initial data.
When we pursue applications of this theory, global propagation results for solutions of Hamilton-Jacobi equations turn out to be necessary. Global propagation for the closure of the singular set was obtained by Albano in [Alb16]. For the propagation of genuine singularities, a global result for a Cauchy-Dirichlet problem with quadratic Hamiltonian was obtained in [CMS15] using an energy method. More global results for weak KAM solutions and Dirichlet problem using intrinsic method can be found in [CC17, CCF17, CCMW19, CCF19].
One important application of the global propagation result is the homotopy equivalence between the complement of Aubry set and the singular set of any weak KAM solution, and the local contractibility of the singular set ( [CCF17,CCF19]). An earlier result for such homotopy equivalence for the distance function on Riemannian manifolds was obtained in [ACNS13] based on invariance properties of the generalized gradients flow. Moreover, global propagation result in [CCF19] can also be applied to some basic problem in Riemmanian geometry such as the analysis of the set of points which can be joined by at least two minimizing geodesics. There are also some applications of this theory to Hamiltonian dynamical systems, mainly in the frame of Mather theory and weak KAM theory ([CCZ14, CC15, CCC19, Zha20]). Above evidence suggests that the story of singularities will continue and further applications to various topics will appear in the near future.
The paper is organized as follows. In section 2, we introduce some necessary materials on Hamilton-Jacobi equations and semiconcavity. In section 3 and 4, we will review the progress in local and global propagation of singularities for various kinds of problem. In section 5, we will concentrate on the setting of the weak KAM theory, especially the applications to the topological and dynamical applications. There is also a short concluding remark in section 6. We also provide a new proof of the Lipschitz regularity for the intrinsic singular characteristics in the appendix, which looks quite natural and intuitive comparing to the original one in [CC17].
Acknowledgements. Piermarco Cannarsa was supported in part by the National Group for Mathematical Analysis, Probability and Applications (GNAMPA) of the Italian Istituto Nazionale di Alta Matematica "Francesco Severi" and by Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Wei Cheng is partly supported by National Natural Science Foundation of China (Grant No. 11871267, 11631006 and 11790272). The second author also thanks to Jiahui Hong for helpful discussion on some part of the appendix.

PRELIMINARIES
Let Ω ⊂ ℝ be a convex open set. We recall that a function ∶ Ω → ℝ is semiconcave (with linear modulus) if there exists a constant > 0 such that for any , ∈ Ω and ∈ [0, 1].
For any continuous function ∶ Ω ⊂ ℝ → ℝ and any ∈ Ω, the closed convex sets are called the subdifferential and superdifferential of at , respectively. The following statement characterizes semiconcavity (with linear modulus) for a continuous function by using superdifferentials.
Let ∶ Ω → ℝ be locally Lipschitz. We recall that a vector ∈ ℝ is called a reachable (or limiting) gradient of at if there exists a sequence { } ⊂ Ω ⧵ { } such that is differentiable at for each ∈ ℕ, and The set of all reachable gradients of at is denoted by * ( ).
The relation between continuous viscosity solution and its semiconcavity is , considered as a function of loc (Ω, ) and is a countably  −1 -rectifiable set. Apart from earlier contributions for distance functions as in [Erd45], to our knowledge the first general results about the rectifiability of the singular sets of concave functions are due to Zajíček [Zaj78,Zaj79] and Veselý [Ves86,Ves87]. Similar properties were later extended to semiconcave functions with general modulus in [AAC92].
To obtain a fine description of Sing ( ) for a semiconcave function on Ω, it is convenient to introduce a hierarchy of subsets of Sing ( ) according to the dimension of the superdifferential. The magnitude of a point ∈ Ω (with respect to u) is the integer ( ) = dim + ( ). Given an integer ∈ {0, … , } we set Now, we turn to the analyze of rectifiability of Sing ( ), with the value function to the classical one free endpoint problem from calculus of variation, i.e., where is a Tonelli Lagrangian of class +1 ( ⩾ 1) and 0 is of class +1 , and  , is the set of all absolutely continuous curves ∶ [0, ] → ℝ such that ( ) = ∈ ℝ .
