# On the geometric evolution and second order dynamics of level-sets of fields defined by partial differential equations

Aarne Pohjonen

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Aarne Pohjonen. On the geometric evolution and second order dynamics of level-sets of fields defined by partial differential equations. 2025. hal-05042668

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# On the geometric evolution and second order dynamics of level-sets of fields defined by partial differential equations

Aarne Pohjonen, Aarne.Pohjonen@Oulu.fi, Aarne.Pohjonen@iki.fi

4th September 2024

## Abstract

Mathematical analysis of the physical principles governing phase morphology evolution is important for the fundamental understanding of the microstructure formation in materials science. Since the physical laws describing the full field evolution of phases can be written as partial differential equations (PDEs), it is of interest to examine the general principles of fields evolution as defined by PDEs. Previously, the connection between first order time partial differential equations and the movement of level-sets (i.e. iso-contours or iso-surfaces) of a general field was established, and demonstrations of the initial calculation with a lagrangian mesh based on moving level-sets was made [1]. Although the first order approximation is capable of estimating field evolution, the stability of numerical methods could be improved by considering second order approximations in time. Furthermore, the theoretical connection between second order time derivative of the field, the differential geometry of the field, velocity and acceleration of level-sets, and their shape evolution provide insights into the dynamical evolution of the field geometry, as defined by partial differential equations in general.

## Introduction

The mathematical analysis of the physical principles of phase morphology evolution involves several different phenomena and there are different approaches to modelling them using continuous fields. In the phase field approach, the evolution of non-conservative order parameter is modelled with the Allen-Cahn equation, also known as the time-dependent Ginzburg-Landau equation [2, 3, 4], Eq. (1).

$$ \partial_t \phi = -L(\partial_\phi f_0 - \nabla \cdot \kappa_g \nabla \phi) \qquad (1) $$

where $L$ is kinetic parameter, $\phi$ is the order parameter phase field, $f_0$ is the free energy and $\kappa_g$ is the gradient energy coefficient. The evolution of the conserved variables is modelled with the Cahn-Hilliard equation [2, 5, 6], Eq. (2).

$$ \partial_t c = \nabla \cdot M \nabla (\partial_c f_0 - \nabla \cdot \epsilon_g \nabla c) \qquad (2) $$

where $c$ is the conserved field, $M$ is kinetic parameter and $\epsilon_g$ is gradient energy coefficient. On the other hand, the propagation of the phase or grain interface can also be modelled using the advection equation, Eq. (1), where the propagation speed is considered as thermally activated kinetic process [7, 8, 9]. Such approach is often called as the level-set method. For considering the movement of phase front in materials, the advection speed in this case can be based on the concept of activation energy [10, 7] for thermally activated processes. It is noted that both the level-set and the phase field methods involve the concepts of phase field (describing the location of the phase density) and the level-set (the iso-contour or -surface of the field) and both are first order equations in time.

In connection to the current topic, it is also noted that the time-varying propagation speed of a phase field has been considered in [11]. The full field modeling of the phase evolution is subject to an intense current research, as envisaged by several recent publications [12, 13, 14, 15, 16, 7, 17], where the geometric description of the field evolution considered in the current article could be of interest, since the analysis can provide interpretation of the modelled results, as well as aid in development of new models.

While the current research has been mainly motivated by the desire to theoretically describe the morphological evolution of phases in materials, it was thought to be of interest to take a broader approach and consider the morphological evolution of fields in general, since naturally the evolution of phase fields and phase interface propagation is subset of the general mathematical description of field evolution. The general results obtained in this study can then provide understanding of the geometry evolution of many different physical fields while also directly providing benefit to the research considering the phase interface evolution described in [7].

The interface movement depending on geometric partial differential equations has been widely researched subject, especially focusing on curvature dependent flows, see the review presented in [18]. The closely related
numerical method for calculating interface movement is the level-set method [19, 20]. It is convenient to apply the method for implicit description of the interfaces, since complex topology evolution, such as merging of interfaces is handled relatively easily this way. The interface movement is also referred to as geometric flow [21]. Reviews of the earlier development of the method are given in [22] and [23]. The curvature and its effect on front propagation has been examined in [24, 25]. In [22] the level-set type formulation was applied for calculating the curvature dependent front propagation, and algorithms for achieving computational stability were presented. Currently, level-set methods have been included in commercial softwares [26]. The conservative level-set method was examined in [27, 28]. The references [20, 29] provides comprehensive overview of the level-set method calculations of moving interfaces.

