Mathematical analysis of modified level-set equations

The linear transport equation allows to advect level-set functions to represent moving sharp interfaces in multiphase flows as zero level-sets. A recent development in computational fluid dynamics is to modify the linear transport equation by introducing a nonlinear term to preserve certain geometrical features of the level-set function, where the zero level-set must stay invariant under the modification. The present work establishes mathematical justification for a specific class of modified level-set equations on a bounded domain, generated by a given smooth velocity field in the framework of the initial/boundary value problem of Hamilton-Jacobi equations. The first main result is the existence of smooth solutions defined in a time-global tubular neighborhood of the zero level-set, where an infinite iteration of the method of characteristics within a fixed small time interval is demonstrated; the smooth solution is shown to possess the desired geometrical feature. The second main result is the existence of time-global viscosity solutions defined in the whole domain, where standard Perron's method and the comparison principle are exploited. In the first and second main results, the zero level-set is shown to be identical with the original one. The third main result is that the viscosity solution coincides with the local-in-space smooth solution in a time-global tubular neighborhood of the zero level-set, where a new aspect of localized doubling the number of variables is utilized.


Introduction
The linear transport equation ∂ t f + v • ∇f = 0 describes the passive advection of a scalar quantity f by a velocity field v.We start with a brief overview of the fundamental role of the linear transport equation in fluid dynamics.Suppose that a domain Ω ⊂ R 3 is occupied by a fluid.The Lagrangian specification of a fluid flow is to look at the position of each fluid element, i.e., for each time t ∈ [0, ∞) the position of the fluid element being at ξ ∈ Ω at time τ ∈ [0, ∞), which is called the "fluid element (τ, ξ)", is denoted by X(t, τ, ξ).
Assuming that X(t, τ, ξ) = x is equivalent to ξ = X(τ, t, x) for all t, τ ∈ [0, ∞) and x, ξ ∈ Ω, one obtains the Eulerian specification of the fluid flow, i.e., the velocity field v defined as , which leads to Hence, X can be seen as the flow of the kinematic ordinary differential equation (ODE) x ′ (s) = v(s, x(s)), s ≥ 0.
With (1.1), we find that f satisfies the linear transport equation where ∇ = (∂ x 1 , ∂ x 2 , ∂ x 3 ).It is intuitively clear (and mathematically true as well) that the solution of (1.3) is given as f (t, x) = φ 0 (X(0, t, x)).(1.4)This observation leads to the method of characteristics for more general first order PDEs.Note that if v and φ 0 are not C 1 -smooth, the meaning of solution must be generalized.
A typical example of F being preserved by each fluid element is the density in an incompressible fluid, where the velocity v comes from the incompressible Navier-Stokes equations.We refer to [22] and [13] for recent development of mathematical analysis for the system of the linear transport equation and the incompressible Navier-Stokes equations.We refer also to [16] and [4] for generalization of ODE-based classical theory of the linear transport equations to the case with velocity fields being less regular.
