1 Introduction

The linear transport equation \(\frac{\partial f}{\partial t}+v\cdot \nabla f=0\) 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 \(\Omega \subset {{\mathbb {R}}}^3\) 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 \(t\in [0,\infty )\) the position of the fluid element being at \(\xi \in \Omega \) at time \(\tau \in [0,\infty )\), which is called the “fluid element \((\tau ,\xi )\)”, is denoted by

$$\begin{aligned} X(t,\tau ,\xi ). \end{aligned}$$

Then, the velocity of each fluid element \((\tau ,\xi )\) is defined for each \(t\ge 0\) as

$$\begin{aligned} \frac{\partial }{\partial t}X(t,\tau ,\xi ). \end{aligned}$$

Assuming that \(X(t,\tau ,\xi )=x\) is equivalent to \(\xi =X(\tau ,t,x)\) for all \(t,\tau \in [0,\infty )\) and \(x,\xi \in \Omega \), one obtains the Eulerian specification of the fluid flow, i.e., the velocity field v defined as

$$\begin{aligned} v:[0,\infty )\times \Omega \rightarrow {{\mathbb {R}}}^3,\quad v(t,x):=\frac{\partial }{\partial t}X(t,0,\xi )\Big |_{\xi =X(0,t,x)}, \end{aligned}$$

which leads to

$$\begin{aligned} \frac{\partial }{\partial t}X(t,0,\xi )=v(t,X(t,0,\xi )),\quad \forall \, t\ge 0,\,\,\,\forall \,\xi \in \Omega . \end{aligned}$$

Hence, X can be seen as the flow of the kinematic ordinary differential equation (ODE)

$$\begin{aligned} x'(s)=v(s,x(s)),\quad s\ge 0. \end{aligned}$$

Let \(F(t,\xi )\) be a scalar quantity at time t that is associated with the fluid element \((0,\xi )\). The Eulerian description f of F is defined as

$$\begin{aligned} f: [0,\infty )\times \Omega \rightarrow {{\mathbb {R}}},\quad f(t,x):=F(t,\xi )\Big |_{\xi =X(0,t,x)}. \end{aligned}$$

Suppose that \(F(t,\xi )\equiv \phi ^0(\xi )\), i.e., each fluid element \((0,\xi )\) preserves the quantity \(F(0,\xi )=\phi ^0(\xi )\). Then, noting that \(f(t,X(t,0,\xi ))\equiv F(t,\xi )\), we have the following identity

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} f(t,X(t,0,\xi ))=0,\quad \forall \, t\ge 0,\,\,\,\forall \,\xi \in \Omega . \end{aligned}$$

With (1.1), we find that f satisfies the linear transport equation

$$\begin{aligned} \left\{ \begin{array}{lll} &{} \frac{\partial f}{\partial t}(t,x)+v(t,x)\cdot \nabla f(t,x)=0 \quad \text{ in } (0,\infty )\times \Omega , \\ &{}f(0,\cdot )=\phi ^0\quad \text{ on } \Omega , \end{array} \right. \end{aligned}$$

where \(\nabla =(\partial _{x_1},\partial _{x_2},\partial _{x_3})\). It is intuitively clear (and mathematically true as well) that the solution of (1.3) is given as

$$\begin{aligned} f(t,x)=\phi ^0(X(0,t,x)). \end{aligned}$$

This observation leads to the method of characteristics for more general first order partial differential equations (PDEs). Note that if v and \(\phi ^0\) 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 [24] 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 [17] 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 two-phase flow problems. Suppose that \(\Omega \) is occupied by two immiscible fluids (distinguished by the superscript ±) in such a way that at \(t=0\) the domain \(\Omega \) is divided into two disjoint connected open sets and their interface: \(\Omega ^+(0)\subsetneq \Omega \) with \(\partial \Omega ^+(0)\cap \partial \Omega =\emptyset \) is a connected open set filled by fluid\({}^+\), \(\Omega ^-(0):=\Omega {\setminus } \overline{\Omega ^+(0)}\) is filled by fluid\({}^-\) and \(\Sigma (0):=\partial \Omega ^+(0)\cap \partial \Omega ^-(0)=\partial \Omega ^+(0)\) is the interface. Note that, for now, we discuss the case where \(\Sigma (0)\) does not touch \(\partial \Omega \), while in Sect. 2.2 we will consider the other case. We suppose that for each \(t>0\) the open set

$$\begin{aligned} \Omega ^+(t):=X(t,0,\Omega ^+(0))\qquad (\text{ resp. } \Omega ^-(t):=X(t,0,\Omega ^-(0))) \end{aligned}$$

is occupied by fluid\({^+}\) (resp. fluid\(^{-}\)) and the common interface of fluid\(^{\pm }\) is given as

$$\begin{aligned} \Sigma (t):=X(t,0, \Sigma (0)), \end{aligned}$$

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 \(\Sigma (0)\). 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 \({\bar{\Omega }}\) to itself with \(\partial \Omega \) being flow invariant; see Sect. 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 \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be a smooth function such that \(\phi ^0>0\) on \(\Omega ^+(0)\) and \(\phi ^0<0\) on \(\Omega ^-(0)\), which implies that \(\phi ^0=0\) only on \(\Sigma (0)\). We assign to each fluid element \((0,\xi )\) the number \(\phi ^0(\xi )\). Let \(F(t,\xi )\) be the label of the fluid element \((0,\xi )\) at time t, which must be equal to \(\phi ^0(\xi )\) for any \(t\ge 0\). Then, the Eulerian description f of F, i.e., \(f(t,x):=F(t,\xi )|_{\xi =X(0,t,x)}=\phi ^0(X(0,t,x))\), satisfies the transport equation (1.3) with the representation (1.4). In particular, we have

$$\begin{aligned} \Omega ^+(t)= & {} \{x\in \Omega \,|\, f(t,x)>0\},\quad \Omega ^-(t)=\{x\in \Omega \,|\, f(t,x)<0\},\\ \Sigma (t)= & {} \{x\in \Omega \,|\, f(t,x)=0 \},\quad \forall \, t\ge 0. \end{aligned}$$

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 \(\Sigma (0)\) is equal to the level-set of a \(C^2\)-function \(\phi ^0\) such that \(\nabla \phi ^0\ne 0\) on \(\Sigma (0)\), where \(\Sigma (0)\) is a \(C^2\)-smooth closed surface (compact manifold without boundary). If \(X(t,\tau ,\cdot ):\Omega ^+(\tau )\cup \Sigma (\tau )\cup \Omega ^-(\tau )\rightarrow \Omega ^+(t)\cup \Sigma (t)\cup \Omega ^-(t)\) is a \(C^2\)-diffeomorphism for each \(t,\tau \ge 0\), we see that

$$\begin{aligned} \nabla f(t,x)\ne 0 \quad \text{ on } \Sigma (t), \forall \,t\ge 0, \end{aligned}$$

and \(\Sigma (t)\) keeps being a \(C^2\)-smooth closed surface for all \(t>0\). In particular, the unit normal vector \( \nu (t,x)\) and the total (twice the mean) curvature \(\kappa (t,x)\) of \(\Sigma (t)\) at each point x are well-defined and represented as

$$\begin{aligned} \nu (t,x)=\frac{\nabla f(t,x)}{|\nabla f(t,x)|},\quad \kappa (t,x)= -\nabla \cdot \nu (t,x), \end{aligned}$$

where \(|\cdot |\) 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 \(\nu \) and \(\kappa \). We refer to [1] and [25] for recent developments of mathematical analysis of multiphase flow problems and to [20] 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 \(\phi ^0\) that coincides locally with the signed distance function of a given closed surface \(\Sigma (0)\), where \(\phi ^0\) is characterized by \(|\nabla \phi ^0| \equiv 1\) in a neighborhood of \(\Sigma (0)\). However, it is known that the local signed distance property is not preserved by (1.3), i.e., \(f(t,\cdot )\) does not coincide even locally with the signed distance function of \(\Sigma (t)\) for \(t>0\) in general. In fact, a short calculation [19] shows that, along each curve \(x(\cdot )\) determined by (1.2) (it is called a characteristic curve) such that \(x(t)\in \Sigma (t)\),

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}|\nabla f(t,x(t))| = -|\nabla f|\langle (\nabla v) \nu , \nu \rangle \, (t,x(t)) \end{aligned}$$

holds for a classical solution f of the standard level-set equation (1.3). Here, \(\langle \cdot , \cdot \rangle \) stands for the inner product of \({{\mathbb {R}}}^3\). Unfortunately, problems with the numerical accuracy emerge if \(|\nabla f|\) becomes too small or too large, which is the case in general, even though the non-degeneracy condition (1.7) is mathematically guaranteed. This is an important point in practice: on the one hand, it must be possible to resolve \(|\nabla f|\) by the computational mesh, which implies an upper limit for \(|\nabla f|\) related to the mesh size; on the other hand, too small values of \(|\nabla f|\) 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 [29, 30] have been developed. Typically, an additional PDE is solved that computes a new function \({\tilde{f}}\) with the same zero contour but with a predefined norm of the gradient (e.g., \(|\nabla {\tilde{f}}|=1\) on the level-set). We refer to Section 11.6 in [28] for a critical assessment of this approach and to [21] for rigorous mathematical analysis of the reinitialization process. In [27], 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 \(\partial \Omega \) [16] (i.e., if a so-called “contact line” is formed; see Sect. 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 introduced in the literature of computational fluid dynamics (see [19, 22] for details):

$$\begin{aligned}{} & {} \left\{ \begin{array}{lll} &{} \frac{\partial \phi }{\partial t}(t,x)+v(t,x)\cdot \nabla \phi (t,x)=\phi (t,x)R(t,x,\nabla \phi (t,x)) \quad \text{ in } \Theta \subseteq (0,\infty )\times \Omega , \\ &{} \phi (0,\cdot )=\phi ^0 \quad \text{ on } \Omega . \end{array} \right. \end{aligned}$$

Note that, from here on, we rather use \(\phi \) 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 \(\phi \), the modification term vanishes on the zero interface and, as seen in Sect. 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 \(|\nabla \phi |\) (at least locally at the level-set). A formal calculation [19] shows that, by choosing,Footnote 1

$$\begin{aligned} R(t,x,p) := \left\langle \nabla v(t,x) \frac{p}{|p|} , \frac{p}{|p|} \right\rangle , \end{aligned}$$

we indeed obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}|\nabla \phi (t,x(t))|\equiv 0, \quad \forall \,t\ge 0 \end{aligned}$$

along each characteristic curve x(t) of (1.9) such that \(x(t)\in \Sigma (t)\) for all \(t\ge 0\) or, equivalently,

$$\begin{aligned} \forall \, t\in [0, \infty ),\,\,\,\forall \,x\in \Sigma (t),\,\,\,\exists \, \xi \in \Sigma (0) \text{ such } \text{ that } |\nabla \phi (t,x)|=|\nabla \phi ^0(\xi )|. \end{aligned}$$

We will prove this statement rigorously using the method of characteristics (see problem (2.3) in Sect. 2). Notice that, in general, the property (1.11) only holds locally at the level-set. The signed distance function of \(\Sigma (t)\) itself does not solve (1.3) nor (1.9) in general, but another nonlinear PDE, cf. Lemma 3.1 in [22]. 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 (3.2) below) and another simpler modified level-set equation, which only keeps the norm of the gradient within given bounds; see the initial value problem (2.24) below. We refer to [19] 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 (2.5)–(2.7) 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:

  1. (i)

    (1.9) provides a level-set that is identical to the original one provided by (1.3) for all \(t\ge 0\);

  2. (ii)

    (1.9) admits a unique classical solution \(\phi \) at least in a t-global tubular neighborhood of the level-set (see its definition in Sect. 2) so that the normal field and mean curvature field are well-defined by \(\phi \) and the property (1.11) (or, less restrictively, an a priori bound of \(|\nabla \phi |\)) holds on the level-set for all \(t\ge 0\);

  3. (iii)

    (1.9) admits a unique global-in-time viscosity solution defined on \([0,\infty )\times {\bar{\Omega }}\);

  4. (iv)

    If initial data is \(C^2\)-smooth, the viscosity solution \({\tilde{\phi }}\) coincides with the local-in-space classical solution \(\phi \) in a t-global tubular neighborhood of the level-set, i.e., partial \(C^2\)-regularity of \({\tilde{\phi }}\).

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 \(|\nabla \phi |\) 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 number 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 reasoning similar to localized doubling the number 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 [22]. In [22], 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 level-set remained open. Our current paper provides a stronger partial regularity property of viscosity solutions in the same context as [22].

2 \(C^2\)-solutions on tubular neighborhood of level-set

Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded connected open set and \(v=v(t,x)\) be a given smooth function defined in \([0,\infty )\times {\bar{\Omega }}\). Our investigation relies on certain properties of the flow generated by v that satisfies certain conditions; in particular, we require flow invariance of \({\bar{\Omega }}\) and \(\partial \Omega \), i.e., the solution \(x(s)=x(s;s_0,\xi )\) of (1.2) with initial condition \(x(s_0)=\xi \), \((s_0,\xi )\in [0,\infty )\times {\bar{\Omega }}\) (resp. \((s_0,\xi )\in [0,\infty )\times \partial \Omega \)) uniquely exists and stays in \({\bar{\Omega }}\) (resp. \(\partial \Omega \)) for all \(s\in [0,\infty )\). Note that since \({\bar{\Omega }}\) is compact, local-in-time invariance implies global-in-time invariance. Typical examples of v fulfilling our requirements are

  1. (i)

    \(\partial \Omega \) is smooth and \(v(t,x)\cdot \nu (x)=0\) for all \(x\in \partial \Omega \) and \(t\ge 0\), where \(\nu \) is the unit normal of \(\partial \Omega \) (cf., non-penetration condition in fluid dynamics),

  2. (ii)

    \(v(t,x)=0\) for all \(x\in \partial \Omega \) and \(t\ge 0\) (cf., non-slip condition in fluid dynamics).

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 \(K\subset {{\mathbb {R}}}^d\) at \(x\in K\) as

$$\begin{aligned} T_K(x):=\left\{ z \in {{\mathbb {R}}}^d\,\Big |\, \liminf \limits _{h \rightarrow 0+} \frac{ \textrm{dist}\, (x+ hz,K)}{h}=0\right\} \text { for } x \in K. \end{aligned}$$

We say that \(y\in {{\mathbb {R}}}^d\) is subtangential to K at a point \(x\in K\), if \(y\in T_K(x)\). Note that \(T_K(x)={{\mathbb {R}}}^d\) for x in the interior of K. We shall employ the following result on flow invariance.