We have already seen in Proposition 3.1 that Sing ( ) is countably ( − 1)-rectifiable. Recall that, under the assumption on and 0 , where Conj ( ) is the set of conjugate points of problem (CV , ) (see [CS04,Page 155] or [CMS97] for the definition). So, we only need to analyze the rectifiability of Conj ( ). By a Sard type argument ( [Fle69]) one has  +1∕ (Conj ( )) = 0. However, the above estimate does not imply the rectifiability of Conj ( ) even if 0 is of class ∞ .
If the initial datum 0 has weaker regularity than 2 , then Sing ( ) can fail to be countably  -rectifiable, see [CS04, Example 6.6.13]. Notice that, in the mentioned example, is of class ∞ . For the further progress along this line, see [Pig02,Men04].

Generalized characteristics.
Let Ω ⊂ ℝ be open and let ∶ Ω → ℝ be a Lipschitz and semiconcavity viscosity solution of the Hamilton-Jacobi equation The notion of generalized characteristics with respect to ( , ) plays a central rôle in the study the phenomenon that the singularities propagates along a Lipschitz curve from an initial point 0 ∈ Sing ( ). 3.2.1. Propagation of singularities for general semiconcave functions. Before dealing with viscosity solutions of (3.1), we begin with a result concerning propagation of singularities for semiconcave functions with linear modulus.
The key idea of the proof of Proposition 3.3 is to construct a function Being strictly concave for small > 0, has a unique maximizer in a small neighborhood of 0 in Ω. The curve ↦ is exactly the local singular arc constructed in Proposition 3.3. It is rather surprising that a similar idea also works with the intrinsic singular characteristics, for the study of which the term 1 2 | − 0 | 2 will be replaced by the fundamental solution.
3.2.2. Generalized characteristics. Applying the basic idea from [AC99] to the viscosity solutions of (3.1), Albano and the first author introduced the notion of generalized characteristic in [AC02]. Suppose ∶ Ω × ℝ × ℝ → ℝ is a continuous function satisfying the following conditions: is differentiable with respect to and, for any 1 , 2 ∈ Ω, 1 , 2 ∈ ℝ, 1 , 2 ∈ ℝ Let be a locally semiconcave solution of (3.1) and let 0 ∈ Sing ( ) be such that 0 ∉ co ( 0 , ( 0 ), + ( 0 )). (1) is a generalized characteristic with respect to ( , ) from 0 , that is, The proof of Proposition 3.4 uses the result in Proposition 3.3 together with an Euler segment approximation method. Moreover, one can also derive the useful energy estimate This kind of energy estimate can be used to deduce global propagation results.

3.2.
3. An approximation method and singular characteristics. Needless to say, the proof of Proposition 3.3 and Proposition 3.4 utilises techniques from nonsmooth analysis and control theory. A simpler method was introduced in [Yu06] and [CY09]. The following approximation lemma, proved in [CY09], will be frequently used in what follows.
Lemma 3.5. Given a semiconcave function on Ω, we assume there are positive constants , = 0, 1, 2, such that | ( )| ⩽ 0 for all ∈ Ω, | ( )| ⩽ 1 for almost all ∈ Ω, and has semiconcavity constant 2 . Let 0 ∈ Ω and let be an open subset of Ω such that The following result can be regarded as a refinement of Proposition 3.4.
Proposition 3.6 ([CY09]). Suppose is semiconcave function on Ω and is a function of class 1 satisfying (A1) and (A2') for any ∈ Ω, ∈ ℝ and ∈ ℝ, the -level set { ∈ ℝ ∶ ( , , ) = } contains no straight line. Let 0 ∈ Sing ( ) and 0 = arg min ∈ + ( 0 ) ( 0 , ( 0 ), ). Then, there exists a Lipschitz arc Remark 3.7. For what follows we need further details related to Proposition 3.6. -The semiconcave function is not required to be a solution of (3.1). But, if Ω is bounded, being Lipschitz, must be a subsolution of (3.1) with Hamiltonian − for some ∈ ℝ. -Observe that, in general, a generalized characteristic may well be a constant arc. But, for solutions of (3.1), it was proved in [AC02] that singularities propagate along genuine shocks (injective generalized characteristics) under assumption (3.4). If is a solution of (3.1), as a corollary, one can show that the generalized characteristics in Proposition 3.6 propagates singularities under the more natural condition 0 ∉ ( 0 , ( 0 ), + ( 0 )). -For the generalized characteristic , constructed in Proposition 3.6, the right-continuity oḟ at 0 is important for further applications. Later, we will call a singular generalized characteristic satisfying properties (i)-(iv) in Proposition 3.6 a singular characteristic.