Level-set type formulation is also common within fluid dynamics methods to simulate two-phase flows [30, 31]. As Navier-Stokes equations yield the velocity field of the fluid, the particle tracking methods are naturally used for calculating the fluid flow. Contour dynamics [32, 33] is an approach and a research field within fluid dynamics where the movement of the points is calculated based on the velocity field obtained from the fluid dynamics equations.

It was recently proposed to apply the advection equation based approach to calculate the evolution of a general field level sets, defined by partial differential equations, where first order time derivative can be estimated [1]. This approach lead to Lagrangian type numerical solution method, where the points contained in the level-sets are used for generating the computational grid, where the spatial derivatives are evaluated. This has several potential benefits: The computational grid is naturally adaptive, as the density of level-sets is high at the locations where gradient is high. The field level-sets often have physical meaning, it is a good way of representing the field, and it may be the desired outcome from the simulations. The initial approach described in [1] could be improved in terms of stability, although some attempts were already made for enhancing this aspect in that publication, such as refining the grid uniformity by moving the points along the level-set curve as well as applying the averaged time-derivative in the calculations.

The natural extension of the previous works is to consider the second order formulation connecting the field evolution and the movement of level-sets. To formulate a solid basis for future numerical developments, it is the aim of current article to explore mathematically the equations relating the time-evolution of field and the movement and shape evolution of the level-sets in the second order approximation in time. In addition, the connection between conserved quantities, flux and isocontour movement is considered.

# 1 Theory

In the present article we examine the evolution equations for the level-sets of a field, which is defined by a partial differential equation. The field is assumed to be sufficiently continuous, so that all the discussed derivatives are defined at the points specified in the text.

## 1.1 Equations describing level-set movement and field evolution

**definition 1.1** (First order time PDE). First order time PDE is a partial differential equation

$$ \frac{\partial}{\partial t} u = A(u) $$

where $A(u)$ is partial differential operator (e.g. for diffusion $A(u) = \nabla \cdot (D\nabla u)$, where $D$ is diffusion coefficient).

The time derivative of the field $\partial_t u$ is related to the local movement of the field $u$ in the direction of the local velocity $\vec{v}$ by the advection equation:

**theorem 1** (Advection equation). If local velocity of a field $u$ is $\vec{v}$, then

$$ \frac{\partial}{\partial t} u = -(\nabla u) \cdot \vec{v} $$

*Proof.* Assume that the local velocity of the field $u$ is $\vec{v}$. Then locally, for sufficiently small time interval $\Delta t = t - t_0$, the field value at point $(x, y, z)$ at time $t_0$ is related to the field value at point $(x - v_x \Delta t, y - v_y \Delta t, z - v_z \Delta t)$ by $u(x, y, z, t) \approx u(x - v_x \Delta t, y - v_y \Delta t, z - v_z \Delta t, t_0 + \Delta t)$. Evoking the chain rule yields

$$ \frac{\partial u}{\partial t} = \left( \frac{\partial u}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial t} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial t} \right) = -(\nabla u) \cdot \vec{v} $$

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Considering the additional rate of adding quantity $u$ to the system can be included by adding a source term, as described by the following corollary 1.1, which directly follows from theorem 1 by adding the source term.
**corollary 1.1.** If local velocity of a field $u$ is $\vec{v}$ as in theorem 1 and in addition to the movement of the field, amount of $u$ is locally introduced to the system at the rate $w$ (amount of $u$ per time), then

$$ \frac{\partial}{\partial t} u = w - (\nabla u) \cdot \vec{v} $$

*remark 1.* For phase field and level set models the source term $w$ in the previous corollary can be used for describing nucleation of a phase, as the field $u$ can be non-conserved phase field.

**definition 1.2 (level-set).** Level-set $L_c$ of a function $f: \mathbb{R}^3 \to \mathbb{R}$ is defined as a set of points $\vec{x}$ where the function obtains the value $c$, i.e. $L_c = \{\vec{x} \in \mathbb{R}^3 | f(\vec{x}) = c\}$.