Lemma 2.1

Let \(J=(a,b)\subset {{\mathbb {R}}}\), \(K \subset {{\mathbb {R}}}^d\) compact and \(g: J \times K \rightarrow {{\mathbb {R}}}^d\) (jointly) continuousFootnote 2 and locally Lipschitz in \(x\in K\). Then, the following holds true:

  1. (a)

    Suppose that \(\pm g\) are subtangential to K, i.e.,

    $$\begin{aligned} \pm g(s,x) \in T_K(x), \quad \forall \, s \in J, \,\,\,\forall \, x \in K. \end{aligned}$$

    Then, given any \(s_0 \in J\) and \(x_0 \in K\), the initial value problem

    $$\begin{aligned} x'(s)=g(s,x(s)),\quad x(s_0)=x_0 \end{aligned}$$

    has a unique solution defined on J that stays in K.

  2. (b)

    Suppose that \(\pm g\) are subtangential to \(\partial K\), i.e.,

    $$\begin{aligned} \pm g(s,x) \in T_{\partial K}(x), \quad \forall \, s \in J, \,\,\,\forall \, x \in \partial K. \end{aligned}$$

    Then, the sets K, \(\partial K\) and \(K{\setminus } \partial K\) 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:

  1. (H1)

    v belongs to \(C^0([0,\infty )\times {\bar{\Omega }};{{\mathbb {R}}}^3)\cap C^1([0,\infty )\times \Omega ;{{\mathbb {R}}}^3)\); v is Lipschitz continuous in x on \([0,\infty )\times {\bar{\Omega }}\); v is three times partially differentiable in x; all of the partial derivatives of v belong to \(C^0([0,\infty )\times \Omega ;{{\mathbb {R}}}^3)\),

  2. (H2)

    \(\pm v(s,x) \in T_{\partial \Omega }(x)\) for all \(s \in [0,\infty ),\,\, x \in \partial \Omega \).

We remark that the x-Lipschitz continuity mentioned in (H1) implies that

$$\begin{aligned} \left| \frac{\partial v_i}{\partial x_j}\right| \quad (i,j=1,2,3) \text { are bounded on } [0,\infty )\times \Omega . \end{aligned}$$

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, \({\bar{\Omega }}\), \(\partial \Omega \) and \(\Omega \) are flow invariant with respect to the flow X of (1.2); \(X(s,\tau ,\cdot )\) is continuous on \( {\bar{\Omega }}\) and \(C^3\)-smooth in \(\Omega \). If \({\bar{\Omega }}\) is a cube, for instance, (H2) implies: at each vertex, v must be equal to zero, while on each edge, v may take non-zero values parallel to the edge. In Sect. 3, we relax the regularity assumption of (H1) as


v belongs to \(C^0([0,\infty )\times {\bar{\Omega }};{{\mathbb {R}}}^3)\cap C^1([0,\infty )\times \Omega ;{{\mathbb {R}}}^3)\); v is Lipschitz continuous in x on \([0,\infty )\times {\bar{\Omega }}\).

2.1 Case 1: problem with level-set being away from \(\partial \Omega \)

Let \(\Sigma (0)\subset \Omega \) be a closed \(C^2\)-smooth surface. Let \(\phi ^0:\Omega \rightarrow {{\mathbb {R}}}\) be a \(C^2\)-smooth function such that

$$\begin{aligned} \left. \begin{array}{lll} &{}&{} \phi ^0>0 \text{ on } \Omega ^+(0), \quad \phi ^0<0 \text{ on } \Omega ^-(0), \quad \{x\in \Omega \,|\,\phi ^0(x)=0\}=\Sigma (0),\\ &{}&{} \nabla \phi ^0\ne 0 \text{ on } \Sigma (0). \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \Sigma (t):=X(t,0,\Sigma (0))=\{x\in \Omega \,|\, f(t,x)=0\}. \end{aligned}$$

Note that the second equality in the line above holds for all \(t\ge 0\) due to (H2). For each \(t\ge 0\), let \(\Sigma _\varepsilon (t)\) with \(\varepsilon >0\) be the \(\varepsilon \)-neighborhood of \(\Sigma (t)\), i.e.,

$$\begin{aligned} \Sigma _\varepsilon (t):=\bigcup _{x\in \Sigma (t)}\{y\in {{\mathbb {R}}}^3\,|\,|x-y|<\varepsilon \}, \end{aligned}$$

where we always consider \(\varepsilon >0\) such that \(\Sigma _\varepsilon (t)\subset \Omega \). We say that a set \(\Theta \subset [0,\infty )\times \Omega \) is a (t-global) tubular neighborhood of the level-set \(\{\Sigma (t)\}_{t\ge 0}\), if \(\Theta \) contains

$$\begin{aligned} \bigcup _{t\ge 0}\left( \{t\}\times \Sigma (t)\right) , \end{aligned}$$

and there exists a nonincreasing function \(\varepsilon :[0,\infty )\rightarrow {{\mathbb {R}}}_{>0}\) such that

$$\begin{aligned} \{t\}\times \Sigma _{\varepsilon (t)}(t)\subset \Theta ,\quad \forall \,t\ge 0. \end{aligned}$$

The problem under consideration is to find a tubular neighborhood \(\Theta \) of \(\{\Sigma (t)\}_{t\ge 0}\) and a \(C^2\)-function \(\phi \) satisfying

$$\begin{aligned}{} & {} \left\{ \begin{array}{lll} &{}&{}\frac{\partial \phi }{\partial t} +v\cdot \nabla \phi =\phi \left\langle \big ( \nabla v \big ) \frac{\nabla \phi }{|\nabla \phi |}, \frac{\nabla \phi }{|\nabla \phi |}\right\rangle \quad \text{ in } \Theta |_{t>0},\\ &{}&{}\phi (0,\cdot )=\phi ^0\quad \text{ on } \Theta |_{t=0}. \end{array} \right. \end{aligned}$$

Note that this problem makes sense with \(\phi ^0\) being defined only in a neighborhood of \(\Sigma (0)\), e.g., \(\phi ^0\) is given as the local signed distance function of \(\Sigma (0)\). We state the first main result of this paper.

Theorem 2.2

Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be \(C^2\)-smooth satisfying (2.2). Then, there exists a tubular neighborhood \(\Theta \) of the level-set \(\{\Sigma (t)\}_{t\ge 0}\) for which (2.3) admits a unique \(C^2\)-solution \(\phi \) satisfying

$$\begin{aligned} \Sigma ^\phi (t):= & {} \{x\in \Omega \,|\, \phi (t,x)=0\}=\Sigma (t),\quad \forall \,t\in [0,\infty ),\\ \forall \, t\in & {} [0, \infty ),\,\,\,\forall \,x\in \Sigma (t),\,\,\,\exists \, \xi \in \Sigma (0) \text{ such } \text{ that } |\nabla \phi (t,x)|=|\nabla \phi ^0(\xi )|. \end{aligned}$$

Before starting our proof of Theorem 2.2, we introduce a system of ODEs for the method of characteristics. We treat the PDE in (2.3) as the Hamilton–Jacobi equation

$$\begin{aligned} \frac{\partial \phi }{\partial t}+H(t,x,\nabla \phi ,\phi )=0, \end{aligned}$$

generated by the Hamiltonian \(H: [0,\infty )\times \Omega \times {{\mathbb {R}}}^3\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) defined as

$$\begin{aligned} H(t,x,p,\Phi ):= & {} v(t,x)\cdot p-\Phi \left\langle \nabla v(t,x) \frac{p}{|p|}, \frac{p}{|p|}\right\rangle \\= & {} v(t,x)\cdot p-\frac{1}{2} \Phi \left\langle \left( \nabla v)+(\nabla v)^\textsf{T}\right) \frac{p}{|p|}, \frac{p}{|p|}\right\rangle . \end{aligned}$$


$$\begin{aligned} D(v):=\frac{\nabla v+(\nabla v)^\textsf{T}}{2},\quad B(v,p):=\frac{\partial }{\partial x}\left\langle \big ( \nabla v(t,x) \big ) \frac{p}{|p|}, \frac{p}{|p|}\right\rangle , \end{aligned}$$

where we also use the symbol “\(\textrm{D} \varphi \)” for the derivative or Jacobian matrix of \(\varphi \). The characteristic ODEs of (2.4) are given as (see Appendix 1 for more details)

$$\begin{aligned} x'(s)= & {} \frac{\partial H}{\partial p}(s,x(s),p(s),\Phi (s))\nonumber \\= & {} v(s,x(s))\nonumber \\{} & {} -2\frac{\Phi (s)}{|p(s)|}\left[ D(v(s,x(s)))\frac{p(s)}{|p(s)|}-\left\langle D(v(s,x(s)))\frac{p(s)}{|p(s)|},\frac{p(s)}{|p(s)|}\right\rangle \frac{p(s)}{|p(s)|} \right] ,\nonumber \\ \end{aligned}$$
$$\begin{aligned} p'(s)= & {} -\frac{\partial H}{\partial x} (s,x(s),p(s),\Phi (s))-\frac{\partial H}{\partial \Phi }(s,x(s),p(s),\Phi (s))p(s)\nonumber \\= & {} -(\nabla v(s,x(s)))^\textsf{T}p(s) +\left\langle D(v(s,x(s))) \frac{p(s)}{|p(s)|}, \frac{p(s)}{|p(s)|}\right\rangle p(s)\nonumber \\{} & {} +\Phi (s) B(v(s,x(s)),p(s)), \end{aligned}$$
$$\begin{aligned} \Phi '(s)= & {} \frac{\partial H}{\partial p}(s,x(s),p(s),\Phi (s))\cdot p(s)-H(s,x(s),p(s),\Phi (s)) \nonumber \\= & {} \Phi (s)\left\langle D( v(s,x(s))) \frac{p(s)}{|p(s)|}, \frac{p(s)}{|p(s)|}\right\rangle , \end{aligned}$$
$$\begin{aligned} x(0)= & {} \xi ,\quad p(0)=\nabla \phi ^0(\xi ),\quad \Phi (0)=\phi ^0(\xi )\quad \text{(except } \xi \text{ such } \text{ that } p(0)=0). \end{aligned}$$

Note that B(v(tx), p) is still \(C^1\)-smooth in (xp) because of (H1), which is required for the method of characteristics. We sometimes use the notation \(x(s;\xi ),p(s;\xi ),\Phi (s;\xi ) \) to specify the initial point. The upcoming proof is based on the investigation (a priori estimates) of the variational equations of the characteristic ODEs (2.5)–(2.7) for each \(\xi \in \Sigma (0)\) to ensure the invertibility of \(x(s;\cdot )\) in a small neighborhood of \(\Sigma (s)\) for each \(s\ge 0\). 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 \(\Sigma (s)\) as s becomes larger.

Proof of Theorem 2.2

Our proof consists of four steps.

Step 1. We demonstrate a priori estimates for the above system of ODEs. For each \(\xi \in \Sigma (0)\), as long as \(x(s;\xi ),p(s;\xi ),\Phi (s;\xi ) \) exist, it holds that

$$\begin{aligned} \Phi (s;\xi )\equiv & {} \Phi (0;\xi )=0, \end{aligned}$$
$$\begin{aligned} x'(s)= & {} v(s,x(s)), \quad x(s)\in \Sigma (s),\end{aligned}$$
$$\begin{aligned} p'(s)= & {} -(\nabla v(s,x(s)))^\textsf{T}p(s) +\left\langle D(v(s,x(s))) \frac{p(s)}{|p(s)|}, \frac{p(s)}{|p(s)|}\right\rangle p(s),\nonumber \\ \end{aligned}$$
$$\begin{aligned} p'(s)\cdot p(s)= & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d} s}|p(s)|^2=0,\quad |p(s)|^2\equiv |p(0)|^2=|\nabla \phi ^0(\xi )|^2\ne 0, \end{aligned}$$

where we note that due to (H2) applied to (2.10), the above equalities (2.9)–(2.12) hold for all \(s\ge 0\) together with

$$\begin{aligned} 0<\inf _{\Sigma (0)}|\nabla \phi ^0|\le |p(s)|\le \sup _{\Sigma (0)}|\nabla \phi ^0|<\infty ,\quad \forall \,s\ge 0. \end{aligned}$$

Hence, for each \(\xi \in \Sigma (0)\), there exist \(\frac{\partial x}{\partial \xi }(s)=\frac{\partial x}{\partial \xi }(s;\xi )\) and \(\frac{\partial \Phi }{\partial \xi }(s)=\frac{\partial \Phi }{\partial \xi }(s;\xi )\) for all \(s\ge 0\) satisfying

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d} s}\frac{\partial x}{\partial \xi }(s)=\nabla v(s,x(s))\frac{\partial x}{\partial \xi }(s)-\frac{2}{|p(s)|} b(s)\otimes \frac{\partial \Phi }{\partial \xi }(s),\quad \frac{\partial x}{\partial \xi }(0)=id, \end{aligned}$$
$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d} s}\frac{\partial \Phi }{\partial \xi }(s)\!\!=\!\!\frac{\partial \Phi }{\partial \xi }(s)\left\langle D( v(s,x(s))) \frac{p(s)}{|p(s)|}, \frac{p(s)}{|p(s)|}\right\rangle , \quad \frac{\partial \Phi }{\partial \xi }(0)\!\!=\!\!\nabla \phi ^0(\xi ), \end{aligned}$$


$$\begin{aligned}{} & {} b(s):=D(v(s,x(s)))\frac{p(s)}{|p(s)|} -\left\langle D(v(s,x(s)))\frac{p(s)}{|p(s)|},\frac{p(s)}{|p(s)|}\right\rangle \frac{p(s)}{|p(s)|},\\{} & {} |p(s)|\equiv |p(0)|. \end{aligned}$$