We now discuss the transport equation in the context of the level-set method in twophase flow problems.Suppose that Ω is occupied by two immiscible fluids (distinguished by the superscript ±) in such a way that at t = 0 the domain Ω is divided into two disjoint connected open sets and their interface: Ω + (0) Ω with ∂Ω + (0) ∩ ∂Ω = ∅ is a connected open set filled by fluid + , Ω − (0) := Ω \ Ω + (0) is filled by fluid − and Σ(0) := ∂Ω + (0) ∩ ∂Ω − (0) = ∂Ω + (0) is the interface.Note that, for now, we discuss the case where Σ(0) does not touch ∂Ω, while in Subsection 2.2 we will consider the other case.We suppose that for each t > 0 the open set Ω + (t) := X(t, 0, Ω + (0)) (resp.Ω − (t) := X(t, 0, Ω − (0))) (1.5) is occupied by fluid + (resp.fluid − ) and the common interface of fluid ± is given as Σ(t) := X(t, 0, Σ(0)), (1.6) where the continuity of the flow X implies that (1.6) is well-defined even though no unique fluid elements are associated to the points on Σ(0).In other words, the velocity field v coming from the two-phase Navier-Stokes equations is assumed to be such that (1.2) generates a proper flow from Ω to itself; see Section 2 below for more details.The interface given as (1.6) is called a material interface, as opposed to a non-material interface formed by a two-phase flow with phase change (see [10] for investigations of (1.2) in the case of non-material interfaces).Let φ 0 : Ω → R be a smooth function such that φ 0 > 0 on Ω + (0) and φ 0 < 0 on Ω − (0), which implies that φ 0 = 0 only on Σ(0).We assign to each fluid element (0, ξ) the number φ 0 (ξ).Let F (t, ξ) be the label of the fluid element (0, ξ) at time t, which must be equal to φ 0 (ξ) for any t ≥ 0.Then, the Eulerian description f of F , i.e., f (t, x) := F (t, ξ)| ξ=X(0,t,x) = φ 0 (X(0, t, x)), satisfies the transport equation (1.3) with the representation (1.4).In particular, we have We call f a level-set function and the linear transport equation for level-set functions the level-set equation.Throughout the paper, the level-set means the zero level of a level-set function.Suppose that Σ(0) is equal to the level-set of a C 2 -function φ 0 such that ∇φ 0 = 0 on Σ(0), where Σ(0) is a C 2 -smooth closed surface (compact manifold without boundary).If X(t, τ, •) : Ω + (τ ) ∪ Σ(τ ) ∪ Ω − (τ ) → Ω + (t) ∪ Σ(t) ∪ Ω − (t) is a C 2 -diffeomorphism for each t, τ ≥ 0, we see that ∇f (t, x) = 0 on Σ(t), ∀ t ≥ 0, (1.7) and Σ(t) keeps being a C 2 -smooth closed surface for all t > 0. In particular, the unit normal vector ν(t, x) and the total (twice the mean) curvature κ(t, x) of Σ(t) at each point x are well-defined and represented as where | • | denotes the Euclidean norm.In a two-phase flow problem, the Navier-Stokes equations for the velocity field are coupled with the level-set equation on the interface through ν and κ.We refer to [1] and [23] for recent developments of mathematical analysis of multiphase flow problems and to [19] for mathematical analysis of level-set methods beyond fluid dynamics.
In computational fluid dynamics, the level-set equation is often used to represent a moving interface.In this context, the level-set approach has several advantages, such as a very accurate approximation of the mean curvature and a straightforward handling of topological changes of the interface (e.g., breakup and coalescence of droplets).In a numerical simulation, it is common to choose an initial level-set function φ 0 that coincides locally with the signed distance function of a given closed surface Σ(0), where φ 0 is characterized by |∇φ 0 | ≡ 1 in a neighborhood of Σ(0).However, it is known that the local signed distance property is not preserved by (1.3), i.e., f (t, •) does not coincide even locally with the signed distance function of Σ(t) for t > 0 in general.In fact, a short calculation [18] shows that, along each curve holds for a classical solution f of the standard level-set equation (1.3).Here, •, • stands for the inner product of R 3 .Unfortunately, problems with the numerical accuracy emerge if |∇f | becomes too small or too large, which is the case in general, even though the nondegeneracy condition (1.7) is mathematically guaranteed.This is an important point in practice: on the one hand, it must be possible to resolve |∇f | by the computational mesh, which implies an upper limit for |∇f | related to the mesh size; on the other hand, too small values of |∇f | lead to an inaccurate positioning of the interface, the normal field, the mean curvature field, etc. in the numerical algorithm.In order to keep the norm of the gradient approximately constant, so-called "reinitialization" methods [28,27,29] have been developed.Typically, an additional PDE is solved that computes a new function f with the same zero contour but with a predefined gradient norm (e.g., |∇ f | = 1 on the level-set).In [26], the authors developed an alternative numerical method to control the size of the gradient based on the level-set equation with a suitable source term that is determined by an extra equation, where the reinitialization procedure was no longer necessary.These methods might be computationally expensive.Moreover, it is known that many reinitialization methods struggle with extra difficulties if the interface touches the domain boundary ∂Ω [25] (i.e., if a so-called "contact line" is formed; see Section 2.2).