**definition 1.3 (level-set normal).** level-set normal $\hat{n}$ is defined for each point where $\nabla u \neq 0$ as the unit vector, which is oriented in the perpendicular direction to the level-set, in the direction of the negative gradient. [27, 34, 7]

$$ \hat{n} = - \frac{\nabla u}{|\nabla u|} $$

*remark 2.* The sign convention is the same as in [27, 7], however, it is possible to choose the opposite sign convention.

By differentiating $\hat{n}$ w.r.t time one obtains the following theorem:

**theorem 2.**

$$ \frac{\partial \hat{n}}{\partial t} = - \frac{1}{|\nabla u|} \left[ \frac{\partial \nabla u}{\partial t} + \hat{n} \frac{\partial |\nabla u|}{\partial t} \right] $$

*Proof.* Differentiating $\hat{n}$, of definition 1.3

$$ \partial_t \hat{n} = - \frac{(\partial_t \nabla u) |\nabla u| - \nabla u \partial_t |\nabla u|}{|\nabla u|^2} = - \frac{1}{|\nabla u|} \left[ \partial_t \nabla u - \frac{\nabla u}{|\nabla u|} \partial_t |\nabla u| \right] $$

Substituting $\hat{n} = -\nabla u/|\nabla u|$ yields the proof of theorem 2. $\square$

**corollary 2.1.** $\frac{\partial \hat{n}}{\partial t} = 0$ if and only if $\frac{\partial \nabla u}{\partial t} = -\hat{n} \frac{\partial |\nabla u|}{\partial t}$

*Proof.* The corollary follows directly from theorem 2. $\square$

**definition 1.4 (level-set propagation speed).** Assume a point where $\nabla u \neq 0$. Level-set propagation speed, $s$, is the speed of movement of the point contained on an level-set in direction to the level-set normal.

*remark 3.* Assume that $\nabla u \neq 0$. If the level-set propagates locally in the direction of the level-set normal with speed $s$, then $\vec{v} = s\hat{n}$ and according to theorem 1

$$ \frac{\partial u}{\partial t} = - \nabla u \cdot s \hat{n} $$

**theorem 3.** Assume that $|\nabla u| \neq 0$. If the field $u$ moves in the direction of the level-set normal then

$$ \frac{\partial u}{\partial t} = s |\nabla u| $$

where $s$ is the level-set speed.

*Proof.* Since the field moves in the direction of the level-set normal, $\vec{v} = s\hat{n}$. From the advection equation (theorem 1) $\partial_t u = -(\nabla u) \cdot s\hat{n} = s\nabla u \cdot \nabla u/|\nabla u| = s|\nabla u|$. $\square$

The connection between the general PDE and the level-set propagation speed was proposed in [1, 7]:

**corollary 3.1.** If $|\nabla u| \neq 0$, the propagation speed of level-set of a function defined by first order time PDE $\partial_t u = A(u)$ is

$$ s = \frac{A(u)}{|\nabla u|} $$

*Proof.* The corollary follows from definition 1.1 and theorem 3. $\square$

*remark 4.* The corollary 3.1 could be used in aid for identifying the operator $A(u) = |\nabla u|s$ if the field $u$ and the local propagation speed $s$ are known, for example from experimental data.
**corollary 3.2.** To describe the movement of the level-sets of the field $u$ defined in 1.1 as $\partial_t u = A(u)$, where $|\nabla u| \neq 0$, it is necessary to consider the movement of level-sets only in the level-set normal direction, $\hat{n} = -\nabla u / |\nabla u|$.

*Proof.* The corollary follows from definition 1.1, theorem 3 and corollary 3.1, since $\partial_t u = A(u)$ and $\partial_t u = -\nabla u \cdot s\hat{n} = s|\nabla u|$ and $s$ can be calculated when $A(u)$ is defined for any point on a level-set, where $|\nabla u| \neq 0$. $\square$

**remark 5.** According to the continuity equation [35], if $u$ is per volume quantity (i.e. volumetric density of some quantity), and $f = u\vec{v}$ is the flux of $u$ due to the movement of the field $u$ with velocity $\vec{v}$ and $w$ is source of $u$, then

$$ \partial_t u = w - \nabla \cdot \vec{f} = w - \nabla u \cdot \vec{v} - u \nabla \cdot \vec{v} $$

**theorem 4.** Assume that $u$ is per volume quantity, $\vec{f} = u\hat{n}$ is the flux of $u$ corresponding to the field movement with velocity $\hat{n}$ to the level set normal direction $\hat{n}$, $w$ is the source of $u$, and $|\nabla u| \neq 0$.