Hereafter, \(V_0,V_1,V_2,V_3,V_4\) will denote constants depending only on v, \(\sup _{\Sigma (0)}|\nabla \phi ^0|<\infty \) and \(\inf _{\Sigma (0)}|\nabla \phi ^0|>0\). Due to (2.1), it holds that for any \(\xi \in \Sigma (0)\),

$$\begin{aligned}{} & {} |\nabla v(s,x(s)) y|\le V_0|y|,\quad \forall \,y\in {{\mathbb {R}}}^3,\\{} & {} \left| D(v(s,x(s)))\frac{p(s)}{|p(s)|} \right| \le V_1,\quad \\{} & {} \left| \left\langle D(v(s,x(s)))\frac{p(s)}{|p(s)|},\frac{p(s)}{|p(s)|}\right\rangle \frac{p(s)}{|p(s)|} \right| \le V_1,\quad |b(s)|\le V_1. \end{aligned}$$

Observe that for each \(\xi \in \Sigma (0)\), we have from (2.14),

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} s}\frac{\partial \Phi }{\partial \xi }(s)= & {} \frac{\partial \Phi }{\partial \xi }(s)\left\langle D( v(s,x(s))) \frac{p(s)}{|p(s)|}, \frac{p(s)}{|p(s)|}\right\rangle ,\\ \left| \frac{\partial \Phi }{\partial \xi }(s)\right|= & {} |\nabla \phi ^0(\xi ) e^{\int _0^s \langle D(v(s,x(\tau ))) p(\tau ), p(\tau )\rangle |d\tau }|\le V_2e^{V_1s }, \quad \forall \,s\ge 0,\\ \left| \frac{\partial \Phi }{\partial \xi _i}(s) b(s)\right|\le & {} V_1V_2e^{V_1s }\quad (i=1,2,3),\quad \forall \,s\ge 0. \end{aligned}$$

For each \(i=1,2,3\) and \(\xi \in \Sigma (0)\), we have from (2.13),

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} s}\frac{\partial x}{\partial \xi _i}(s)= & {} \nabla v(s,x(s))\frac{\partial x}{\partial \xi _i}(s)-\frac{2}{|p(0)|}\frac{\partial \Phi }{\partial \xi _i}(s) b(s),\quad \frac{\partial x}{\partial \xi _i}(0)=e^i,\\ \frac{\textrm{d}}{\textrm{d} s}\frac{\partial x}{\partial \xi _i}(s)\cdot \frac{\partial x}{\partial \xi _i}(s)= & {} \nabla v(s,x(s))\frac{\partial x}{\partial \xi _i}(s)\cdot \frac{\partial x}{\partial \xi _i}(s)-\frac{2}{|p(0)|}\frac{\partial \Phi }{\partial \xi _i}(s) b(s)\cdot \frac{\partial x}{\partial \xi _i}(s),\\ \frac{1}{2} \frac{\textrm{d}}{\textrm{d} s}\left| \frac{\partial x}{\partial \xi _i}(s)\right| ^2\le & {} V_1\left| \frac{\partial x}{\partial \xi _i}(s)\right| ^2+ \frac{2}{ \inf _{\Sigma (0)}|\nabla \phi ^0|} V_1V_2e^{V_1s}\left| \frac{\partial x}{\partial \xi _i}(s)\right| \\\le & {} (V_1+ 1) \left| \frac{\partial x}{\partial \xi _i}(s)\right| ^2+ \left( \frac{V_1V_2e^{V_1s}}{ \inf _{\Sigma (0)}|\nabla \phi ^0|}\right) ^2. \end{aligned}$$

Gronwall’s inequality implies

$$\begin{aligned} \left| \frac{\partial x}{\partial \xi _i}(s;\xi )\right| ^2\le & {} e^{2(1+V_1)s}\left\{ 1+V_1\left( \frac{V_2}{ \inf _{\Sigma (0)}|\nabla \phi ^0|}\right) ^2(e^{2V_1s} -1)\right\} \nonumber \\\le & {} V^2_3,\quad \forall \, s\in [0,1],\,\,\,\forall \,\xi \in \Sigma (0). \end{aligned}$$

It follows from (2.15) that

$$\begin{aligned} \left| \frac{\textrm{d}}{\textrm{d} s}\frac{\partial x}{\partial \xi _i}(s;\xi )\right|\le & {} V_0 \left| \frac{\partial x}{\partial \xi _i}(s;\xi )\right| +2V_1V_2 e^{V_1s}\nonumber \\\le & {} V_4, \quad \forall \, s\in [0,1],\,\,\,\forall \,\xi \in \Sigma (0). \end{aligned}$$

Due to the continuity in (2.5)–(2.8) and (2.15)–(2.16), we find \(\varepsilon _1>0\) such that

$$\begin{aligned}{} & {} \Sigma _{\varepsilon _1}(0)\subset \Omega ,\quad (2.5){-}(2.8) \text{ is } \text{ solvable } \text{ for } \text{ all } s\in [0,1] \text{ and } \xi \in \Sigma _{\varepsilon _1}(0),\nonumber \\{} & {} \quad \left| \frac{\textrm{d}}{\textrm{d} s}\frac{\partial x}{\partial \xi _i}(s;\xi )\right| < 2V_4, \quad \forall \, s\in [0,1],\,\,\,\forall \,\xi \in \Sigma _{\varepsilon _1}(0). \end{aligned}$$

Fix a number \(t_*\in (0,1]\) such that

$$\begin{aligned} 1- \{3\cdot (2V_4 t_*)+6\cdot (2V_4 t_*)^2+6\cdot (2V_4 t_*)^3\}>0. \end{aligned}$$

Step 2. We show the injectivity of \(\xi \mapsto x(s;\xi )\), i.e.,

$$\begin{aligned}&\exists \, {\tilde{\varepsilon }}_1\in (0,\varepsilon _1]\textit{ such that }\Sigma _{{\tilde{\varepsilon }}_1}(0)\ni \xi \mapsto x(s;\xi )\textit{ is injective for each }0\le s\le t_*.\nonumber \\ \end{aligned}$$

First, we give an auxiliary explanation on how to prove (2.19). In order to prove the injectivity of \(x(s;\cdot ):\Sigma _{\varepsilon }(0)\rightarrow x(s;\Sigma _{\varepsilon }(0))\) for each fixed \(s\in [0,s_*]\) (\(\varepsilon >0\) and \(s_*>0\) are some constants), we would take one of the following strategies:

  1. (i)

    Fix \(\varepsilon >0\) and take a sufficiently small \(s_*>0\),

  2. (ii)

    Fix \(s_*>0\) and take a sufficiently small \(\varepsilon >0\),

where we note that \(\det \frac{\partial x}{\partial \xi }(s;\xi )\ne 0\) everywhere on \(\Sigma _{\varepsilon }(0)\) is not enough for the injectivity in general. In our proof of Theorem 2.2, we need to repeat the argument from \([0,s_*]\) within \([s_*,2s_*]\) to demonstrate infinite iteration with the fixed \(s_*>0\). Hence, we take the strategy (ii) with \(s_*=t_*\) given from (2.18) and sufficiently small \(\varepsilon ={\tilde{\varepsilon }}_1\in (0,\varepsilon _1]\). Then, what we need to show is that

$$\begin{aligned} x(s;{\tilde{\zeta }})=x(s;\zeta ) \text{ for } \zeta ,{\tilde{\zeta }}\in \Sigma _{\varepsilon }(0)\Rightarrow {\tilde{\zeta }}=\zeta . \end{aligned}$$

If the line segment joining \(\zeta \) and \({\tilde{\zeta }}\) is included in \(\Sigma _{\varepsilon }(0)\), we may immediately apply Taylor’s approximation to have

$$\begin{aligned} 0=x(s;{\tilde{\zeta }})-x(s;\zeta )=\Lambda ({\tilde{\zeta }}-\zeta ) \quad \text{ with } \text{ some } (3\times 3)\text{-matrix } \Lambda \end{aligned}$$

and get \({\tilde{\zeta }}=\zeta \) via \(\det \Lambda >0\), where \(t_*\) in (2.18) is given so that \(\det \Lambda >0\) holds in \(\Sigma _{\varepsilon _1}(0)\). However, it is not a priori clear if the line segment joining \(\zeta \) and \({\tilde{\zeta }}\) is included in \(\Sigma _{\varepsilon }(0)\) for each \(s\in [0,t_*]\). This is a major difficulty to prove (2.19) directly.

Now, we prove (2.19) by contradiction. Suppose that (2.19) does not hold. Then, we find a sequence \(\{\delta _j\}_{j\in {{\mathbb {N}}}}\subset (0,\varepsilon _1]\) with \(\delta _j\rightarrow 0\) as \(j\rightarrow \infty \) and \(\{t_j\}_{j\in {{\mathbb {N}}}}\subset (0,t_*]\) for which

$$\begin{aligned} \exists \,\zeta ,{\tilde{\zeta }}\in \Sigma _{\delta _j}(0) \text{ such } \text{ that } \zeta \ne {\tilde{\zeta }},\,\,\,x(t_j;\zeta )=x(t_j;{\tilde{\zeta }}) \text{ for } \text{ each } j\in {{\mathbb {N}}}. \end{aligned}$$

Define \(d_j:=\sup \{ |{\tilde{\zeta }}-\zeta |\,|\, (2.20) \text{ holds } \}\). We prove that \(d_j\rightarrow 0\) as \(j\rightarrow \infty \). If not, we find \(\eta >0\) and a subsequence of \(\{d_j\}_{j\in {{\mathbb {N}}}}\) (still denoted by the same symbol) such that \(d_j\ge \eta \) for all j. We further take out subsequences so that \(t_j\rightarrow s\in [0,t_*]\) as \(j\rightarrow \infty \). Then, for each \(j\in {{\mathbb {N}}}\), there exist \(\zeta _j\) and \({\tilde{\zeta }}_j\) such that

$$\begin{aligned} |\zeta _j-{\tilde{\zeta }}_j|\ge \frac{\eta }{2},\quad \textrm{dist}(\zeta _j,\Sigma (0))\le \delta _j,\quad \textrm{dist}({\tilde{\zeta }}_j,\Sigma (0))\le \delta _j. \end{aligned}$$

Taking subsequences if necessary, we see that \(\zeta _j\), \({\tilde{\zeta }}_j\) converge to some \(\xi ,{\tilde{\xi }}\in \Sigma (0)\) as \(j\rightarrow \infty \), respectively, where \(|\xi -{\tilde{\xi }}|\ge \frac{\eta }{2}\). The limit \(j\rightarrow \infty \) in \(x(t_j;\zeta _j)=x(t_j;{\tilde{\zeta }}_j)\) implies \(x(s;\xi )=x(s;{\tilde{\xi }})\) with \(\xi ,{\tilde{\xi }}\in \Sigma (0)\) and \(\xi \ne {\tilde{\xi }}\). This is a contradiction, because \(x(s;\cdot )|_{\Sigma (0)}\) is given by the flow of (1.2). Consequently, \(d_j\rightarrow 0\) as \(j\rightarrow \infty \). Hence, for all j sufficiently large, we find that \(\zeta ,{\tilde{\zeta }}\) in (2.20), denoted by \(\zeta _j\), \({\tilde{\zeta }}_j\), are included in the \(\varepsilon _1\)-neighborhood of some \(\xi _j\in \Sigma (0)\); for \(i=1,2,3\), Taylor’s approximation within the \(\varepsilon _1\)-neighborhood of \(\xi _j\) yields

$$\begin{aligned} 0= & {} x_i(t_j;{\tilde{\zeta }}_j)-x_i(t_j,\zeta _j)\\= & {} x_i(0;{\tilde{\zeta }}_j)-x_i(0;\zeta _j)+(x_i(t_j;{\tilde{\zeta }}_j)- x_i(t_j,\zeta _j))-(x_i(0;{\tilde{\zeta }}_j)-x_i(0;\zeta _j))\\= & {} ({\tilde{\zeta }}_j-\zeta _j)_i+\frac{dx_i}{ds}(\lambda _{ji}t_j;{\tilde{\zeta }}_j)t_j- \frac{dx_i}{ds}(\lambda _{ji}t_j,\zeta _j))t_j, \quad \lambda _{ji}\in (0,1),\\= & {} ({\tilde{\zeta }}_j-\zeta _j)_i+\frac{\textrm{d}}{\textrm{d} s}\frac{\partial x_i}{\partial \xi }(\lambda _{ji}t_j;{\tilde{\zeta }}_j +\lambda '_{ji}({\tilde{\zeta }}_j -\zeta _j ) )t_j\cdot ({\tilde{\zeta }}_j-\zeta _j), \quad \lambda '_{ji}\in (0,1). \end{aligned}$$

Therefore, by (2.17), we obtain

$$\begin{aligned}{} & {} (I+A)({\tilde{\zeta }}_j-\zeta _j)=0,\\{} & {} I \text{ is } \text{ the } (3\times 3)\text{-identity } \text{ matrix, } A \text{ is } \text{ a } (3\times 3)\text{-matrix } \text{ with } |A_{kl}|<2V_4 t_j\le 2V_4 t_*,\\{} & {} \det (I+A)\ge 1- \{3\cdot (2V_4 t_*)+6\cdot (2V_4 t_*)^2+6\cdot (2V_4 t_*)^3\}>0, \end{aligned}$$

which leads to \({\tilde{\zeta }}_j=\zeta _j\). This is a contradiction and we conclude (2.19). Note that we also obtain \(\det \frac{\partial x}{\partial \xi }(s;\xi )>0\) for all \(s\in [0,t_*]\) and \(\xi \in \Sigma _{{\tilde{\varepsilon }}_1}(0)\); this follows from \( \frac{\partial x}{\partial \xi }(s;\xi )= \frac{\partial x}{\partial \xi }(0;\xi )+ \frac{\partial x}{\partial \xi }(s;\xi )- \frac{\partial x}{\partial \xi }(0;\xi )=I+[\frac{\textrm{d}}{\textrm{d} s} \frac{\partial x_k}{\partial \xi _l}(\lambda _{kl}s;\xi )s]\), \(\lambda _{kl}\in (0,1)\).