In order to control the norm of the gradient within a single PDE, the following nonlinear modification of the level-set equation has been introduced1 in the literature of computational fluid dynamics (see [18,20] for details): (1.9) Note that, from here on, we rather use φ instead of f to stress the fact that we deal with a modified level-set equation.Since the source term on the right-hand side is chosen proportional to the level-set function φ, the modification term vanishes on the zero interface and, as seen in Section 2, one can show that the evolution of the zero level-set is unaffected by the modification (in fact, the configuration component of the characteristic ODEs for (1.9) becomes (1. 2) on the level-set).Moreover, a suitable choice of the nonlinear function R allows to control the evolution of |∇φ| (at least locally at the level-set).A formal calculation [18] shows that, by choosing along each characteristic curve x(t) of (1.9) such that x(t) ∈ Σ(t) for all t ≥ 0 or, equivalently, We will prove this statement rigorously using the method of characteristics (see problem (2.1) in Section 2).Notice that, in general, the property (1.11)only holds locally at the level-set.The signed distance function of Σ(t) itself does not solve (1.3) nor (1.9) in general, but rather a non-local PDE, cf.Lemma 3.1 in [20].From the numerical perspective, it is of interest to study a formulation like (1.9) because the advection of the interface and the preservation of the norm of the gradient are combined into one single PDE, i.e., into a monolithic approach.In addition to the choice (1.10), we will also study a variant in which a cut-off function is applied such that the nonlinear source term is only active in a neighborhood of the level-set (see problem (3.1) below) and another simpler modified level-set equation, which only keeps the norm of the gradient within given bounds; see the initial value problem (2.21) below.We refer to [18] for a numerical investigation of (1.9).Now, we move to the mathematical analysis of (1.9).It is important to note that, due to the nonlinear source term in (1.9), the ODE (1.2) is no longer the characteristic ODE of (1.9); instead, the system of ODEs (2.3)-(2.5)defines the characteristic curves of (1.9).See Appendix 1 for more details on the method of characteristics as applied to Hamilton-Jacobi equations.Furthermore, since (1.9) is a first order fully nonlinear PDE, the mathematical analysis of (1.9) is not at all as simple as that of (1.3), even if v and R are smooth enough.Existence of a classical solution on the whole domain within an arbitrary time interval is no longer possible in general, i.e., the notion of viscosity solutions is necessary.Then, it is expected that the following statements hold true for R given by (1.10) or its variants: (i) (1.9) provides a level-set that is identical to the original one provided by (1.3) for all t ≥ 0; (ii) (1.9) admits a unique classical solution φ at least in a t-global tubular neighborhood of the level-set (see its definition in Section 2) so that the normal field and mean curvature field are well-defined by φ and the property (1.11) (or, less restrictively, an a priori bound of |∇φ|) holds on the level-set for all t ≥ 0; (iii) (1.9) admits a unique global-in-time viscosity solution defined on [0, ∞) × Ω; (iv) If initial data is C 2 -smooth, the viscosity solution φ coincides with the local-inspace classical solution φ in a t-global tubular neighborhood of the level-set, i.e., partial C 2 -regularity of φ.
The purpose of the current paper is to provide full proofs of (i)-(iv) for the problem (1.9) with a given smooth velocity field v and the above-mentioned R, where mathematical analysis on the system of (1.9) and Navier-Stokes type equations for v is an interesting future work.We will exploit the method of characteristics to show (ii) and (i) for the smooth solution; usually, the method of characteristics works only within a short time interval; however, since the nonlinearity of (1.9) becomes arbitrarily small near the level-set, on which |∇φ| is appropriately controlled as well, we may iterate the method of characteristics countably many times with a shrinking neighborhood of the level-set to construct a time global solution defined in a t-global tubular neighborhood of the level-set.To show (iii) and (i) for the viscosity solution, we will apply the standard theory of viscosity solutions to (1.9) with a boundary condition arising formally from the classical solutions.To prove (iv), we adapt the idea of localized doubling the numbers of variables for the comparison principle of viscosity solutions within a cone of dependence; the difficulty is that we cannot have a cone of dependence that contains a t-global tubular neighborhood of the level-set; we will demonstrate a new version of localized doubling the numbers of variables with an unusual choice of a penalty function in a t-global tubular neighborhood of the level-set.We emphasize that the result (iv) is particularly important from application points of view in the sense that, once a continuous viscosity solution is obtained, it provides the level-set, its normal field and mean curvature field with the necessary regularity being guaranteed; numerical construction of a viscosity solution on the whole domain would be easier than that of a local-in-space smooth solution; there is huge literature pioneered by [12] on rigorous numerical methods of viscosity solutions.