Owing to Lemma 3.5, there is a sequence of smooth functions { } enjoying properties (a) and (b) in the lemma for = 0 . It is easy to see that, for every ⩾ 1, the Cauchy problem has a 1 solution ∶ [0, ] → Ω. Without loss of generality, we can assume that uniformly converges to on [0, ] as → ∞. A standard argument (see, for instance, [Yu06]) shows that is a generalized characteristic for ( , ) starting at 0 .
3.3. Strict singular characteristics. The rôle of the convex hull in the definition of generalized characteristic is quite mysterious. This is a big obstacle for us to reveal more information about the propagation of singularities and related Hamiltonian dynamics. The next notion gets rid of such a convexity operator. The existence of strict singular characteristics for equation (HJ) was proved in [KS16] (see also the appendix of [CC20]), where additional regularity properties of such curves were established, including the right-differentiability of for every . However, the intrinsic nature of the strict singular characteristics is still unclear. One of the most important issues of the theory is to establish the uniqueness of solutions to (3.6). We describe below a partial answer to such a fundamental problem, following the paper [CC20].

GLOBAL PROPAGATION OF SINGULARITIES
In this section, we will discuss the global behavior of the propagation of singularities along generalized characteristics. 4.1. Propagating structure of the 1 singular support. A typical problem is the following evolutionary Hamilton-Jacobi equation The proof of Proposition 4.2 is based on an improvement of some classic results when 0 is of class 2 (see, for instance, Chapter 6 of [CS04]). In fact, even if 0 is just continuous, one can show that if ( , ) ∉ sing supp 1 ( ), then the associated optimal curve ending at must satisfies the property that ( , ( )) ∉ sing supp 1 ( ) for ∈ (0, ]. Now, suppose there exists ( , ( )) ∉ sing supp 1 ( ) for some ∈ ( 0 , ). Then there exists a tubular neighborhood where is the optimal curve such that ( ) = ( ). Moreover, ∩ sing supp 1 ( ) = ∅. On the open set , and are essentially identified because both solve the same ordinary differential equation (4.2) (by the claim above) and satisfy the same endpoint condition. This leads to a contradiction and it follows that the 1 singular support must propagate to ( , ( )) along the generalized characteristic .
Remark 4.3. We should emphasize that the proof of Proposition 4.2 is based on an intrinsic approach, i.e., the argument just uses the analysis of the associated characteristics system.

Generalized gradients. Let
⊂ ℝ be closed and denote by the distance function from . It is well known that = satisfies the eikonal equation If Ω is a bounded open subset of ℝ , it was shown in [ACNS13] that the generalized gradient flow given by (4.5) propagates singularities for all > 0. This is also true for the case of Riemannian manifolds. A significant application of this global propagation result to geometry is that the singular set of has the same homotopy type as Ω. Further deep extension of this topological result to the weak KAM context will be discussed later. We will also discuss more general Dirichlet problem in Section 5.3.2. for all ( , ), ( , ) ∈ such that > ⩾ 0.
The proof of the above result relies on two main ideas that are converted in two technical results, respectively. The first one is a sharp semiconcavity estimate for a suitable transform of the solution in [ACNS13]. The second one is an inequality established showing that the full Hamiltonian associated with (4.6), that is, decreases along a selection of the superdifferential of , evaluated at any point of a suitable arc.
Remark 4.5. We remark that if Ω = ℝ , Proposition 4.4 directly leads to a global propagation result. For general case, the statement ensures that the singularities will have global propagation or hit the boundary (see also Section 5.3.2).