The quantity $u$ is conserved if and only if the level-set propagation speed

$$ s = \frac{w - \nabla \cdot \vec{f}}{|\nabla u|} $$

*Proof.* First, let us assume that $u$ is conserved. According to the continuity equation, $\partial_t u = w - \nabla \cdot \vec{f}$, and according to theorem 3 $\partial_t u = s|\nabla u|$. Equating these and solving for $s$ yields $s = \frac{w - \nabla \cdot \vec{f}}{|\nabla u|}$.

Next, conversely, assume that $s = \frac{w - \nabla \cdot \vec{f}}{|\nabla u|}$. Then, through the theorem 3, $\partial_t u = s|\nabla u| = w - \nabla \cdot \vec{f}$, i.e. the continuity equation holds, and therefore $u$ is conserved. $\square$

**corollary 4.1.** Assume that $u$ is conserved, $\vec{f}$ is flux of $u$ over the surface $\partial\Omega$ bounding volume $\Omega$ and $w$ is the source of $u$. Assume also that inside the volume $\Omega$ the flux $\vec{f} = u\hat{n}$ is the flux of the field $u$ moving with velocity $\hat{n}$ towards level-set normal direction $\hat{n}$. Then from divergence theorem and theorem 4 it follows that $\int_{\Omega}(\partial_t u)dV = \int_{\Omega} w - s|\nabla u|dV = \int_{\Omega} \nabla \cdot f dV = \int_{\partial\Omega} \vec{f} \cdot \hat{n} dS$, where $\int_{\Omega} dV$ and $\int_{\partial\Omega} dS$ denote integration over volume and surface, respectively.

*Proof.* Since $u$ is conserved, $\partial_t u = w - \nabla \cdot \vec{f}$. According to divergence theorem $\int_{\Omega} \nabla \cdot f dV = \int_{\partial\Omega} \vec{f} \cdot \hat{n} dS$ and, according to theorem 4, $w - s|\nabla u| = \nabla \cdot \vec{f}$. Therefore $\int_{\Omega} \partial_t u dV = \int_{\Omega} w - s|\nabla u|dV = \int_{\Omega} \nabla \cdot f dV = \int_{\partial\Omega} \vec{f} \cdot \hat{n} dS \square$

**definition 1.5 (Curvature).** Curvature $\kappa$ is defined similar to [22, 34, 7] as $\kappa = -\nabla \cdot \hat{n}$, where the negative sign convention is adopted in the current work, similar to [27, 7].

**remark 6.** The minus sign follows from the definition 1.3 of the level set normal $\hat{n}$. Opposite sign convention is possible.

**corollary 4.2.** Assume that per volume quantity $u$ is conserved, $\vec{f}$ is the flux of $u$ due to the movement of the field in the level-set unit normal direction $\hat{n}$ with speed $s$, i.e. $\vec{f} = u\hat{n}$ and $|\nabla u|$ is non-zero. Then the speed of the level-set $s = \frac{1}{|\nabla u|} [w - (\nabla(\hat{n}) \cdot \hat{n} - (\hat{n})\kappa)]$ where $\kappa$ is curvature.

*Proof.* Since $\vec{f} = u\hat{n}$, substituting $\nabla \cdot \vec{f} = \nabla(\hat{n}) \cdot \hat{n} + (\hat{n})\nabla \cdot \hat{n} = \nabla(\hat{n}) \cdot \hat{n} - s\kappa$ to the equation of Theorem 4 yields the proof of the corollary. $\square$

**remark 7.** The corollary 4.2 provides interpretation of the factors affecting the movement (i.e. flow) of the level-sets of conserved quantity when flux is directed along the level-set normal and the source term $w = 0$. We can distinguish between two extremes: 1. the flow affected purely by flux gradient along the level-set normal where $\kappa = 0$, and 2. the purely curvature affected flow where $\nabla(\hat{n}) \cdot \hat{n} = 0$. The case 1 applies for such condition where the level-set is locally flat and the case 2 applies to the condition that the derivative of $\hat{n}$ is zero along the level-set normal direction, i.e. when $\nabla\hat{n}(\hat{n}) = 0$.