Finally, we compare our results on (i)-(iv) with the work [20].In [20], the author formulated a modification of the initial value problem of a general Hamilton-Jacobi equation with an autonomous Hamiltonian (including the linear transport equation with v = v(x)) on the whole space and proved the existence of a unique viscosity solution, where the modification is essentially the same as (1.10); owing to the modification, he showed that the (continuous) viscosity solution of the modified equation stays close to the signed distance function of its own level-set with good upper/lower estimates, from which he obtained differentiability of the viscosity solution on the level-set with the norm of the derivative to be one.Additional regularity of the viscosity solution away from the levelset remained open.Our current paper provides a stronger partial regularity property of viscosity solutions in the same context as [20].
In the current paper, we consider a more general situation based on the theory of ODEs on closed sets and flow invariance.For this purpose, we introduce the so-called Bouligand contingent cone T K (x) of an arbitrary closed set K ⊂ R d at x ∈ K as We say that y ∈ R d is subtangential to K at a point x ∈ K, if y ∈ T K (x).Note that T K (x) = R d for x in the interior of K. We shall employ the following result on flow invariance.
continuous and locally Lipschitz in x ∈ K.Then, the following holds true: (a) Suppose that ±g are subtangential to K, i.e., ±g(s, x) ∈ T K (x), ∀ s ∈ J, ∀ x ∈ K.
Then, given any s 0 ∈ J and x 0 ∈ K, the initial value problem x ′ (s) = g(s, x(s)), x(s 0 ) = x 0 has a unique solution defined on J that stays in K.
(b) Suppose that ±g are subtangential to ∂K, i.e., Then, the sets K, ∂K and K \ ∂K are flow invariant.
See Appendix 2 for more on flow invariance and Lemma 2.1.
Now we state the hypothesis on the velocity field v: ) is locally Lipschitz in x ∈ Ω; v is three times partially differentiable in x; all of the partial derivatives of v belong to We remark that the upcoming nonlinear modification of the linear transport equation requires C 3 -smoothness of v in x so that its characteristic ODEs are properly defined; due to Lemma 2.1, Ω, ∂Ω and Ω are flow invariant with respect to the flow X of (1.2); X(s, τ, •) is continuous on Ω and C 3 -smooth in Ω.If Ω is a cube, for instance, (H2) implies: at each vertex, v must be equal to zero, while on each edge, v may take nonzero values parallel to the edge.
2.1 Case 1: problem with level-set being away from ∂Ω Let f be the solution of the original level-set equation (1.3).We keep the notation and configuration in (1.5) and (1.6), where we repeat Note that due to (H2) we have where we always consider ε > 0 such that Σ ε (t) ⊂ Ω.We say that a set and there exists a nonincreasing function ε The problem under consideration is to find a tubular neighborhood Θ of (2.1) Note that this problem makes sense with φ 0 being defined only in a neighborhood of Σ(0), e.g., φ 0 is given as the local signed distance function of Σ(0).We state the first main result of this paper.
Theorem 2.2.There exists a tubular neighborhood Θ of the level-set {Σ(t)} t≥0 for which (2.1) admits a unique C 2 -solution φ satisfying Proof.We treat the PDE in (2.1) as the Hamilton-Jacobi equation The characteristic ODEs of (2.2) are given as (see Appendix 1 for more details) smooth because of (H1), which is required for the method of characteristics.We sometimes use the notation x(s; ξ), p(s; ξ), Φ(s; ξ) to specify the initial point.Our proof is based on the investigation of the variational equations of the characteristic ODEs (2.3)-(2.5)for each ξ ∈ Σ(0) to ensure the invertibility of x(s; •) in a small neighborhood of Σ(s) for each s ≥ 0. In a general argument of the method of characteristics, such invertibility is proven only within a small time interval.Below, we will show an iterative scheme to extend the time interval in which the invertibility holds with a shrinking neighborhood of Σ(s) as s becomes larger.