Step 3. We construct a local in space solution within \([0,t_*]\). Define

$$\begin{aligned}{} & {} O_{{\tilde{\varepsilon }}_1}(t_*) :=\bigcup _{0\le s\le t_*}\left( \{s\}\times \{x(s;\xi )\,|\,\xi \in \Sigma _{{\tilde{\varepsilon }}_1}(0)\}\right) ,\\{} & {} \psi _1: [0,t_*]\times \Sigma _{{\tilde{\varepsilon }}_1}(0)\rightarrow O_{{\tilde{\varepsilon }}_1}(t_*),\quad \psi _1(s,\xi ):=(s, x(s;\xi )),\\{} & {} \psi _1(t,\cdot ): \Sigma _{{\tilde{\varepsilon }}_1}(0)\rightarrow O_{{\tilde{\varepsilon }}_1}(t_*)|_{s=t}=\{t\}\times \{x(t;\xi )\,|\,\xi \in \Sigma _{{\tilde{\varepsilon }}_1}(0)\},\\{} & {} \quad 0\le t\le t_*. \end{aligned}$$

The invertibility of \(\xi \mapsto x(s;\xi )\) and \(\det \frac{\partial x}{\partial \xi }(s;\xi )>0\) imply that \(\psi _1\) and \(\psi _1(t,\cdot )\) are \(C^1\)-diffeomorphic and that there exists \(\varepsilon _2>0\) such that

$$\begin{aligned} \{x(t_*;\xi )\,|\,\xi \in \Sigma _{{\tilde{\varepsilon }}_1}(0)\}\supset \Sigma _{\varepsilon _2}(t_*). \end{aligned}$$

Let \(\varphi _1: O_{{\tilde{\varepsilon }}_1}(t_*)\rightarrow \Sigma _{{\tilde{\varepsilon }}_1}(0)\) be defined as

$$\begin{aligned} \psi _1(t,\xi )=(t,x)\Leftrightarrow (t,\xi )=\psi ^{-1}_1(t,x)=:(t,\varphi _1(t,x)), \end{aligned}$$

where we note that \(\varphi _1(t,\Sigma (t))=\Sigma (0)\). Then, it follows from the general results of the method of characteristics (see Appendix 1) that the function

$$\begin{aligned} \phi _1: O_{{\tilde{\varepsilon }}_1}(t_*) \rightarrow {{\mathbb {R}}},\quad \phi _1(t,x):=\Phi (t; \varphi _1(t,x)) \end{aligned}$$

is \(C^2\)-smooth and satisfies

$$\begin{aligned} \frac{\partial \phi _1}{\partial t} +v\cdot \nabla \phi _1= & {} \phi _1 \left\langle \big ( \nabla v \big )\frac{\nabla \phi _1}{|\nabla \phi _1|}, \frac{\nabla \phi _1}{|\nabla \phi _1|}\right\rangle ,\quad (t,x)\in O_{{\tilde{\varepsilon }}_1}(t_*),\\ \phi _1(0,x)= & {} \phi ^0(x),\quad x\in \Sigma _{{\tilde{\varepsilon }}_1}(0),\\ \{x\,|\,\phi _1(s,x)=0\}= & {} \Sigma (s),\quad \forall \,s\in [0,t_*],\\ \nabla \phi _1(s,x)= & {} p(s;\varphi _1(s,x)),\quad \forall \,(s,x)\in O_{{\tilde{\varepsilon }}_1}(t_*),\\ |\nabla \phi _1(s,x)|= & {} |p(s;\varphi _1(s,x))|=|\nabla \phi ^0(\varphi _1(s,x))|,\quad \forall \, s\in [0,t_*],\,\,\, \forall \, x\in \Sigma (s). \end{aligned}$$

Step 4. We demonstrate an iteration of Step 1–3 with the common \(t_*\). The argument up to now, in particular Step 1, has used the \(C^2\)-smoothness of \(\phi ^0\) on a neighborhood of \(\Sigma (0)\) and the upper/lower bound of \(|\nabla \phi ^0|\) on \(\Sigma (0)\), where we note that \(|\nabla \phi _1(t_*,\cdot )|\) on \(\Sigma (t_*)\) has exactly the same upper/lower bound as \(|\nabla \phi ^0|\) on \(\Sigma (0)\). Therefore, we may replace \(\phi ^0\) with \(\phi _1(t_*,\cdot )\) to demonstrate the same kind of estimates in terms of the above constants \(V_0,V_1,V_2,V_3,V_4\) and \(t_*\) as well as \(\varepsilon _2 >0\) appropriately chosen in (2.21) for the characteristic ODEs (2.5)–(2.7) for \(s\in [t_*,2t_*]\) with the initial condition

$$\begin{aligned} x(t_*)=\xi ,\quad p(t_*)=\nabla \phi _1(t_*,\xi ),\quad \Phi (t_*)= \phi _1(t_*,\xi ),\quad \xi \in \{x(t_*;{\tilde{\xi }})\,|\,{\tilde{\xi }}\in \Sigma _{{\tilde{\varepsilon }}_1}(0)\}. \end{aligned}$$

Furthermore, we find a constant \({\tilde{\varepsilon }}_2\in (0,\varepsilon _2]\) such that

$$\begin{aligned}{} & {} O_{{\tilde{\varepsilon }}_2}(2t_*) :=\bigcup _{t_*\le s\le 2t_*} \left( \{s\}\times \{x(s;\xi )\,|\,\xi \in \Sigma _{{\tilde{\varepsilon }}_2}(t_*)\}\right) ,\\{} & {} \psi _2: [t_*,2t_*]\times \Sigma _{{\tilde{\varepsilon }}_2}(t_*)\rightarrow O_{{\tilde{\varepsilon }}_2}(2t_*),\quad \psi _2(s,\xi ):=(s, x(s;\xi )),\\{} & {} \psi _2(t,\cdot ): \Sigma _{{\tilde{\varepsilon }}_2}(t_*)\rightarrow O_{{\tilde{\varepsilon }}_2}(2t_*)|_{s=t}=\{t\}\times \{x(t;\xi )\,|\,\xi \in \Sigma _{{\tilde{\varepsilon }}_2}(t_*)\},\\{} & {} \quad t_*\le t\le 2t_*, \end{aligned}$$

are \(C^1\)-diffeomorphic. Also, there exists \(\varepsilon _3>0\) such that

$$\begin{aligned} \{x(2t_*;\xi )\,|\,\xi \in \Sigma _{{\tilde{\varepsilon }}_2}(t_*)\}\supset \Sigma _{\varepsilon _3}(2t_*). \end{aligned}$$

Let \(\varphi _2: O_{{\tilde{\varepsilon }}_2}(2t_*)\rightarrow \Sigma _{{\tilde{\varepsilon }}_2}(t_*)\) be defined as

$$\begin{aligned} \psi _2(t,\xi )=(t,x)\Leftrightarrow (t,\xi )=\psi _2^{-1}(t,x)=:(t,\varphi _2(t,x)), \end{aligned}$$

where we note that \(\varphi _2(t,\Sigma (t))=\Sigma (t_*)\). Then, the function

$$\begin{aligned} \phi _2: O_{{\tilde{\varepsilon }}_2}(2t_*) \rightarrow {{\mathbb {R}}},\quad \phi _2(t,x):=\Phi (t; \varphi _2(t,x)) \end{aligned}$$

is \(C^2\)-smooth and satisfies

$$\begin{aligned} \frac{\partial \phi _2}{\partial t} +v\cdot \nabla \phi _2= & {} \phi _2 \left\langle \big ( \nabla v \big ) \frac{\nabla \phi _2}{|\nabla \phi _2|}, \frac{\nabla \phi _2}{|\nabla \phi _2|}\right\rangle ,\quad (t,x)\in O_{{\tilde{\varepsilon }}_2}(2t_*),\\ \phi _2(0,x)= & {} \phi _1(t_*, x),\quad x\in \Sigma _{{\tilde{\varepsilon }}_2}(t_*),\\ \{x\,|\,\phi _2(s,x)=0\}= & {} \Sigma (s),\quad \forall \,s\in [t_*,2t_*],\\ \nabla \phi _2(s,x)= & {} p(s;\varphi _2(s,x)),\quad \forall \,(s,x)\in O_{{\tilde{\varepsilon }}_2}(2t_*),\\ |\nabla \phi _2(s,x)|= & {} |p(s;\varphi _2(s,x))|=|\nabla \phi _1(t_*, \varphi _2(s,x))|\\= & {} |\nabla \phi ^0(\varphi _1(t_*,\varphi _2(s,x)))|,\quad \forall \, s\in [t_*,2t_*],\,\,\, \forall \, x\in \Sigma (s). \end{aligned}$$

Note that \(\phi _1\) and \(\phi _2\) are smoothly connected at \(t=t_*\). With the common constant \(t_*\), we may repeat this process with \(\varepsilon _1,{\tilde{\varepsilon }}_1, \varepsilon _2,{\tilde{\varepsilon }}_2,,\varepsilon _3, {\tilde{\varepsilon }}_3, \cdots \) (note that it is possible that \(\varepsilon _k,{\tilde{\varepsilon }}_k,\rightarrow 0\) as \(k\rightarrow \infty \)). We conclude the proof with defining \(\Theta := \cup _{l\in {{\mathbb {N}}}} O_{{\tilde{\varepsilon }}_l}(lt_*)\). \(\square \)

If we choose \(\phi ^0\) which coincides with the (local) signed distance function of \(\Sigma (0)\), then the solution \(\phi \) obtained in Theorem 2.2 satisfies

$$\begin{aligned} |\nabla \phi (t,x)|\equiv 1, \quad \forall \, t\ge 0,\,\,\,\forall \,x\in \Sigma (t). \end{aligned}$$

2.2 Case 2: problem with level-set touching \(\partial \Omega \)

We consider the problem (2.3) with the level-set touching the boundary of \(\Omega \). Let \(K\subset {{\mathbb {R}}}^3\) be a bounded connected open set such that \(K\cap \Omega \ne \emptyset \), \(\partial K\cap \partial \Omega \ne \emptyset \) and \(\partial K\) is a closed \(C^2\)-smooth surface. Define

$$\begin{aligned} \Sigma (0):=\partial K \cap {\bar{\Omega }}. \end{aligned}$$

Let \(\phi ^0\) be a \(C^2\)-smooth \({{\mathbb {R}}}\)-valued function defined in an open set containing \({\bar{K}}\cup {\bar{\Omega }}\) such that

$$\begin{aligned} \phi ^0>0 \text{ in } K,\quad \phi ^0<0 \text{ outside } {\bar{K}},\quad \nabla \phi ^0\ne 0 \text{ on } \partial K. \end{aligned}$$

Then, we have

$$\begin{aligned} \phi ^0>0\quad \text{ in } K\cap \Omega ,\quad \{x\in {\bar{\Omega }}\,|\,\phi ^0(x)=0\}=\Sigma (0),\quad \nabla \phi ^0\ne 0 \text{ on } \Sigma (0). \end{aligned}$$

Let f be the solution of the original level-set equation (1.3). Define

$$\begin{aligned} \Sigma (t):=X(t,0,\Sigma (0))=\{x\in {\bar{\Omega }}\,|\, f(t,x)=0\},\quad \forall \,t\ge 0, \end{aligned}$$

where we note that \(\partial \Omega \) is invariant under the flow X of (1.2), and hence \(\Sigma (t)\) always touches \(\partial \Omega \). If we follow the same argument as given in Sect. 2.1, we would face non-trivial issues at/near \(\Sigma (t)\cap \partial \Omega \) coming from the behavior of the variational equations of the characteristic ODEs on \(\partial \Omega \). Hence, we modify the reasoning of Sect. 2.1 so that \(\Sigma (t)\cap \partial \Omega \) is not involved.

Let \(\{\Omega ^k\}_{k\in {{\mathbb {N}}}}\) be a monotone approximation of \(\Omega \), i.e., each \(\Omega ^k\) is an open subset of \(\Omega \); \(\Omega ^k\subset \Omega ^{k+1}\) for all \(k\in {{\mathbb {N}}}\); for any \(G\subset \Omega \) compact, there exists \(k=k(G)\) such that \(G\subset \Omega ^k\). Introduce

$$\begin{aligned} \{ \Sigma ^k(0) \}_{k\in {{\mathbb {N}}}}, \quad \Sigma ^k(0):=\Sigma (0)\cap \Omega ^k,\quad \Sigma ^k(t):=X(t,0,\Sigma ^k(0)), \end{aligned}$$

where for each \(k\in {{\mathbb {N}}}\) we have \(\varepsilon >0\) depending on k such that

$$\begin{aligned} \Sigma ^k_{\varepsilon }(0):=\bigcup _{x\in \Sigma ^k(0)}\{ y\in {{\mathbb {R}}}^3\,|\, |y-x|<\varepsilon \}\subset \Omega . \end{aligned}$$

Now, we may follow the same argument as given in Sect. 2.1 with \(\Sigma ^k(0)\) in place of \(\Sigma (0)\) to obtain the following objects:

$$\begin{aligned} O_{{\tilde{\varepsilon }}_1(k)}(t_*(k))= & {} \bigcup _{0\le s\le t_*(k)} \left( \{s\}\times \{x(s;\xi )\,|\,\xi \in \Sigma ^k_{{\tilde{\varepsilon }}_1(k)}(0)\}\right) ,\\ O_{{\tilde{\varepsilon }}_2(k)}(2t_*(k))= & {} \bigcup _{t_*(k)\le s\le 2t_*(k)} \left( \{s\}\times \{x(s;\xi )\,|\,\xi \in \Sigma ^k_{{\tilde{\varepsilon }}_2(k)}(t_*(k))\}\right) ,\ldots ,\\ \Theta ^k:= & {} \bigcup _{l\in {{\mathbb {N}}}} O_{{\tilde{\varepsilon }}_l(k)}(lt_*(k)),\\{} & {} \text{ a } \text{ unique } C^2\text{-solution } \phi ^k \text{ of } (2.3)|_{\Theta =\Theta ^k}. \end{aligned}$$

The method of characteristics implies that \(\phi ^k\equiv \phi ^{k'}\) on \(\Theta ^k\cap \Theta ^{k'}\) for every \(k,k'\in {{\mathbb {N}}}\). Therefore, setting

$$\begin{aligned} \Theta :=\bigcup _{k\in {{\mathbb {N}}}} \Theta ^k, \end{aligned}$$

we obtain a unique \(C^2\)-solution \(\phi \) of (2.3). We remark that

$$\begin{aligned} \Theta \cap (\{t\}\times \Sigma (t))=\{t\}\times (\Sigma (t){\setminus }\partial \Omega ),\quad \forall \,t\ge 0. \end{aligned}$$

We summarize the result:

Theorem 2.3

Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be \(C^2\)-smooth satisfying (2.22). Then, there exists a tubular neighborhood \(\Theta \) in the sense of (2.23) of the level-set \(\{\Sigma (t)\}_{t\ge 0}\) touching \(\partial \Omega \) for which (2.3) admits a unique \(C^2\)-solution \(\phi \) satisfying

$$\begin{aligned} \Sigma ^\phi (t):= & {} \{x\in \Omega \,|\, \phi (t,x)=0\}=\Sigma (t){\setminus } \partial \Omega ,\quad \forall \,t\in [0,\infty ),\\ \forall \, t\in & {} [0, \infty ),\,\,\,\forall \,x\in \Sigma (t){\setminus } \partial \Omega ,\,\,\,\exists \, \xi \in \Sigma (0){\setminus } \partial \Omega \text{ such } \text{ that } |\nabla \phi (t,x)|=|\nabla \phi ^0(\xi )|. \end{aligned}$$

Let us note in passing that another method to investigate a problem with the level-set touching \(\partial \Omega \) would run via smooth extension of v outside \(\Omega \). The following steps would suffice:

Step 1. Extend the velocity field v to \({{\mathbb {R}}}\times {{\mathbb {R}}}^3\) as a \(C^3\)-function by means of Whitney’s extension theorem [31] 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 \(\Omega \) are required accordingly.