**remark 8.** According to Schwarz theorem [36, 37]: If derivatives of time $t$ and position $x$, $\partial_x f \partial_t f$, and mixed derivatives $\partial_{xt} f$ and $\partial_{tx} f$ are defined for function $f$ at a given point, then

$$ \partial_t \partial_x f = \partial_x \partial_t f $$

at that point.

**theorem 5.** Consider points, where $\nabla u \neq 0$ and assume that the field $u$ is propagating in the level-set normal direction $\hat{n}$. The second order time derivative of $u$ is related to the movement of the level-set by equations (3), (4) and (5):

$$ \frac{\partial^2 u}{\partial t^2} = (\nabla[\nabla u \cdot s\hat{n}] \cdot s\hat{n}) - \nabla u \cdot \vec{a} \quad (3) $$
$$ \frac{\partial^2 u}{\partial t^2} = \frac{\partial s}{\partial t} |\nabla u| + s \frac{\partial |\nabla u|}{\partial t} \qquad (4) $$

$$ \frac{\partial^2 u}{\partial t^2} = -(\partial_t \nabla u) \cdot s\hat{n} - \nabla u \cdot \vec{a} = -s \nabla_{\hat{n}} \partial_t u - \nabla u \cdot \vec{a} \qquad (5) $$

where $s$ is the local propagation speed of the level-set, $\nabla_{\hat{n}} \partial_t u = \nabla(\partial_t u) \cdot \hat{n}$ is the directional derivative of $\partial_t u$ in the direction $\hat{n}$, and the acceleration $\vec{a} = \partial_t(s\hat{n}) = \hat{n}\partial_t s + s\partial_t \hat{n}$.

If $\partial_t u = A(u)$, as defined in 1.1, then

$$ \frac{\partial^2 u}{\partial t^2} = -s \nabla_{\hat{n}} A(u) - \nabla u \cdot \vec{a} \qquad (6) $$

*Proof.* Consider first the proof of Eq. (3): Since $\partial_t u = -\nabla u \cdot s\hat{n}$ and $\partial_t \nabla u = \nabla \partial_t u$ (see remark 8 (Schwarz)), $\partial_{tt} u = \partial_t(-\nabla u \cdot s\hat{n}) = -\nabla(\partial_t u) \cdot s\hat{n} - \nabla u \cdot \partial_t(s\hat{n}) = \nabla(\nabla u \cdot s\hat{n}) \cdot s\hat{n} - \nabla u \cdot \partial_t(s\hat{n})$, which proves Eq. (3).

Consider then the proof of Eq. (4): $\partial_{tt} u = \partial_t(\partial_t u)$. According to Theorem 3 $\partial_t u = s|\nabla u|$ and therefore $\partial_{tt} u = \partial_t(s|\nabla u|) = (\partial_t s)|\nabla u| + s\partial_t|\nabla u|$, which proves Eq. (4).

Consider then the proof of Eq. (5): Again, $\partial_{tt} u = \partial_t(\partial_t u)$. According to corollary 3.2 and remark 3 $\partial_t u = -\nabla u \cdot s\hat{n}$ hence $\partial_{tt} u = -(\partial_t \nabla u) \cdot s\hat{n} - \nabla u \cdot (\partial_t(s\hat{n}))$. According to remark 8 (Schwarz), $\partial_t \nabla u = \nabla \partial_t u$, and therefore $\partial_{tt} u = -s(\nabla \partial_t u) \cdot \hat{n} - \nabla u \cdot (\partial_t(s\hat{n}))$ which proves Eq. (5).

Finally, substituting $\partial_t u = A(u)$ to Eq. (5) yields Eq. (6).

□

*remark 9.* The equations of theorem 5 provide the connection between the second order time partial differential equations $\partial_{tt} u = B(u)$ and the movement of level-sets. Here $B(u)$ is the corresponding partial differential operator similar to $A(u)$ in definition 1.1.