If we choose φ 0 which coincides with the (local) signed distance function of Σ(0), then the solution φ obtained in Theorem 2.2 satisfies Let φ 0 be a C 2 -smooth R-valued function defined in an open set containing K ∪ Ω such that φ 0 > 0 in K, φ 0 < 0 outside K, ∇φ 0 = 0 on ∂K.
Then, we have Let f be the solution of the original level-set equation (1.3).Define where we note that ∂Ω is invariant under the flow X of (1.2), and hence Σ(t) always touches ∂Ω.If we follow the same argument as given in Subsection 2.1, we would face non-trivial issues at/near Σ(t)∩∂Ω coming from the behavior of the variational equations of the characteristic ODEs on ∂Ω.Hence, we modify the reasoning of Subsection 2.1 so that Σ(t) ∩ ∂Ω is not involved.
Let {Ω k } k∈N be a monotone approximation of Ω, i.e., each Ω k is an open subset of Ω; Ω k ⊂ Ω k+1 for all k ∈ N; for any G ⊂ Ω compact, there exists where for each k ∈ N we have ε > 0 depending on k such that Now, we may follow the same argument as given in Subsection 2.1 with Σ k (0) in place of Σ(0) to obtain the following objects: The method of characteristics implies that we obtain a unique C 2 -solution φ of (2.1).We remark that We summarize the result: Theorem 2.3.There exists a tubular neighborhood Θ in the sense of (2.20) of the level-set {Σ(t)} t≥0 touching ∂Ω for which (2.1) admits a unique C 2 -solution φ satisfying Let us note in passing that another method to investigate a problem with the level-set touching ∂Ω would run via smooth extension of v outside Ω.The following steps would suffice: Step 1. Extend the velocity field v to R × R 3 as a C 3 -function by means of Whitney's extension theorem [30] or the extension operators in Sobolev spaces (see, e.g., Chapter 5 of [2]) together with the Sobolev embedding theorem, where additional conditions on v and Ω are required accordingly.
Step 3. Solve the extended problem in the same way as Subsection 2.1, where each characteristic curve starting at a point of Ω stays inside Ω forever due to the flow invariance of Ω and ∂Ω under X.

Simpler nonlinear modification
The nonlinear modification in (2.1) is designed to preserve |∇φ| along each characteristic curve on the level-set.If we relax the requirement, i.e., we only ask for an a priori bound of |∇φ| on the level-set, we may use a much simpler modification.
We take the same configuration of the level-set considered in Subsection 2.1.Let α > 0 be a constant such that With a constant β > α, we consider which is seen as the Hamilton-Jacobi equation (2.2) with The characteristic ODEs in this case are given as As long as the above characteristic ODEs have solutions, |p(s)| 2 evolves as which leads to which is a contradiction; a fully analogous argument yields the lower bound.Therefore, we may apply the reasoning of Subsection 2.1 to the following problems: find a tubular neighborhood Θ of {Γ(t)} t≥0 and a (2.21) We obtain the following result.
Theorem 2.4.Suppose that β −α ≤ |∇φ 0 | ≤ β +α on Σ(0).Then, there exists a tubular neighborhood Θ of the level-set {Σ(t)} t≥0 for which (2.21) admits a unique C 2 -solution φ satisfying Note that Theorem 2.4 is interesting for numerical purposes because it is usually not important to exactly keep |∇φ| ≡ 1, but rather to stay away from extreme values of the gradient norm.Therefore, the problem (2.21) should be investigated in more detail from the numerical perspective in the future.