Step 2. Extend the flow \(X(t,\tau ,\xi )\) to \(t,\tau \in {{\mathbb {R}}}\), \(\xi \in {{\mathbb {R}}}^3\) and the problem (2.3) to a tubular neighborhood \({\tilde{\Theta }}\) of the level-set \(\{ X(t,0,\partial K) \}_{t\ge 0}\).

Step 3. Solve the extended problem in the same way as Sect. 2.1, where each characteristic curve starting at a point of \(\Omega \) stays inside \(\Omega \) forever due to the flow invariance of \({\bar{\Omega }}\) and \(\partial \Omega \) under X.

The restriction of the solution obtained in Step 3 to \(\Theta :={\tilde{\Theta }}\cap ([0,\infty )\times {\bar{\Omega }})\) then is the desired object.

2.3 Simpler nonlinear modification

The nonlinear modification in (2.3) is designed to preserve \(|\nabla \phi |\) along each characteristic curve on the level-set. If we relax the requirement, i.e., if we only ask for an a priori bound of \(|\nabla \phi |\) on the level-set, we may use a much simpler modification.

We take the same configuration of the level-set considered in Sect. 2.1. Let \(\alpha >0\) be a constant such that

$$\begin{aligned} -\alpha |p^2|\le \langle (\nabla v(t,x))p,p\rangle \le \alpha |p|^2,\quad \forall \,p\in {{\mathbb {R}}}^3. \end{aligned}$$

With a constant \(\beta >\alpha \), we consider

$$\begin{aligned} \frac{\partial \phi }{\partial t}+v\cdot \nabla \phi =\phi (\beta -|\nabla \phi |),\quad \phi (0,\cdot )=\phi ^0, \end{aligned}$$

which is seen as the Hamilton–Jacobi equation (2.4) with

$$\begin{aligned} H(t,x,p,\Phi ):=v\cdot p-\Phi (\beta -|p|). \end{aligned}$$

The characteristic ODEs in this case are given as

$$\begin{aligned} x'(s)= & {} \frac{\partial H}{\partial p}(s,x(s),p(s),\Phi (s))=v(s,x(s))+\Phi (s)\frac{p(s)}{|p(s)|},\\ p'(s)= & {} -\frac{\partial H}{\partial x}(s,x(s),p(s),\Phi (s))-\frac{\partial H}{\partial \Phi }(s,x(s),p(s),\Phi (s))p(s)\\= & {} (\nabla v(s,x(s)))^\textsf{T} p(s)-(|p(s)|-\beta )p(s),\\ \Phi '(s)= & {} \frac{\partial H}{\partial p}(s,x(s),p(s),\Phi (s))\cdot p(s)-H(s,x(s),p(s),\Phi (s))=\Phi (s),\\ x(0)= & {} \xi ,\quad p(0)=\nabla \phi ^0(\xi ),\quad \Phi (0)=\phi ^0(\xi )\quad \text{(except } \xi \text{ such } \text{ that } p(0)=0). \end{aligned}$$

As long as the above characteristic ODEs have solutions, \(|p(s)|^2\) evolves as

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d} s}|p(s)|^2=\langle (\nabla v(s,x(s)))p(s),p(s)\rangle -(|p(s)|-\beta )|p(s)|^2, \end{aligned}$$

which leads to

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} s}|p(s)|^2\le -2(|p(s)|-\beta -\alpha )|p(s)|^2, \quad \frac{\textrm{d}}{\textrm{d} s}|p(s)|^2\ge -2(|p(s)|-\beta +\alpha )|p(s)|^2. \end{aligned}$$

If \(\beta -\alpha \le |\nabla \phi ^0|\le \beta +\alpha \) on \(\Sigma (0)\), we have for each \(\xi \in \Sigma (0)\),

$$\begin{aligned} \beta -\alpha \le |p(s)|\le \beta +\alpha \quad \text{ as } \text{ long } \text{ as } p(s) \text{ exists }. \end{aligned}$$

In fact, suppose that there exists \(\tau >0\) such that \(|p(\tau )|>\beta +\alpha \); set \(s^*:=\sup \{ s\le \tau \,|\, |p(s)|=\beta +\alpha \}\); the continuity of \(|p(\cdot )|\) implies that \(|p(s^*)|=\beta +\alpha \) and \(|p(s)|> \beta +\alpha \) for all \(s\in (s^*,\tau ]\); then, we necessarily have

$$\begin{aligned} |p(\tau )|^2\le |p(s^*)|^2-2\int ^\tau _{s^*} (|p(s)|-\beta -\alpha )|p(s)|^2ds <|p(s^*)|^2, \end{aligned}$$

which is a contradiction; a fully analogous argument yields the lower bound. Therefore, we may apply the reasoning of Sect. 2.1 to the following problems: find a tubular neighborhood \(\Theta \) of \(\{\Gamma (t)\}_{t\ge 0}\) and a \(C^2\)-function \(\phi \) satisfying

$$\begin{aligned}{} & {} \left\{ \begin{array}{lll} &{}&{} \frac{\partial \phi }{\partial t} +v\cdot \nabla \phi =\phi (\beta -|\nabla \phi |)\quad \text{ in } \Theta |_{t>0},\\ &{}&{} \phi (0,\cdot )=\phi ^0\quad \text{ on } \Theta |_{t=0}. \end{array} \right. \end{aligned}$$

We obtain the following result.

Theorem 2.4

Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be \(C^2\)-smooth satisfying (2.2) and \(\beta -\alpha \le |\nabla \phi ^0|\le \beta +\alpha \) on \(\Sigma (0)\), where \(\Sigma (0)\) does not touch \(\partial \Omega \). Then, there exists a tubular neighborhood \(\Theta \) of the level-set \(\{\Sigma (t)\}_{t\ge 0}\) for which (2.24) admits a unique \(C^2\)-solution \(\phi \) satisfying

$$\begin{aligned}{} & {} \Sigma ^\phi (t):= \{x\,|\, \phi (t,x)=0\}=\Sigma (t),\quad \forall \,t\in [0,\infty ),\\{} & {} \beta -\alpha \le |\nabla \phi (t,x)|\le \beta +\alpha ,\quad \forall \, t\in [0, \infty ),\,\,\,\forall \,x\in \Sigma (t). \end{aligned}$$

Note that Theorem 2.4 is interesting for numerical purposes because it is usually not important to exactly keep \(|\nabla \phi | \equiv 1\), but rather to stay away from extreme values of the gradient norm. Therefore, the problem (2.24) should be investigated in more detail from the numerical perspective in the future.

3 Viscosity solution on the whole domain

Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded connected open set. Let \(v=v(t,x)\) be a given function satisfying (H1)\('\) and (H2). Let \(\Sigma (0)\subset \Omega \) be a closed 2-dimensional surface (topological manifold) and let \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be a \(C^0\)-function such that

$$\begin{aligned} \phi ^0>0 \text{ on } \Omega ^+(0), \quad \phi ^0<0 \text{ on } \Omega ^-(0), \quad \{x\in \Omega \,|\,\phi ^0(x)=0\}=\Sigma (0), \end{aligned}$$

where \(f(t,x):=\phi ^0(X(0,t,x))\) 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 \(\phi ^0\), where we repeat

$$\begin{aligned} \Sigma (t):=X(t,0,\Sigma (0))=\{x\in \Omega \,|\, f(t,x)=0\},\quad \Sigma (t)\cap \partial \Omega =\emptyset ,\quad \forall \,t\ge 0. \end{aligned}$$

If we deal with the Hamilton–Jacobi equation in (2.3) on the whole domain \(\Omega \), the nonlinear term would suffer from singularity, i.e., \(\langle (\nabla v) p, p \rangle |p|^{-2} \) cannot be continuous at \(p=0\). Hence, we remove the singularity by a multiplying cut-off with respect to p; furthermore, for a simpler structure near \(\partial \Omega \), we also introduce a smooth cut-off with respect to x so that the nonlinearity is effective near the level-set but ineffective near \(\partial \Omega \) (note that \(\nabla v\) is not required on \(\partial \Omega \) after cut-off). Apparently, such a cut-off with respect to x requires a priori information on the unknown level-set (i.e., distance between the level-set and \(\partial \Omega \)). However, we will see that it is sufficient to introduce a cut-off only with the information of the original level-set \(\{\Sigma (t)\}_{t\ge 0}\) defined as (1.6).

We explain two ways to introduce suitable cut-off functions. The first way is simpler but available only for the problem within each finite time interval. Fix an arbitrary terminal time \(T>0\). For \(\varepsilon >0\), let \(K_{\varepsilon } \subset {\bar{\Omega }}\) denote the intersection of \({\bar{\Omega }}\) and the \(\varepsilon \)-neighborhood of \(\partial \Omega \). Then, since \(\Sigma (t)\) never touches \(\partial \Omega \) within [0, T], we find \(\varepsilon >0\) such that

$$\begin{aligned} K_{3\varepsilon } \cap \Sigma (t)=\emptyset \quad (\text{ or, } \text{ equivalently, } K_{3\varepsilon }\subset \Omega ^-(t) ),\quad \forall \, t \in [0,T]. \end{aligned}$$

Let \(0<a_0<a_1\) be arbitrary constants. Define the continuous function \(R_T:[0,T]\times {\bar{\Omega }}\times {{\mathbb {R}}}^3\rightarrow {{\mathbb {R}}}\) as

$$\begin{aligned} R_T(t,x,p):= & {} \eta _0(x)\eta _2(|p|)\left\langle \big ( \nabla v(t,x)\big ) \frac{p}{|p|},\frac{p}{|p|}\right\rangle , \\ \eta _0(x):= & {} \left\{ \begin{array}{ll} 1&{}\quad \text{ for } x\not \in K_{3\varepsilon }, \\ 0&{}\quad \text{ for } x\in K_{2\varepsilon }, \\ \text{ non-negative } \text{ smooth } \text{ transition } \text{ from } 1 \text{ to } 0&{}\quad \text{ otherwise }, \end{array} \right. \\ \eta _2(r):= & {} \left\{ \begin{array}{ll} 1&{}\quad \text{ for } a_1\le r, \\ 0&{}\quad \text{ for } r\le a_0, \\ \text{ monotone } \text{ smooth } \text{ transition } \text{ from } 1 \text{ to } 0&{}\quad \text{ otherwise }. \end{array} \right. \end{aligned}$$

This choice of cut-off is uncomplicated since we do not need detailed information on the asymptotics of dist\((\Sigma (t),\partial \Omega )\) as \(t\rightarrow \infty \); however, \(R_T\) depends on the terminal time and a time global analysis is impossible. The second way is to allow t-dependency for the cut-off. Since \(\Sigma (t)\) never touches \(\partial \Omega \) within \([0,\infty )\), we find a smooth nonincreasing function \(\varepsilon :[0,\infty )\rightarrow {{\mathbb {R}}}_{>0}\) (possibly \(\varepsilon (t)\rightarrow 0\) as \(t\rightarrow \infty \)) such that

$$\begin{aligned} K_{3\varepsilon (t)} \cap \Sigma (t)=\emptyset \quad (\text{ or, } \text{ equivalently, } K_{3\varepsilon (t)}\subset \Omega ^-(t) ),\quad \forall \, t \in [0,\infty ). \end{aligned}$$

Then, we take a smooth function \(\eta _1:[0,\infty )\times {\bar{\Omega }}\rightarrow [0,1]\) such that

$$\begin{aligned} \eta _1(t,x)= & {} \left\{ \begin{array}{ll} 1&{}\quad \text{ for } x\not \in K_{3\varepsilon (t)}, \\ 0&{}\quad \text{ for } x\in K_{2\varepsilon (t)}, \\ \text{ non-negative } \text{ smooth } \text{ transition } \text{ from } 1 \text{ to } 0&{}\quad \text{ otherwise }, \end{array} \right. \end{aligned}$$

where we omit an explicit formula of such \(\eta _1\), and define the continuous function R as

$$\begin{aligned}{} & {} R(t,x,p):=\eta _1(t,x)\eta _2(|p|)\left\langle \big ( \nabla v(t,x)\big ) \frac{p}{|p|},\frac{p}{|p|}\right\rangle , \end{aligned}$$

where \(\eta _2\) is the same as in \(R_T\) above.