*remark 10.* Procedure for obtaining $\partial_{tt} u$ with expression involving $A(u)$ when $\partial_t u = A(u)$ as in definition 1.1: by applying the Schwarz theorem (see remark 8), the second order time derivative can be obtained from an expression involving $A(u)$ as used in definition 1.1 when $\partial_t u = A(u)$. Let us take somewhat arbitrary but suitable example to demonstrate the procedure: if $\partial_t u = A(u) = u\partial_x u + \partial_y u$, then $\partial_{tt} u = (\partial_t u)\partial_x u + u\partial_t(\partial_x u) + \partial_t(\partial_y u)$. According to Schwarz theorem, $\partial_t(\partial_x u) = \partial_x(\partial_t u) = \partial_x(A(u))$ and $\partial_t(\partial_y u) = \partial_y(\partial_t u) = \partial_y(A(u))$. Therefore in this case $\partial_{tt} u = A(u)\partial_x u + u\partial_x A(u) + \partial_y A(u)$. This approach has the benefit that if $A(u)$ involves only spatial derivatives, then replacing $\partial_{tt} u$ in Eqs. (3) or (6) yields an equation which does not contain time derivatives of the field $u$, but only the spatial derivatives and speed and acceleration of level-sets.

The general procedure for replacing $\partial_{tt} u$ with expressions involving $A(u)$ and the derivatives of $u$ contained in $A(u)$ is demonstrated by the example in remark 10. For a special case, when $A(u)$ does not contain products of functions and its coefficients are independent of time, the theorem 7 is proven using the following lemma:

*lemma 6.* Assume $\frac{\partial u}{\partial t} = A(u)$ as in definition 1, the coefficients of $A(u)$ are independent of time, and $A(u)$ does not include products of $u$ or its spatial derivatives (e.g. $A(u) = (\partial_x u)u + \partial_y u$ includes product but $A(u) = \partial_x u + \partial_y u$ does not). If the n:th order time derivatives of $A(u)$ exist at given point then

$$ \frac{\partial^n u}{\partial t^n} = \underbrace{A \circ A \circ \dots \circ A}_{n}(u) $$

at that point, where $\circ$ is the composition of differential operators (e.g. $A \circ A(u) = A(A(u))$).

*Proof.* According to Schwarz theorem (remark 8) the order of differentiation can be changed. Since $A(u)$ does not involve products and the coefficients of $A(u)$ are independent of time, $\partial_t A(u) = A(\partial_t u)$. Since $\partial_t u = A(u)$, $\partial_t A(u) = A((A(u))) = A \circ A(u)$. The general theorem can be obtained by induction: Assume that $\frac{\partial^{k-1} u}{\partial t^{k-1}} = \underbrace{A \circ A \circ \dots \circ A}_{k-1}(u)$ then, by applying Schwarz theorem, $\frac{\partial}{\partial t} \frac{\partial^{k-1} u}{\partial t^{k-1}} = \underbrace{A \circ A \circ \dots \circ A}_{k-1}(\partial_t u) = \underbrace{A \circ A \circ \dots \circ A}_{k-1}(A(u)) = \underbrace{A \circ A \circ \dots \circ A}_{k}(u)$, hence the lemma holds for any natural number $k$. □

**theorem 7.** If $\partial_t u = A(u)$ as in definition 1.1, the coefficients of $A(u)$ are independent of time, $A(u)$ does not involve products of $u$ and its derivatives, and the time derivatives of $u$ and $A(u)$ are defined, then

$$ A(A(u)) = -s \nabla_{\hat{n}} A(u) - \nabla u \cdot (\hat{n} \partial_t s + s \partial_t \hat{n}) \qquad (7) $$

*Proof.* The theorem follows from Eq. (6) and lemma 6. □
**theorem 8.** Assume that level-set of $u$ propagates in direction $\vec{v}_0$, the level-set of $\partial_t u$ propagates in direction $\vec{v}_1$, and the level-set of $\partial_{tt} u$ propagates in direction $\vec{v}_2$. Then

$$ \begin{aligned} \partial_t u &= -\nabla u \cdot \vec{v}_0 \\ \partial_{tt} u &= (-1)^2 \nabla(\nabla u \cdot \vec{v}_0) \cdot \vec{v}_1 \\ \partial_{ttt} u &= (-1)^3 \nabla(\nabla(\nabla u \cdot \vec{v}_0) \cdot \vec{v}_1) \cdot \vec{v}_2 \end{aligned} $$

etc.