Viscosity solution on the whole domain
Let Ω ⊂ R 3 be a bounded connected open set.Let v = v(t, x) be a given function belonging to C 0 ([0, ∞) × Ω; R 3 ) ∩ C 1 ([0, ∞) × Ω; R 3 ) and satisfying (H2)-(H3) (we do not need C 3 -regularity in x).Let Σ(0) ⊂ Ω be a closed 2-dimensional surface (topological manifold) and let φ 0 : Ω → R be a C 0 -function such that {x ∈ Ω | φ 0 (x) = 0} = Σ(0), where f (t, x) := φ 0 (X(0, t, x)) is not necessarily a classical solution of the original linear transport equation.We keep the notation and configuration in (1.5) and (1.6) with the current X and φ 0 , where we repeat If we deal with the Hamilton-Jacobi equation in (2.1) on the whole space Ω, the nonlinear term would suffer from singularity, i.e., (∇v)p, p |p| −2 cannot be continuous at p = 0. Hence, also for a simpler structure near ∂Ω, we introduce a smooth cut-off so that the nonlinearity is smooth and effective only around the level-set {Σ(t)} t≥0 , where we note carefully that the level-set {Σ(t)} t≥0 is not the one determined by the upcoming viscosity solution, i.e., such cut-off can be given independently from the unknown function.We also note that ∇v is not required on ∂Ω due to the cut-off.We explain two ways to introduce a suitable cut-off.
The first way is simpler but available only for the problem within each finite time interval.Fix an arbitrary terminal time T > 0. For ε > 0, let K ε ⊂ Ω denote the intersection of Ω and the ε-neighborhood of ∂Ω.Then, since Σ(t) never touches ∂Ω within [0, T ], we find ε > 0 such that non-negative smooth transition from 1 to 0 otherwise, monotone smooth transition from 1 to 0 otherwise.
This choice of cut-off is uncomplicated since we do not need detailed information on asymptotics of dist(Σ(t), ∂Ω) as t → ∞; however, R T depends on the terminal time and a time global analysis is impossible.Note that the above specific choice of 1/3, 2/3, etc., in η 2 is not essential.
The second way is to allow t-dependency for cut-off.Since Σ(t) never touches ∂Ω within [0, ∞), we find a smooth function ε Then, we take a smooth function η non-negative smooth transition from 1 to 0 otherwise, where we omit an explicit formula of such η 1 , and define the continuous function R as where η 2 is the one in R T .
Due to (H3), there exists a constant V 0 > 0 such that In the rest of the paper, we take R given as in (3.1).Note that all upcoming results hold also for R T as long as the terminal time T is unchanged.For an arbitrary T > 0, we discuss existence of a unique viscosity solution φ of Before going on with (3.3), we recall the definition of viscosity (sub/super)solutions of a general first order Hamilton-Jacobi equation of the form is a given continuous function and u : O → R is the unknown function.Our evolutional Hamilton-Jacobi equation is also seen in this form with z = (t, x).In the literature, (3.4) is often treated as a typical example of degenerate second order PDEs (i.e., the second order term is completely degenerate to be 0).To state the definition, we introduce the upper semicontinuous envelope Note that u * is upper semicontinuous and u * is lower semicontinuous; if u is upper semicontinuous (resp.lower semicontinuous), we have u = u * (resp.u = u * ).
Definition.A function u : O → R is a viscosity subsolution (resp.supersolution) of (3.4), provided • u * is bounded from the above (resp.u * is bounded from the below); A function u : O → R is a viscosity solution of (3.4), if it is both a viscosity subsolution and supersolution of (3.4).
It is well-known that max, min in the definition can be replaced by the local maximum, local minimum, respectively.

Existence of viscosity solution
We state the main result of this subsection.Theorem 3.1.Let T > 0 be arbitrary.There exists a viscosity solution φ ∈ C 0 ([0, T ) × Ω; R) of (3.3), i.e., φ satisfies the first equation in (3.3) in the sense of the above definition and the initial/boundary condition strictly.Such a viscosity solution is unique.Furthermore, it holds that

Remark. Theorem 3.1 implies the existence of a unique viscosity solution
non-negative smooth transition from V 0 to 0 otherwise, where ε(t) ≥ ε(T ) for all t ∈ [0, T ] and S is such that S ≡ 0 near ∂Ω for all t ∈ [0, T ].Note that S is smooth except on ∪ 0≤t≤T ({t} × Σ(t)).