Due to (2.1), there exists a constant \(V_0>0\) such that

$$\begin{aligned} \sup |R_T|\le V_0,\quad \sup |R|\le V_0. \end{aligned}$$

In the rest of the paper, we take R given as (3.2). 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 \(\phi \) of

$$\begin{aligned}{} & {} \left\{ \begin{array}{rll} \frac{\partial \phi }{\partial t}(t,x)+v(t,x)\cdot \nabla \phi (t,x)&{}=&{} \phi (t,x)R(t,x,\nabla \phi (t,x)) \text{ in } (0,T)\times \Omega ,\\ \phi (0,x)&{}=&{}\phi ^0(x) \text{ on } \Omega ,\\ \phi (t,x)&{}=&{}\phi ^0(X(0,t,x)) \text{ on } [0,T)\times \partial \Omega . \end{array} \right. \end{aligned}$$

Before going on with (3.4), we recall the definition of viscosity (sub/super)solutions of a general first order Hamilton–Jacobi equation of the form

$$\begin{aligned} G(z,u(z),\nabla _z u(z))=0 \quad \text{ in } O, \end{aligned}$$

where \(O\subset {{\mathbb {R}}}^N\) is an open set, \(G=G(z,u,q):O\times {{\mathbb {R}}}\times {{\mathbb {R}}}^N\rightarrow {{\mathbb {R}}}\) is a given continuous function and \(u:O\rightarrow {{\mathbb {R}}}\) is the unknown function. Our evolutional Hamilton–Jacobi equation is also seen in this form with \(z=(t,x)\). Let us note in passing that (3.5) is often treated as a typical example of degenerate second order PDEs (i.e., the second order term is completely degenerate to be 0) in the literature, e.g., in the main reference [11]. To state the definition, we introduce the upper semicontinuous envelope \(u^*:O\rightarrow {{\mathbb {R}}}\) and the lower semicontinuous envelope \(u_*:O\rightarrow {{\mathbb {R}}}\) of a locally bounded function \(u:O\rightarrow {{\mathbb {R}}}\) as

$$\begin{aligned} u^*(z):= & {} \lim _{r\rightarrow 0}\,\sup \{ u(y)\,|\,y\in O,\,\,\,0\le |y-z|\le r\},\\ u_*(z):= & {} \lim _{r\rightarrow 0}\,\inf \{ u(y)\,|\,y\in O,\,\,\,0\le |y-z|\le r\}. \end{aligned}$$

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 \(u:O\rightarrow {{\mathbb {R}}}\) is a viscosity subsolution (resp. supersolution) of (3.5), provided

  • \(u^*\) is bounded from above (resp. \(u_*\) is bounded from below);

  • If \((\varphi ,z)\in C^1(O;{{\mathbb {R}}})\times O\) satisfies

    $$\begin{aligned} \max _{y\in O} (u^*(y)-\varphi (y))=u^*(z)-\varphi (z) \quad (\text{ resp. } \min _{y\in O} (u_*(y)-\varphi (y))=u_*(z)-\varphi (z)), \end{aligned}$$

    we have

    $$\begin{aligned} G(z,u^*(z),\nabla _y \varphi (z))\le 0 \quad (\text{ resp. } G(z,u_*(z),\nabla _y \varphi (z))\ge 0). \end{aligned}$$

A function \(u:O\rightarrow {{\mathbb {R}}}\) is a viscosity solution of (3.5), if it is both a viscosity subsolution and supersolution of (3.5).

It is well-known that \(\max ,\min \) in the definition can be replaced by the local (strict) maximum, local (strict) minimum, respectively.

3.1 Existence of viscosity solutions

We state the main result of this subsection.

Theorem 3.1

Suppose that v satisfy (H1)\('\) and (H2). Let \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be continuous satisfying (3.1). Let \(T>0\) be arbitrary. Then, there exists a unique viscosity solution \(\phi \in C^0([0,T)\times {\bar{\Omega }};{{\mathbb {R}}})\) of (3.4), i.e., \(\phi \) satisfies the first equation in (3.4) in the sense of the above definition and the initial/boundary condition strictly. Furthermore, it holds that

$$\begin{aligned} \Sigma ^\phi (t):=\{x\in \Omega \,|\,\phi (t,x)=0 \}=\Sigma (t),\quad \forall \,t\in [0,T). \end{aligned}$$


Because \(T>0\) is arbitrary, Theorem 3.1implies the existence of a unique viscosity solution \(\phi \in C^0([0,\infty )\times {\bar{\Omega }};{{\mathbb {R}}})\) of (3.4)\(|_{T=\infty }\) with \(\Sigma ^\phi (t)=\Sigma (t)\) for all \(t\in [0,\infty )\).

Proof of Theorem 3.1

Introduce the function \(S:[0,T]\times {\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) as

$$\begin{aligned}{} & {} S(t,x):=\left\{ \begin{array}{ll} -V_0&{}\quad \text{ for } x\in \overline{\Omega ^+(t)}, t\in [0,T],\\ V_0&{}\quad \text{ for } x\in \Omega ^-(t){\setminus } K_{\varepsilon (T)}, t\in [0,T],\\ 0&{}\quad \text{ for } x\in K_{\varepsilon (T)/2}, t\in [0,T],\\ \text{ non-negative } \text{ smooth } \text{ transition } \text{ from }\\ V_0 \text{ to } 0&{}\quad \text{ otherwise }, \end{array} \right. \end{aligned}$$

where \(\varepsilon (t)\ge \varepsilon (T)\) for all \(t\in [0,T]\), S is such that \(S\equiv 0\) near \(\partial \Omega \) for all \(t\in [0,T]\) and S is smooth except on \(\cup _{0\le t\le T}(\{t\}\times \Sigma (t))\). It holds that

$$\begin{aligned} S(t,x)&=-V_0\le R(t,x,p),\quad \forall \,(t,x)\in \bigcup _{0\le t\le T}\big (\{t\}\times \overline{\Omega ^+(t)}\big ), \quad \forall \, p\in {{\mathbb {R}}}^3, \end{aligned}$$
$$\begin{aligned} S(t,x)&\ge R(t,x,p),\quad \forall \,(t,x)\in \bigcup _{0\le t\le T}\big (\{t\}\times \Omega ^-(t)\big ), \quad \forall \, p\in {{\mathbb {R}}}^3, \end{aligned}$$

where we note that \(\eta _1\equiv 0\) on the region on which S has positive transition from \(V_0\) to 0. As a candidate of a viscosity sub/supersolution, introduce the functions \(\rho ,{\tilde{\rho }}:[0,T)\times {\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) as

$$\begin{aligned} \rho (t,x)&:=\phi ^0(X(0,t,x))e^{\int _0^t S(s,X(s,t,x))ds}\quad (X \text{ is } \text{ the } \text{ flow } \text{ of } (1.2)),\\ {\tilde{\rho }}(t,x)&:=\phi ^0(X(0,t,x))e^{\int _0^t -S(s,X(s,t,x))ds}, \end{aligned}$$

where \(\rho ,{\tilde{\rho }}\) are continuous (smooth except on the level-set), \(\rho (0,\cdot )={\tilde{\rho }}(0,\cdot )=\phi ^0\) and \(\rho (t,x)={\tilde{\rho }}(t,x)=\phi ^0(X(0,t,x))\) on \([0,T)\times \partial \Omega \).

Step 1. We prove that \(\rho \) (resp. \({\tilde{\rho }}\)) is a viscosity subsolution (resp. supersolution) of (3.4) satisfying the initial/boundary condition strictly. For this purpose, we first observe:

Lemma 3.2

Suppose that v satisfy (H1)\('\) and (H2). Let \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be continuous. Then, \(f(t,x):=\phi ^0(X(0,t,x))\) satisfies \(\frac{\partial f}{\partial t}+v\cdot \nabla f=0\) on \((0,T)\times \Omega \) in the sense of viscosity solutions.


We first check that f is a viscosity subsolution. Fix any \((t,x)\in (0,T)\times \Omega \). Let \(\varphi \) be any test function satisfying the condition of the test for viscosity subsolutions at (tx), i.e., \(f(s,y)-\varphi (s,y) \le f(t,x)-\varphi (t,x)\) for all (sy) near (tx). Then, we see that

$$\begin{aligned}{} & {} \limsup _{(s,y)\rightarrow (t,x)}\frac{f(s,y)-f(t,x)-(\varphi (s,y)-\varphi (t,x))}{|(s,y)-(t,x)|}\nonumber \\{} & {} \quad =\limsup _{(s,y)\rightarrow (t,x)}\frac{f(s,y)-f(t,x)-\textrm{D}\varphi (t,x)\cdot ((s,y)-(t,x))}{|(s,y)-(t,x)|}\le 0, \end{aligned}$$

i.e., \(\textrm{D}\varphi (t,x)=(\frac{\partial \varphi }{\partial s}(t,x),\nabla _y\varphi (t,x))\in \textrm{D}^+f(t,x)\), where \(\textrm{D}^+f(t,x)\) stands for the superdifferentialFootnote 3 of f at (tx). Since \(f(s,X(s,t,x))\equiv f(t,x)= \phi ^0(X(0,t,x))\) for all \(0\le s\le t\) and

$$\begin{aligned} \lim _{s\rightarrow t-0}\frac{X(s,t,x)-x}{s-t}=\lim _{s\rightarrow t-0}\frac{X(s,t,x)-X(t,t,x)}{s-t}=v(t,x), \end{aligned}$$

(3.8) with \(y=X(s,t,x)\) and \(s\rightarrow t-0\) implies that

$$\begin{aligned} \textrm{D}\varphi (t,x)\cdot (1,v(t,x))\le 0, \end{aligned}$$

from which we obtain

$$\begin{aligned} \frac{\partial \varphi }{\partial s}(t,x)+v(t,x)\cdot \nabla _y \varphi (t,x)\le 0. \end{aligned}$$

Therefore, we conclude that f is a viscosity subsolution. A similar reasoning shows that f is a viscosity supersolution, where we look at the subdifferentialFootnote 4\(\textrm{D}^-f(t,x)\) of f. \(\square \)

We come back to \(\rho ,{\tilde{\rho }}\) in the proof of Theorem 3.1. Fix any \((t,x)\in (0,T)\times \Omega \). We check that \(\rho \) is viscosity subsolution of (3.4). Let \(\varphi \) be any test function satisfying the condition of the test for viscosity subsolutions at (tx):

$$\begin{aligned} \rho (s,y)-\varphi (s,y) \le \rho (t,x)-\varphi (t,x), \quad \forall \, (s,y) \text{ near } (t,x), \end{aligned}$$

Case 1: \((t,x)\in \cup _{0< s<T}(\{s\}\times \Sigma (s))\). Then, \(\rho (s,X(s,t,x))=\rho (t,x)= 0\) for all \(0\le s\le t\). Hence, the same reasoning as the proof of Lemma 3.2 works, i.e., taking \((s,y)=(s,X(s,t,x)\) in (3.9) and sending \(s\rightarrow t-0\), we obtain

$$\begin{aligned} \textrm{D}\varphi (t,x)\cdot (1,v(t,x))\le 0. \end{aligned}$$

This yields the desired inequality:

$$\begin{aligned} \frac{\partial \varphi }{\partial s}(t,x)+v(t,x)\cdot \nabla _y \varphi (t,x)\le 0=\rho (t,x)R(t,x,\nabla \varphi _y(t,x)). \end{aligned}$$

Case 2: \((t,x)\not \in \cup _{0< s<T}(\{s\}\times \Sigma (s))\). Setting \(r:=\rho (t,x)-\varphi (t,x)\) in (3.9), we have

$$\begin{aligned} \rho (s,y)-(\varphi (s,y)+r) \le 0= \rho (t,x)-(\varphi (t,x)+r), \quad \forall \, (s,y) \text{ near } (t,x). \end{aligned}$$

Since \(e^{-\int _0^s S(s',X(s',s,y))ds'}>0\), (3.10)\(\times e^{-\int _0^s S(s',X(s',s,y))ds'}\) yields

$$\begin{aligned}&\phi ^0(X(0,s,y))-(\varphi (s,y)+r)e^{-\int _0^s S(s',X(s',s,y))ds'} \le 0, \end{aligned}$$

while \(0= \rho (t,x)-(\varphi (t,x)+r) = \rho (t,x)e^{-\int _0^t S(s',X(s',t,x))ds'}-(\varphi (t,x)+r)e^{-\int _0^t S(s',X(s',t,x))ds'}\). Hence, setting \(\psi (s,y):=(\varphi (s,y)+r)e^{-\int _0^s S(s',X(s',s,y))ds'}\), we obtain

$$\begin{aligned}&\phi ^0(X(0,s,y))-\psi (s,y) \le \phi ^0(X(0,t,x))-\psi (t,x), \quad \forall \, (s,y) \text{ near } (t,x). \end{aligned}$$

Since \(\psi (s,y)\) is \(C^1\)-smooth near (tx), \(\psi \) serves as a test function for \(f(s,y)=\phi ^0(X(0,s,y))\) at (tx); since Lemma 3.2 confirms that f is a viscosity solution of \(\frac{\partial f}{\partial t}+v\cdot \nabla f=0\) on \((0,T)\times \Omega \), we obtain

$$\begin{aligned} \frac{\partial \psi }{\partial s}(t,x)+v(t,x)\cdot \nabla _y\psi (t,x)\le 0. \end{aligned}$$

Set \(g(s,y):=-\int _0^s S(s',X(s',s,y))ds'\). Then, \(\psi =(\varphi +r) e^g\) and the left hand side of (3.11) is

$$\begin{aligned}&\left( \frac{\partial \varphi }{\partial s}(t,x)+v(t,x)\cdot \nabla _y\varphi (t,x)\right) e^{g(t,x)}\\&\quad +\left( \frac{\partial g}{\partial s}(t,x)+v(t,x)\cdot \nabla _y g(t,x) \right) (\varphi (t,x)+r)e^{g(t,x)}. \end{aligned}$$

A direct calculation yields \(\frac{\partial g}{\partial s}(t,x)+v(t,x)\cdot \nabla _y g(t,x)=-S(t,x)\). Therefore, we obtain

$$\begin{aligned} \frac{\partial \varphi }{\partial s}(t,x)+v(t,x)\cdot \nabla _y\varphi (t,x)\le (\varphi (t,x)+r)S(t,x)=\rho (t,x)S(t,x). \end{aligned}$$

It follows from (3.6) and (3.7) that, if \(x\in \Omega ^+(t)\) (resp. \(x\in \Omega ^-(t)\)), \(\rho (t,x)>0\) and \(\rho (t,x)S(t,x)=-\rho (t,x)V_0\le \rho (t,x)R(t,x,\nabla _y\varphi (t,x))\) (resp. \(\rho (t,x)<0\) and \(\rho (t,x)S(t,x)\le \rho (t,x)R(t,x,\nabla _y\varphi (t,x))\)). Thus, we obtain

$$\begin{aligned} \frac{\partial \varphi }{\partial s}(t,x)+v(t,x)\cdot \nabla _y\varphi (t,x)\le \rho (t,x)R(t,x,\nabla _y\varphi (t,x)). \end{aligned}$$

We conclude that \(\rho \) is a viscosity subsolution. A similar reasoning shows that \({\tilde{\rho }}\) is a viscosity supersolution.