WEAK KAM ASPECTS OF SINGULARITIES
In this section, we will discuss the problem of propagation of singularities in the frame of weak KAM theory ( 2) can be regarded as the value function of some basic problem in the calculus of variation or optimal control. For any , ∈ and > 0, we denote by Γ , the set of all absolutely continuous curves ∶ [0, ] → such that (0) = and ( ) = . We define the fundamental solution of (5.2) as Recall that for any Tonelli Lagrangian, the function ( , ) ↦ ( , ) is locally semiconcave and semiconvex for small > 0. Moreover, the function ↦ ( , ) is convex with constant ∕ for small (see, for instance, [CC17, Proposition B.8]). In symplectic geometry, ( , ) is also known as generating function. The following result is known for generating functions in symplectic geometry (see, for instance, [MS17,Chapter 9]). The readers can compare Proposition B.8 in [CC17] (see also [CF14,Theorem 4.2] for Cauchy problems) and the following lemma for fundamental solutions of Hamilton-Jacobi equations, with two analogous concepts of convexity radius and injectivity radius from Riemannian geometry.
We now claim that We say such an absolutely continuous curve is a ( , )-calibrated curve, or a -calibrated curve for short, if the equality holds in the inequality above. A curve ∶ (−∞, 0] → ℝ is called a -calibrated curve if it is -calibrated on each compact sub-interval of (−∞, 0]. In this case, we also say that is a backward calibrated curve (with respect to ). Recall that a continuous function on is called a weak KAM solution of (5.1) if = for all > 0. The following result explains the relation between the set of all reachable gradients and the set of all backward calibrated curves from (see, e.g., [CS04] or [Rif08] for the proof).
Proposition 5.4. Let ∶ → ℝ be a weak KAM solution of (5.1) and let ∈ . Then ∈ * ( ) if and only if there exists a unique 2 curve ∶ (−∞, 0] → with (0) = and = ( ,̇ (0)), which is a backward calibrated curve with respect to . A confirmative result that no isolated singular point exists for a weak KAM solution of (5.10) was proved in [CCZ14] for mechanical systems using a topological argument.
Hereafter, we refer to the arc defined in (5.13) as the intrinsic characteristic from . Notice that 0 is independent of the initial point. Thus, when ∈ Sing ( ), Proposition (5.5) yields global propagation of singularities.
The reader can compare to the idea of the proof-that we outline below-to the argument used to deduce the propagation of the 1 singular support in Section 4.1. Suppose ∈ Sing ( ) but , ∉ Sing ( ) (0 < ⩽ 0 ). Applying Fermat's rule, we have that ( , , ) = ( , ). Invoking Proposition 5.4 and the differentiability property of the fundamental solution for small time, we conclude that there exist two minimal curves. One is the backward calibrated curve satisfying Take any sequence of ∞ -functions { } such that converging uniformly to on Ω as → ∞ (for instance, the sequence given by Lemma 3.5). As was observed above, the sequence of curves is well defined for some 0 > 0. So, Recall that 0 is chosen that that 2 − 3 ∕ < 0 for ∈ (0, 0 ]. Recall that the family { } is equi-Lipschitz by (5.16). This implies converges to uniformly on [0, 0 ].
Remark 5.8. The method used here is closely related to the Lasry-Lions regularization from convex analysis ( [Att84,AA93]) and PDE ( [LL86]). In a weak KAM context, this method was also widely used as an interaction of the positive-negative Lax-Oleinik operators ( [Ber07,Ber10,Ber12,FZ10]). The relation between Lasry-Lions regularization and generalized characteristics was also studied in [CC16,CCZ18]. This method was applied to minimal homoclinic orbits with respect to the Aubry set ([CC15]). (5.20) where Ω ⊂ ℝ is a bounded Lipschitz domain, is a Tonelli Hamiltonian, and is the boundary datum. For any , ∈ Ω and any < , we define the set of admissible arcs from to as Let be the value function of the following problem: where ∶ Ω → ℝ is a continuous function satisfying the compatibility condition Observe that the function given by (5.21) is the value function of an optimal exit problem (see, for instance, [BCD97]) and a viscosity solution of (5.20). The following result can be regarded as an extension of Proposition 4.4. To exclude the case that the singularities hit the boundary we need more conditions on Ω. We shall suppose the following, where we denote Ω by Γ: (G1) there exists ∈ [0, 1) such that ( 1 ) − ( 2 ) ⩽ Φ Ω ( 2 , 1 ), ∀ 1 , 2 ∈ Γ; (G2) there exists ∈ 1,1 (Γ ) for some > 0 such that = | Γ and for somĕ > 0, where Γ denotes the -neighborhood of Γ.