We remark that the result and reasoning of this subsection hold also for the Hamilton-Jacobi equation corresponding to (2.21), where we need to add suitable cut-off to the nonlinearity in (2.21).To see this, observe that the mapping u → u(β − |p|) is not monotone for all p ∈ R 3 ; hence, we define

C 2 -regularity of viscosity solution near the level-set
We prove that the viscosity solution of (3.3) coincides with the classical solution of (2.1) in a t-global tubular neighborhood of the level-set.Note that |∇φ| is nicely controlled on the level-set in (2.1) and there exists a tubular neighborhood Θ in which η 1 (t, x)η 2 (|∇φ(t, x)|) ≡ 1, i.e., the solution of (2.1) satisfies the Hamilton-Jacobi equation in (3.3) within Θ.
Our proof is based on the technique known as doubling the number of variables, which is standard in proofs of comparison principles for viscosity solutions; more precisely, we adapt the localized version of this technique to our situation with an unusual choice of a penalty function.The outcome of standard "localized doubling the number of variables" (see, e.g., Theorem 3.12 of [6]) for a Hamilton-Jacobi equation states the following: Let u, ũ be viscosity solutions of (3.8) defined in a cone The cone is the region of dependence for general first order Hamilton-Jacobi equations, i.e., the value u(b, z) is determined by the information only on the bottom of C (the speed of propagation is finite).If we directly apply the result to our case, we have to take such a cone contained in the tubular neighborhood Θ, which implies that the time interval [0, b] must be small.This is the nontrivial aspect of this subsection.In order to overcome the difficulty, we will introduce an unusual penalty function (i.e., h(ū 2 ) below) in localized doubling the number of variables that consists of the classical solution itself.Therefore, our technique is specialized for local comparison of a viscosity solution and a classical solution.We emphasize that if both solutions are defined on the whole domain, the issue is obvious, but otherwise not.
We rewrite (3.11) in the form of (3.9) with Due to the condition of v and the definition of R (see (3.1)), there exists a constant For a technical reason, we consider another tubular neighborhood Θ ⊂ Θ of {Σ(t)} t≥0 such that φ is defined on the closure of Θ, and (partly) describe ΘT := Θ| 0≤t≤T as a family of cylinders, i.e., a foliation: • Consider the function ū(t, x) := e αt w(t, x), where ū solves in the classical sense • With a constant m 0 > 0, define Since ∇ū = 0 near ∪ 0≤t≤T ({t}×Σ(t)), we may choose m 0 > 0 (possibly very small) so that Γ T is contained in ΘT and ∂Γ T \ Γ T | t=0,T = (A m 0 ∪ A −m 0 )| 0<t<T , while Γ T contains the [0, T ]-part of a tubular neighborhood of {Σ(t)} 0≤t≤T .Note that such an m 0 depends, in general, on T .Theorem 3.2.It holds that φ ≡ φ on Γ T , where T > 0 is arbitrary.
Remark.Γ T would become narrower as T gets larger.In order to find a more optimal region on which φ ≡ φ, one can iterate Theorem 3.2 for T = T 0 , T = 2T 0 , T = 3T 0 , . .., with a small T 0 > 0.
Proof of Theorem 3.2.We proceed by contradiction.Suppose that the assertion does not hold.We assume max Γ T (φ − φ) > 0 (in the other case, we switch φ and φ).
Since the level-set of w and that of w are identical, we see that (t * , x * ) Let δ > 0 be a constant such that Take a constant M > 0 such that M ≥ max if r ≤ (m * ) 2 + δ, monotone transition between 0 and 3M , otherwise.
We confirmed that the point (t 0 , x 0 ) is an interior point of Γ T , or t 0 = T and x 0 is an interior point of Γ T | t=T ; (s 0 , y 0 ) is an interior point of Γ T , or s 0 = T and y 0 is an interior point of Γ T | t=T , which enables us to test w and w at (t 0 , x 0 ), (s 0 , y 0 ) in terms of viscosity sub-/supersolutions (see Lemma in Section 10.2 of [17] for a remark on the case t 0 = T or s 0 = T ).Observe that the mapping (t, x) → F ε,λ (t, x, s 0 , y 0 ) takes the maximum at the point (t 0 , x 0 ); then, introducing the C 1 -function ψ as we see that the definition of F ε,λ implies that w − ψ takes a maximum at (t 0 , x 0 ).