Note that one can also proceed in Case 2 of Step 1 based on mollification of \(\phi ^0\) and the fact that \(\rho \), \({\tilde{\rho }}\) satisfy \(\frac{\partial \rho }{\partial t}(t,x)+v(t,x)\cdot \nabla \rho (t,x)=\rho (t,x)S(t,x)\), \(\frac{\partial {\tilde{\rho }}}{\partial t}(t,x)+v(t,x)\cdot \nabla {\tilde{\rho }}(t,x)=-{\tilde{\rho }}(t,x)S(t,x)\), provided \(\phi ^0\) is smooth.

Step 2. We apply Perron’s method and comparison principle. Recall that \(\partial \Omega \) is invariant under X and that S is defined to satisfy \(S\equiv 0\) near \(\partial \Omega \). Hence, there exist \(\varepsilon _1,\varepsilon _2\) with \(0<\varepsilon _2\ll \varepsilon _1\ll \varepsilon (T)\) such that \(S\equiv 0\) on \([0,T)\times K_{\varepsilon _1}\) and \(X(s,t,x) \in K_{\varepsilon _1}\) for all \(s\in [0,t]\) and all \((t,x)\in [0,T)\times K_{\varepsilon _2}\). Therefore, the viscosity subsolution and supersolution \(\rho ,{\tilde{\rho }}\in C^0([0,T)\times {\bar{\Omega }})\) are such that

$$\begin{aligned}{} & {} \rho \le {\tilde{\rho }} \text{ in } [0,T)\times {\bar{\Omega }},\quad \rho ={\tilde{\rho }}=\phi ^0(X(0,t,x)) \text{ in } [0,T)\times K_{\varepsilon _2},\\{} & {} \rho (0,x)={\tilde{\rho }}(0,x)=\phi ^0(x) \text{ on } {\bar{\Omega }},\quad \rho ={\tilde{\rho }}=0 \text{ on } \bigcup _{0\le s<T}\left( \{s\}\times \Sigma (s)\right) . \end{aligned}$$

Applying Perron’s method (Theorem 3.1 in [23]), we obtain a viscosity solution \(\phi :(0,T)\times \Omega \rightarrow {{\mathbb {R}}}\) of the first equation in (3.4) such that

$$\begin{aligned} \rho \le \phi \le {\tilde{\rho }} \text{ in } (0,T)\times \Omega . \end{aligned}$$

Since the level-sets of \(\rho (t,\cdot ),{\tilde{\rho }}(t,\cdot )\) are equal to \(\Sigma (t)\), the level-sets of \(\phi (t,\cdot )\) must be equal to \(\Sigma (t)\) as well. It is clear that \(\phi \) can be continuously extended up to \(([0,T)\times \partial \Omega )\cup (\{0\}\times {\bar{\Omega }})\) satisfying

$$\begin{aligned} \rho (t,x)=\phi (t,x)={\tilde{\rho }}(t,x)=\phi ^0(X(0,t,x)) \text{ on } ([0,T)\times \partial \Omega )\cup (\{0\}\times {\bar{\Omega }}); \end{aligned}$$

the same holds for \(\phi ^*,\phi _*\), the lower/upper semicontinuous envelop of \(\phi \):

$$\begin{aligned} \phi ^*(t,x)=\phi _*(t,x)=\phi ^0(X(0,t,x)) \text{ on } ([0,T)\times \partial \Omega )\cup (\{0\}\times {\bar{\Omega }}). \end{aligned}$$

By definition, we have \(\phi _*\le \phi ^*\). On the other hand, \(\phi \) being both a viscosity subsolution and supersolution implies that \(\phi ^*\) is an upper semicontinuous viscosity subsolution (note that \((\phi ^{*})^{*}=\phi ^*\)) and \(\phi _*\) is a lower semicontinuous viscosity supersolution (note that \((\phi _{*})_*=\phi _*\)); the comparison principle (Theorem 8.2 in [11]) implies that \(\phi ^*\le \phi _*\) on \([0,T)\times \Omega \). Here, we remark that (3.4) in the form of (3.5) does not directly satisfy the monotonicity property (\(u\mapsto G(z,u,q)\) must be nondecreasing), but (3.3) implies that one can verify the monotonicity property through the change of variable \(u=e^{V_0t}{\tilde{u}}\) (see the following subsection and Chapter 2 of [20]). Thus, we conclude that \(\phi \) is continuous on \([0,T)\times {\bar{\Omega }}\) satisfying the initial/boundary condition strictly. Furthermore, such a viscosity solution is unique. In fact, if \({\tilde{\phi }}\) is a viscosity solution of (3.4) in the current sense, the comparison principle implies \(\phi \le {\tilde{\phi }}\) by regarding \(\phi \) as a viscosity subsolution and \({\tilde{\phi }}\) as a viscosity supersolution; \(\phi \ge {\tilde{\phi }}\) by regarding \(\phi \) as a viscosity supersolution and \({\tilde{\phi }}\) as a viscosity subsolution. \(\square \)

We remark that the result and reasoning of this subsection hold also for the Hamilton–Jacobi equation corresponding to (2.24), where we need to add suitable cut-off to the nonlinearity in (2.24). To see this, observe that the mapping \(u\mapsto u(\beta -|p|)\) is not monotone for all \(p\in {{\mathbb {R}}}^3\); hence, we define

$$\begin{aligned} R(t,x,p)&:=\eta _1(t,x)\eta _3(|p|)(\beta -|p|),\nonumber \\ \eta _3(r)&:=\left\{ \begin{array}{ll} 1&{}\quad \text{ for } 0\le r\le 2(\beta +\alpha ) , \\ 0&{}\quad \text{ for } r\ge 3(\beta +\alpha ), \\ \text{ monotone } \text{ smooth } \text{ transition } \text{ from } 1 \text{ to } 0&{}\quad \text{ otherwise }, \end{array} \right. \nonumber \\ \beta&>\alpha >0 \text{ are } \text{ the } \text{ constants } \text{ given } \text{ in } \text{ Sect. }~2.3{} \end{aligned}$$

to confirm \(\sup |R|\le {\tilde{V}}_0:=2\beta +3\alpha \). Then, with this R, Theorem 3.1 still holds.

3.2 \(C^2\)-regularity of viscosity solution near the level-set

We prove that the viscosity solution of (3.4) under the stronger regularity assumptions on v and \(\phi ^0\) coincides with the classical solution of (2.3) in a t-global tubular neighborhood of the level-set. Note that \(|\nabla \phi |\) is nicely controlled on the level-set in (2.3) and there exists a tubular neighborhood \(\Theta \) in which \(\eta _1(t,x)\eta _2(|\nabla \phi (t,x)|)\equiv 1\), i.e., the solution of (2.3) satisfies the Hamilton–Jacobi equation in (3.4) within \(\Theta \) and local in space comparison of the solutions to (2.3) and (3.4) makes sense.

Our proof is motivated by the technique known as doubling the number of variables, which is standard in proofs of comparison principles for two viscosity (sub/super)solutions; more precisely, we adapt the localized version of this technique to our situation with an unusual choice of a penalty function, where the reasoning can be much simpler due to the fact that one of the two viscosity solutions is smooth.

The outcome of standard “localized doubling the number of variables” (see, e.g., Theorem 3.12 of [6]) for a Hamilton–Jacobi equation

$$\begin{aligned} \frac{\partial u}{\partial t}+ H(t,x,\nabla u)=0 \end{aligned}$$

states the following:

  • Let \(u,{\tilde{u}}\) be \(C^0\)-viscosity solutions of (3.14) defined in a cone

    $$\begin{aligned} {\mathcal {C}}:= \{ (t,x)\in [0,b]\times {{\mathbb {R}}}^N\,|\, |x-z|\le C(b-t) \}, \end{aligned}$$

    where \(C>0\) is a constant such that

    $$\begin{aligned}&|H(t,x,p)-H(t,x,q)|\le C|p-q| ,\\&|H(t,x,p)-H(s,y,p)|\le C(1+|p|)|(t,x)-(s,y)|. \end{aligned}$$

    If \(u(0,\cdot )={\tilde{u}}(0,\cdot )\) on \(\{x\,|\,|x-z|\le Cb\}\) (the bottom of \({\mathcal {C}}\)), then \(u\equiv {\tilde{u}}\) on \({\mathcal {C}}\).

The cone is the region of dependence for general first order Hamilton–Jacobi equations, i.e., the value u(bz) is determined by the information only on the bottom of \({\mathcal {C}}\) (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 \(\Theta \), 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., \(h({\bar{u}}^2)\) below) that consists of the classical solution itself. In this sense, our technique is specialized for local in space 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.

Suppose that v satisfies (H1) and (H2). We consider (3.4) with initial data \(\phi ^0\) belonging to \(C^2({\bar{\Omega }};{{\mathbb {R}}})\) and satisfying (2.2) with \(a_1<|\nabla \phi ^0|\) on \(\Sigma (0)\) (this ensures \(\eta _1\eta _2\equiv 1\) near the level-set; \(a_1\) appears in the definition of \(\eta _2\)), where \(\Sigma (0)\subset \Omega \) is assumed to be a closed \(C^2\)-smooth surface. Then, Theorem 2.2 yields a tubular neighborhood \(\Theta \) of the level-set \(\{\Sigma (t)\}_{t\ge 0}\) and a unique \(C^2\)-solution \(\phi :\Theta \rightarrow {{\mathbb {R}}}\) of (2.3) with \(\eta _1(t,x)\eta _2(|\nabla \phi (t,x)|)\equiv 1\) in \(\Theta \) (hence, from here, we rewrite (2.3) with R given in (3.2)), while Theorem 3.1 yields a unique continuous viscosity solution \({\tilde{\phi }}:[0,\infty )\times {\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) of (3.4). We fix an arbitrary \(T>0\) and set \(\Theta _T:=\Theta _{0\le t\le T}\).

We need to convert the Hamilton–Jacobi equation in (3.4) into

$$\begin{aligned} \frac{\partial u}{\partial t}+ G(t,x,\nabla u, u)=0 \end{aligned}$$

with G such that \(u\mapsto G(t,x,p,u)\) is nondecreasing. This is done by the change of the functions as

$$\begin{aligned} w(t,x):=e^{-V_0t}\phi (t,x),\quad {\tilde{w}}(t,x):=e^{-V_0t}{\tilde{\phi }}(t,x) \end{aligned}$$

with the constant \(V_0\) given in (3.3). Since \(\phi \) is a \(C^2\)-solution of the original Hamilton–Jacobi equation, it is clear that w satisfies the new Hamilton–Jacobi equation

$$\begin{aligned} \frac{\partial u}{\partial t}(t,x)+ v(t,x)\cdot \nabla u(t,x)+u(t,x)\left( V_0-R(t,x,e^{V_0t}\nabla u(t,x))\right) =0. \end{aligned}$$

Note that w satisfies (3.17) also in the sense of viscosity solutions. In the case of \({\tilde{\phi }}\) being a viscosity solution, one can also show that \({\tilde{w}}:=e^{-V_0t}{\tilde{\phi }}\) is a viscosity solution of (3.17). For the readers’ convenience, we briefly explain how to do it: suppose that a test function \(\psi \) is such that \({\tilde{w}}-\psi \) has a local maximum at \((t_0,x_0)\); then, setting the constant \(r:={\tilde{w}}(t_0,x_0)-\psi (t_0,x_0)\), we have

$$\begin{aligned}&{\tilde{w}}(t,x)-\psi (t,x)\le {\tilde{w}}(t_0,x_0)-\psi (t_0,x_0)=r\quad \text{ near } (t_0,x_0),\\&{\tilde{w}}(t,x)-(\psi (t,x)+r)\le 0= {\tilde{w}}(t_0,x_0)-(\psi (t_0,x_0)+r)\quad \text{ near } (t_0,x_0), \end{aligned}$$

from which we obtain

$$\begin{aligned} e^{V_0t}{\tilde{w}}(t,x)-e^{V_0t}(\psi (t,x)+r)\le 0&={\tilde{w}}(t_0,x_0)-(\psi (t_0,x_0)+r)\quad \text{ near } (t_0,x_0),\\ e^{V_0t}{\tilde{w}}(t,x)-e^{V_0t}(\psi (t,x)+r)\le 0&=e^{V_0t_0}{\tilde{w}}(t_0,x_0)-e^{V_0t_0}(\psi (t_0,x_0)+r)\quad \text{ near } (t_0,x_0),\\ {\tilde{\phi }}(t,x)-e^{V_0t}(\psi (t,x)+r)&\le {\tilde{\phi }}(t_0,x_0)-e^{V_0t_0}(\psi (t_0,x_0)+r)\quad \text{ near } (t_0,x_0); \end{aligned}$$

since \({\tilde{\phi }}\) is a viscosity subsolution, it holds that

$$\begin{aligned}&\frac{\partial }{\partial t}\{e^{V_0t}(\psi (t,x)+r)\}+v(t,x)\cdot \nabla \{e^{V_0t}(\psi (t,x)+r)\} \\&\qquad - {\tilde{\phi }}(t,x)R\left( t,x,\nabla \{e^{V_0t}(\psi (t,x)+r)\}\right) |_{t=t_0,x=x_0}\\&\quad =e^{V_0t_0}V_0(\psi (t_0,x_0)+r) + e^{V_0t_0}\frac{\partial \psi }{\partial t}(t_0,x_0)+e^{V_0t_0}v(t_0,x_0)\cdot \nabla \psi (t_0,x_0) \\&\qquad - {\tilde{\phi }}(t_0,x_0)R\left( t_0,x_0,e^{V_0t_0}\nabla \psi (t_0,x_0)\}\right) \le 0; \end{aligned}$$

hence, noting that \(\psi (t_0,x_0)+r={\tilde{w}}(t_0,x_0)\) and \({\tilde{\phi }}(t_0,x_0)=e^{V_0t_0}{\tilde{w}}(t_0,x_0)\), we obtain

$$\begin{aligned} \frac{\partial \psi }{\partial t}(t_0,x_0)+v(t_0,x_0)\cdot \nabla \psi (t_0,x_0)+{\tilde{w}}(t_0,x_0)\left( V_0 -R(t_0,x_0,e^{V_0t_0}\nabla \psi (t_0,x_0)) \right) \le 0; \end{aligned}$$

therefore, we conclude that \({\tilde{w}}\) is a viscosity subsolution of (3.17); similar argument shows that \({\tilde{w}}\) is a viscosity supersolution of (3.17).