*Proof.* According to theorem 1, $\partial_t u = -\nabla u \cdot \vec{v}_0$. $\partial_{tt} u = \partial_t(\partial_t u) = -\nabla(\partial_t u) \cdot \vec{v}_1 = -\nabla(-\nabla u \cdot \vec{v}_0) \cdot \vec{v}_1$. Similarly, $\partial_{ttt} u = \partial_t(\partial_{tt} u) = (-1)^3 \nabla(\nabla(\nabla u \cdot \vec{v}_0) \cdot \vec{v}_1) \cdot \vec{v}_2$, etc. □

*remark 11.* When distance between two level-sets, $L$, is sufficiently small, then for any point $\vec{r}$ on a level-set, $u(\vec{r}+L\hat{n}) \approx u(\vec{r})+L\hat{n} \cdot \nabla u$. In the following, the distance $L$ is assumed to be such small that this approximation is valid.

**definition 1.6** (Dilation and contraction along level-set normal). Dilation $d = L/L_0$ along level-set normal describes the lengthening of distance $L$ between neighbouring level-sets, along the level-set normal direction, relative to the original length $L_0$. If $d > 1$, dilation has occurred, and if $d < 1$ contraction has occurred.

**theorem 9.** The dilation rate

$$ \frac{\partial d}{\partial t} = \frac{L}{L_0} \nabla s \cdot \hat{n} = d \nabla s \cdot \hat{n} $$

*Proof.* The difference is speed between two points located on neighbouring level-sets, separated by distance $L$ along $\nabla u$, is $L\nabla s \cdot \hat{n}$. For sufficiently small time interval $\Delta t$, $L(t + \Delta t) \approx L(t) + L\nabla s \cdot \hat{n}\Delta t$. Therefore

$$ \frac{\partial L}{\partial t} = \lim_{\Delta t \to 0} \frac{L(t + \Delta t) - L(t)}{\Delta t} = L \nabla s \cdot \hat{n} $$

Since, according to the definition 1.6 $d = L/L_0$, and the initial length $L_0$ is independent of time, it follows that $\frac{\partial d}{\partial t} = \frac{L}{L_0} \nabla s \cdot \hat{n}$ □

**corollary 9.1.** The total dilation along the (possibly curved) path between two points in neighbouring level-sets is obtained from

$$ d = \exp \left( \int_{0}^{t} \nabla s \cdot \hat{n} dt' \right) $$

where $t$ is time and $t'$ is the corresponding integration variable.

*Proof.* The corollary follows by evaluating the time derivative of $d$ which satisfies the equation of theorem 9. □

**definition 1.7** (level-set rotation rate). The rate of level-set rotation describes the change of the orientation of the level-set normal per time,

$$ \frac{\partial \vec{R}}{\partial t} = \frac{\partial \phi}{\partial t} \hat{m} $$

where $\phi$ is the angle of rotation and the unit vector $\hat{m}$ defines the axis of rotation.

**theorem 10.**

$$ \frac{\partial \vec{R}}{\partial t} = \nabla s \times \hat{n} $$

*Proof.* The vectors $\hat{n}$ and $\nabla s$ define a plane $T = \{a\hat{n} + b\nabla s|a, b \in \mathbb{R}\}$. Let $\hat{n}_{\perp}$ be the orthonormal unit vector of $\hat{n}$ on plane $T$. $\nabla s \cdot \hat{n}_{\perp}$ is the difference of speed of points located on the iso-contour tangent on plane $T$, in the direction of $\hat{n}_{\perp}$, per unit length. This difference in speed causes the angle of rotation to grow or decrease by $\Delta\phi$. The axis of rotation is the plane $T$ unit normal $\hat{m} = \nabla s \times \hat{n}/|\nabla s \times \hat{n}|$.

Assume that time interval $\Delta t$ passes. The change of arc corresponding to the change in angle $\Delta\phi$ at unit distance along $\hat{n}_{\perp}$ is $\Delta q = \nabla s \cdot \hat{n}_{\perp} \Delta t = |\nabla s \times \hat{n}| \Delta t$ because $\hat{n}_{\perp}$ and $\hat{n}$ are perpendicular unit vectors. Since the change in angle $\phi$ is related to the arc length $\Delta q$ by $\Delta\phi = \Delta q/|\hat{n}_{\perp}| = \Delta q$, the angle $\Delta\phi = |\nabla s \times \hat{n}| \Delta t$. Taking the limit $\lim_{t \to 0}$ yields $\partial_t \phi = |\nabla s \times \hat{n}|$.