Therefore, we obtain Similarly, the mapping (s, y) → −F ε,λ (t 0 , x 0 , s, y) takes a minimum at the point (s 0 , y 0 ); then, introducing the C 1 -function ψ as we see that w − ψ takes a minimum at (s 0 , y 0 ).
We remark that the result and reasoning of this subsection hold also for the problems (2.21) and (3.3) with R given in (3.7) and C The map F (s, ξ) := Q(s; ξ) = (s, x(s; ξ)) is C 1 -smooth as long as the characteristic ODEs have solutions.Furthermore, it holds that F (0, ξ) = (0, ξ), det D (s,ξ) F (0, ξ) = 1, (A.11) where D (s,ξ) F is the Jacobian matrix of F .Therefore, for each ξ ∈ R N , the inverse map theorem guarantees that there exist two sets, a neighborhood O 1 of (0, ξ) and a neighborhood O 2 of F (0, ξ) = (0, ξ), such that F : O 1 → O 2 is a C 1 -diffeomorphism.We obtain the inverse map of F as where ξ = ϕ(t, x) is the point for which x(s; ξ) passes through x at s = t.The essential point is to obtain open sets O 1 and O 2 = F (O 1 ) such that F : O 1 → O 2 is a C 1diffeomorphism, no matter how we find them.The easiest way is to use the inverse map theorem around a single point (0, ξ) based on (A.11 ).The invertibility of F , or the invertibility of x(s; ξ) for fixed s, on a wider region requires injectivity, which will be an additional issue to be verified.
We proceed, assuming that O 1 ⊂ R × R N is a neighborhood of a subset of {0} × R N and F :   x ′ (t) = g t, x(t) on J, x(t 0 ) = x 0 has a local (classical) solution for every (t 0 , x 0 ) ∈ J × V , i.e. there is ε = ε(t 0 , x 0 ) > 0 and a C 1 -function x : I ε → R N , with I ε = (t 0 − ε, t 0 + ε), such that (A.12 ) is satisfied in every point.
In this situation, a closed set K ⊂ V is said to be positive (negative) flow invariant for (A.12 ), if every solution of (A.12 ) that starts in t = t 0 at a point x 0 ∈ K stays inside K for all (admissible) t > t 0 (t < t 0 ).In this case, one also speaks of forward (backward) invariance of K for the right-hand side g.A closed set K is called flow invariant, or just invariant, for (A.12 ), if K is both positive and negative flow invariant for (A.12 ).
Since classical solutions for (A.12 ) with (only) continuous right-hand side need not be unique, it might happen that, for given closed K ⊂ V , one solution starting in x 0 stays in K, while another solution leaves K.The autonomous standard example for nonuniqueness, namely g(x) = 2 |x| on V = R, already shows this behavior with K := {0} and x 0 = 0.One therefore calls a closed K ⊂ V weakly (positively or negatively) flow invariant for (A.12 ), if t 0 ∈ J and x 0 ∈ K implies the existence of one solution staying in K (for t > t 0 or t < t 0 , respectively).
In this case, the Picard-Lindelöf theorem yields unique solvability of (A.12 ).Therefore, weak flow invariance then is the same as flow invariance.Let us note in passing that forward (backward) unique solvability of (A.12 ) for continuous g holds under weaker additional assumptions such as one-sided Lipschitz continuity in x.More precisely, if for every (t 0 , x 0 ) ∈ J × V there is δ = δ(t 0 , x 0 ) > 0 and a k = k t 0 ,x 0 ∈ L 1 (J) such that (A.14) g(t, x) − g(t, x), x − x ≤ k(t)|x − x| 2 for t ∈ J, x, x ∈ B δ (x 0 ) ∩ V, then forward uniqueness holds for (A.12 ).
Consequently, the condition g(t 0 , x 0 ) ∈ T K (x 0 ) with the so-called "tangent cone" T K (x) = z ∈ X : lim h→0+ h −1 dist (x + hz, K) = 0 for x ∈ K is a necessary condition, where X = R N .Actually, it turns out that the apparently weaker condition (A.15) g(t, x) ∈ T K (x) for t ∈ J, x ∈ K