We rewrite (3.17) in the form of (3.15) with

$$\begin{aligned} G(t,x,p,u):=v(t,x)\cdot p+ u{\tilde{R}}(t,x,p),\quad {\tilde{R}}(t,x,p):= V_0-R(t,x,e^{V_0t}p). \end{aligned}$$

Due to the condition of v and the definition of R (see (3.2)), there exists a constant \(C>0\) such that

$$\begin{aligned} |{\tilde{R}}(t,x,p)-{\tilde{R}}(t,x,q)|&\le C|p-q| ,\quad \forall \,(t,x)\in \Theta _T,\,\,\, \forall \,p,q\in {{\mathbb {R}}}^3,\nonumber \\ |{\tilde{R}}(t,x,p)-{\tilde{R}}(s,y,p)|&\le C(1+|p|)|(t,x)-(s,y)|,\,\,\, \forall \,(t,x)\in \Theta _T,\,\,\, \forall \,p\in {{\mathbb {R}}}^3. \end{aligned}$$

In order to have \(\phi \) defined also on the boundary of a tubular neighborhood, we consider another tubular neighborhood \({\tilde{\Theta }}\subset \Theta \) of \(\{\Sigma (t)\}_{t\ge 0}\) such that \(\phi \) is defined on the closure of \({\tilde{\Theta }}\), and (partly) describe \({\tilde{\Theta }}_T:={\tilde{\Theta }}|_{0\le t \le T}\) as a family of cylinders, i.e., a foliation:

  • Let \(\alpha >0\) be a constant such that

    $$\begin{aligned} \alpha -2V_0-C\sup _{{\tilde{\Theta }}_T}|\nabla w|>0\,\,\, \text{ with } w(t,x)=e^{-V_0t}\phi (t,x); \end{aligned}$$
  • Consider the function

    $$\begin{aligned} {\bar{u}}(t,x):= e^{\alpha t}w(t,x), \end{aligned}$$

    where \({\bar{u}}\) solves in the classical sense

    $$\begin{aligned} \frac{\partial u}{\partial t}+v\cdot \nabla u+u{\tilde{R}}(t,x,e^{-\alpha t}\nabla u)= \alpha u ,\quad u(0,\cdot )=\phi ^0; \end{aligned}$$
  • With a constant \(m_0>0\), define

    $$\begin{aligned} A_m&:=\{ (t,x)\in [0,T]\times \Omega \,|\,{\bar{u}}(t,x)=m\},\quad -m_0\le m\le m_0,\quad \\ \Gamma _T&:= \bigcup _{-m_0\le m\le m_0} A_m. \end{aligned}$$

    Since \(\nabla {\bar{u}}\ne 0\) near \(\cup _{0\le t\le T} (\{t\}\times \Sigma (t))\), we may choose \(m_0>0\) (possibly very small) so that \(\Gamma _T\) is contained in \({\tilde{\Theta }}_T\) and \(\partial \Gamma _T{\setminus } \Gamma _T|_{t=0,T}=(A_{m_0}\cup A_{-m_0})|_{0<t<T}\), while \(\Gamma _T\) contains the [0, T]-part of a tubular neighborhood of \(\{\Sigma (t)\}_{t\ge 0}\). Note that such an \(m_0\) depends, in general, on T.

Theorem 3.3

Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be a function belonging to \(C^2({\bar{\Omega }};{{\mathbb {R}}})\) and satisfying (2.2) with \(a_1<|\nabla \phi ^0|\) on \(\Sigma (0)\), where \(a_1\) appears in the cut-off \(\eta _2\) and \(\Sigma (0)\subset \Omega \) is a closed \(C^2\)-smooth surface. Then, for the \(C^2\)-solution \(\phi \) of (2.3) and the continuous viscosity solution \({\tilde{\phi }}\) of (3.4), it holds that \(\phi \equiv {\tilde{\phi }}\) on \(\Gamma _T\), where \(T>0\) is arbitrary.


\(\Gamma _T\) would become narrower as T gets larger. In order to find a more optimal region on which \(\phi \equiv {\tilde{\phi }}\), one can iterate Theorem 3.3 for \(T=T_0\), \(T=2T_0\), \(T=3T_0\), \(\ldots \), with a small \(T_0>0\).

Proof of Theorem 3.3

We proceed by contradiction. Suppose that the assertion does not hold. We assume

$$\begin{aligned} \max _{\Gamma _T}(\phi -{\tilde{\phi }})>0 \text{(in } \text{ the } \text{ other } \text{ case, } \text{ we } \text{ switch } \phi \text{ and } {\tilde{\phi }}). \end{aligned}$$

From now on, we deal with \(w,{\tilde{w}}\) given as (3.16). Then, we find an interior point \((t^*,x^*)\) of \(\Gamma _T\) such that

$$\begin{aligned} \sigma :=w(t^*,x^*)-{\tilde{w}}(t^*,x^*)>0. \end{aligned}$$

Since the level-set of w and that of \({\tilde{w}}\) are identical, we see that \((t^*,x^*)\not \in A_0=\cup _{0\le t\le T}(\{t\}\times \Sigma (t))\). Let \(m^*\in (-m_0,m_0) \) be such that

$$\begin{aligned} {\bar{u}}(t^*,x^*)=m^*,\,\,\,\text{ or } \text{ equivalently, } (t^*,x^*)\in A_{m^*}. \end{aligned}$$

Let \(\delta >0\) be a constant such that

$$\begin{aligned} (m^*)^2+2\delta <(m_0)^2. \end{aligned}$$

Take a constant \(M>0\) such that

$$\begin{aligned} M\ge \max _{(t,x,s,y)\in \Gamma _T\times \Gamma _T} |w(t,x)-{\tilde{w}}(s,y)| \end{aligned}$$

and a monotone increasing \(C^1\)-function \(h: {{\mathbb {R}}}\rightarrow [0,3M]\) such that

$$\begin{aligned} h(r) =\left\{ \begin{array}{ll} 3M, &{}\quad \text{ if } (m^*)^2 + 2\delta \le r,\\ 0,&{}\quad \text{ if } r\le (m^*)^2+\delta ,\\ \text{ monotone } \text{ transition } \text{ between } 0 \text{ and } 3M,&{}\quad \text{ otherwise }. \end{array} \right. \end{aligned}$$

With this setting, one can proceed with doubling the number of variables in terms of the function \(F_{\varepsilon ,\lambda }:\Gamma _T\times \Gamma _T\rightarrow {{\mathbb {R}}}\) defined as

$$\begin{aligned} F_{\varepsilon ,\lambda }(t,x,s,y)&:=w(t,x)-{\tilde{w}}(s,y)-\lambda (t+s)-\frac{1}{\varepsilon ^2}\left( |x-y|^2+|t-s|^2\right) \\&\quad - h({\bar{u}}(t,x)^2)-h({\bar{u}}(s,y)^2). \end{aligned}$$

However, since w is now a classical solution, we may simplify the argument of doubling the number of variables as follows. We redefine M as

$$\begin{aligned} M\ge \max _{(t,x)\in \Gamma _T} |w(t,x)-{\tilde{w}}(t,x)|. \end{aligned}$$

For each \(\lambda >0\), define \(F_\lambda :\Gamma _T\rightarrow {{\mathbb {R}}}\) as

$$\begin{aligned} F_\lambda (t,x):= w(t,x) -{\tilde{w}}(t,x)-\lambda t -h({\bar{u}}(t,x)^2). \end{aligned}$$

Let \((t_0,x_0)=(t_0(\lambda ),x_0(\lambda ))\in \Gamma _T\) be such that

$$\begin{aligned} F_{\lambda }(t_0,x_0)=\max _{ \Gamma _T}F_{\lambda }. \end{aligned}$$

Step 1. We claim that there exists a sufficiently small \(\lambda >0\) for which \((t_0,x_0)\) is away from the “lateral surface” of \(\Gamma _T\) (i.e., \((t_0,x_0)\not \in A_{\pm m_0}\), or equivalently, \(|{\bar{u}}(t_0,x_0)|\ne m_0\)) and \(t_0\ne 0\). We prove the claim. Since \(F_{\lambda }(t_0,x_0)\ge F_{\lambda }(t^*,x^*)\) and \(h({\bar{u}}(t^*,x^*)^2)=h((m^*)^2)=0\) by definition, we may fix \(\lambda >0\) small enough to obtain

$$\begin{aligned} w(t_0,x_0)-{\tilde{w}}(t_0,x_0)&\ge F_{\lambda }(t_0,x_0) \ge w(t^*,x^*)-{\tilde{w}}(t^*,x^*)-\lambda t^*-h({\bar{u}}(t^*,x^*)^2)\nonumber \\&=\sigma -\lambda t^*\ge \frac{\sigma }{2}>0. \end{aligned}$$

Suppose that \((t_0,x_0)\) is on the “lateral surface” of \(\Gamma _T\), i.e., \((t_0,x_0)\in A_{\pm m_0}\), or equivalently, \(|{\bar{u}}(t_0,x_0)|=m_0\). Then, with the definition of h, we see that

$$\begin{aligned} F_{\lambda }(t_0,x_0)&=w(t_0,x_0)-{\tilde{w}}(t_0,x_0)-\lambda t_0 - h(m_0^2) \le M-h(m_0^2)=-2M <0, \end{aligned}$$

which contradicts to (3.21). Since \(w\equiv {\tilde{w}}\) on \(\Gamma _T|_{t=0}\), (3.21) implies that \(t_0=0\) is impossible.

Step 2. Fixing \(\lambda >0\) as mentioned in Step 1, we demonstrate the test at \((t_0,x_0)\) for w being a viscosity subsolution (this is an obvious issue since w is a classical solution) and \({\tilde{w}}\) being a viscosity supersolution with the test function \(\psi \) given as

$$\begin{aligned} \psi (t,x):=w(t,x)-\lambda t-h({\bar{u}}(t,x)^2) \quad (w \text{ and } {\bar{u}} \text{ are } C^1\text{-smooth) }. \end{aligned}$$

Note that this makes sense, because the point \((t_0,x_0)\) is either an interior point of \(\Gamma _T\), or \(t_0=T\) and \((T,x_0)\not \in A_{\pm m_0}\) (see Lemma in Section 10.2 of [18] for a remark on the case \(t_0=T\)). Since w is a classical solution, we have

$$\begin{aligned} \frac{\partial w}{\partial t}(t_0,x_0)+G(t_0,x_0,\nabla w(t_0,x_0),w(t_0,x_0))\le 0\quad \text{(in } \text{ fact, } ``=\text {'' holds)}. \end{aligned}$$

For the above \(\psi \), we see that \({\tilde{w}}(t,x)-\psi (t,x)\) takes a minimum at \((t_0,x_0)\). Hence, by the test for \({\tilde{w}}\) being a viscosity supersolutions, we obtain

$$\begin{aligned} \frac{\partial {\tilde{\psi }}}{\partial t}(t_0,x_0)+G(t_0,x_0,\nabla {\tilde{\psi }}(t_0,x_0),{\tilde{w}}(t_0,x_0))\ge 0, \end{aligned}$$

that is,

$$\begin{aligned}&-\lambda + \frac{\partial w}{\partial t}(t_0,x_0) -2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)\frac{\partial {\bar{u}}}{\partial t}(t_0,x_0)\nonumber \\&\quad +G\left( t_0,x_0, \nabla w(t_0,x_0)-2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)\nabla {\bar{u}}(t_0,x_0),{\tilde{w}}(t_0,x_0) \right) \ge 0. \end{aligned}$$

Since G(txpu) is nondecreasing with respect to u, (3.21) and (3.23) yield

$$\begin{aligned}&-\lambda + \frac{\partial w}{\partial t}(t_0,x_0) -2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)\frac{\partial {\bar{u}}}{\partial t}(t_0,x_0)\nonumber \\&\quad +G\left( t_0,x_0, \nabla w(t_0,x_0)-2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)\nabla {\bar{u}}(t_0,x_0),w(t_0,x_0) \right) \ge 0. \end{aligned}$$

By (3.22) and (3.24), we obtain

$$\begin{aligned} \lambda&\le -2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)\left( \frac{\partial {\bar{u}}}{\partial t}(t_0,x_0) +v(t_0,x_0)\cdot \nabla {\bar{u}}(t_0,x_0)\right) \\&\quad + w(t_0,x_0){\tilde{R}}\left( t_0,x_0, \nabla w(t_0,x_0) -2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)\nabla {\bar{u}}(t_0,x_0) \right) \\&\quad - w(t_0,x_0){\tilde{R}}\left( t_0,x_0, \nabla w(t_0,x_0)\right) . \end{aligned}$$

Since \({\bar{u}}\) solves (3.20), we obtain with (3.18),

$$\begin{aligned} \lambda&\le -2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0) \times {\bar{u}}(t_0,x_0)\left( \alpha -{\tilde{R}}(t_0,x_0,e^{-\alpha t_0}\nabla {\bar{u}}(t_0,x_0))\right) \\&\quad +|w(t_0,x_0)|\times C|2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)\nabla {\bar{u}}(t_0,x_0)|. \end{aligned}$$

Since \(|{\tilde{R}}|\le 2V_0\), \({\bar{u}}(t,x)=e^{\alpha t}w(t,x)\) and \(h'\ge 0\), we find that

$$\begin{aligned} 0<\lambda \le -2h'({\bar{u}}(t_0,x_0)^2){\bar{u}}(t_0,x_0)^2\left( \alpha -2V_0-C|\nabla w(t_0,x_0)|\right) . \end{aligned}$$

Due to the choice of \(\alpha \) in (3.19), we reach a contradiction.

The case of \(\max _{\Gamma _T}({\tilde{\phi }}-\phi )>0\) can be treated in the same way. \(\square \)

We remark that the result and reasoning of this subsection hold also for the problems (2.24) and (3.4) with R given in (3.13) and \(C^2\)-initial data \(\phi ^0\) such that \(\beta -\alpha \le |\nabla \phi ^0|\le \beta +\alpha \).