Remark 5.11. We note that the energy condition Ω ( ) < 0 (which is implicitly assumed even in the above proposition as a consequence of the hypothesis ⩾ > 0) ensures that any optimal curve touches the boundary in finite time in the associated optimal exit time problem. On the other hand, the case of Ω ( ) = 0 is still open, especially the analysis of the Aubry set on the boundary. For a state constrained problem, weak KAM aspects of the boundary behaviour of solutions were studied in [CCMW20]. 5.4.1. Aubry set and cut locus. Let be compact and be a weak KAM solution of (5.1). We define the projected Aubry set ( ) of as the subset of such that ∈ ( ) if there exists a -calibrated curve ∶ (−∞, +∞) → passing though . We also define the cut locus of , denoted by Cut ( ), as the set of points ∈ where no backward -calibrated curve ending at can be extended to a -calibrated curve beyond . In general we have the following inclusions: Notice that we just assume the solution of (5.24) to be uniformly continuous without any extra conditions on the initial data. So, there are a lot of technical points one needs to clear in order to deal with arbitrary initial conditions (see [Fat20]). 5.4.3. Applications to Riemmanian geometry. Now, suppose ( , ) is a complete Riemannian manifold, and is the distance function to a closed subset ⊂ . We denote by Sing * ( ) the set of points in ⧵ where is not differentiable.
In classical Riemmanian geometry, for any ∈ one denotes by Cut ( , ) ( ) the cut locus with respect to . It is well-known that, when is compact, such a cut locus Cut The difficulty for all these studies is an unavoidable dichotomy for cut points: the mixture of points with two different segments and conjugate points.
We now proceed to explain how to distinguish the study of these two sets by using the above methods. We will begin with another consequence of Proposition 5.16, for which we need the following definition: for a complete Riemannian manifold ( , ), we define   ( , ) = ( × ) ⧵  ( , ). The set  ( , ) contains a neighborhood of the diagonal Δ ⊂ × . In fact, we have   ( , ) = Sing * ( Δ ), the set of singularities in ( × ) ⧵ Δ of the distance function of points in × to the closed subset Δ . Therefore, Proposition 5.16 implies: Proposition 5.17 ([CCF19]). For every complete Riemannian manifold ( , ), the set is locally contractible. In particular, the set   ( , ) is locally path connected.
For a closed subset ⊂ , we define its Aubry set  * ( ) as the set of points ∈ ⧵ such that there exists a curve ∶ [0, +∞) → parameterized by arc-length such that ( ( )) = and = ( 0 ) for some 0 > 0. We remark that if is a bounded connected component of ⧵ , then ∩  * ( ) = ∅, and Sing * ( ) ∩ ⊂ is a homotopy equivalence (see also [Lie04] and Section 4.2.2). As for unbounded components, see also [CP01] for the Euclidean case.

CONCLUDING REMARKS
The study of singularities of solutions to HJ equation has made remarkable progress in the past decades. Many results that seemed impossible have been obtained, and connections with other domains have been established. Nevertheless, many interesting problems remain open. Some open problems were proposed in [CC18].
In [CC20], the uniqueness of strict singular characteristic on = ℝ 2 is proved when the initial point is not a critical point. However, the uniqueness issue is still open for higher dimensional manifolds. Recalling some results in [CCC19], assuming uniqueness for generalized characteristics, one can bridge the Aubry set (Mather set) and the invariant set of the associated semi-flow of generalized characteristics. Recently, relations between propagation of singularities and global dynamics of lower dimensional Hamiltonian systems have also been pointed out in [Zha20]. More concrete applications to problems from Hamiltonian dynamical systems in the scheme of Mather theory and weak KAM theory are expected, including applications to the study of Burgers turbulence as noted in [KS16]. [Zha20] Jianlu Zhang. Global behaviors of weak KAM solutions for exact symplectic twist maps. J. Differential Equations, 269 (7)