Since we have concluded that $\hat{m} = \nabla s \times \hat{n}/|\nabla s \times \hat{n}|$ and $\partial_t \phi = |\nabla s \times \hat{n}|$, the rotation rate $\frac{\partial \vec{R}}{\partial t} = (\partial_t \phi)\hat{m} = \nabla s \times \hat{n}$ □

*remark 12.* From the definition 1.5 and the Schwarz theorem (remark 8) it follows that

$$ \frac{\partial \kappa}{\partial t} = -\frac{\partial}{\partial t} \nabla \cdot \hat{n} = -\nabla \cdot \frac{\partial \hat{n}}{\partial t} $$
The first order approximation for the movement of a point on the level-set can be calculated using the advection equation (theorem 1) when the velocity of the point $\vec{v}$ is known. The second order approximation for the movement of a point contained on the level-set can be calculated if additionally it's acceleration is known. If the $\vec{r}(t)$ is the position of the point on time instant $t$, then the position of the point after sufficiently small time-step $\Delta t$ is described by Eq. (8)

$$ \vec{r}(t + \Delta t) \approx \vec{r}(t) + \vec{v}\Delta t + \frac{1}{2}\vec{a}\Delta t^2 \qquad (8) $$

remark 13. If the point on the level-set only moves in the direction of level-set normal (as is the case for the when the field is defined by first order time PDE $\partial_t u = A(u)$, see corollary 3.2), then it's acceleration is given by

$$ \vec{a} = \frac{\partial \vec{v}}{\partial t} = \frac{\partial s}{\partial t} \hat{n} + s \frac{\partial \hat{n}}{\partial t} $$

From the corollary 3.1 ($s = A(u)/|\nabla u|$), it directly follows that:

remark 14. If the time-evolution of the field $u$ is defined by first order time PDE

$$ \frac{\partial u}{\partial t} = A(u) $$

and $|\nabla u| \neq 0$ then, since $s = \partial_t u/|\nabla u|$ (theorem 3) and $s = A(u)/|\nabla u|$ (corollary 3.1), the time derivative of the propagation speed of the level-set is

$$ \frac{\partial s}{\partial t} = \frac{\partial}{\partial t} \frac{\partial_t u}{|\nabla u|} = \frac{\partial}{\partial t} \frac{A(u)}{|\nabla u|} \qquad (9) $$

For the PDE $\partial_t u = A(u)$, the following expressions for the time derivative can be obtained:

**theorem 11.** If $\partial_t u = A(u)$, where $A(u)$ is a differential operator acting on $u$ and $|\nabla u| \neq 0$, then the time derivative of propagation speed $s$ can be obtained from either of the following equations

$$ \frac{\partial s}{\partial t} = \frac{1}{|\nabla u|} [\partial_{tt} u - s \partial_t |\nabla u|] \qquad (10) $$

$$ \frac{\partial s}{\partial t} = \frac{1}{|\nabla u|} [\partial_t A(u) - s \partial_t |\nabla u|] \qquad (11) $$

*Proof.* Expanding the time differentiation of the right hand side of Eq. (9) yields

$$ \frac{\partial s}{\partial t} = \frac{(\partial_{tt} u) |\nabla u| - (\partial_t u) (\partial_t |\nabla u|)}{|\nabla u|^2} = \frac{1}{|\nabla u|} \left[ \partial_{tt} u - \frac{\partial_t u}{|\nabla u|} \partial_t |\nabla u| \right] $$

Since according to theorem 3, $s = \partial_t u/|\nabla u|$, equation (10) follows.

By assumption $\partial_t = A(u)$. Since $\partial_{tt} u = \partial_t (\partial_t u) = \partial_t A(u)$. Substituting this to Eq. (10) yields Eq. (11). $\square$

From theorem 11 it directly follows that:

remark 15. Assume that $|\nabla u| \neq 0$. The level-set propagates with constant speed ($\partial_t s = 0$) if and only if $\partial_t A(u) = s \partial_t |\nabla u| \iff \partial_{tt} u = s \partial_t |\nabla u|$.

## 2 Conclusions

Exploratory derivations of the first and second order time equations connecting the field evolution to the movement of field level-sets were described starting from the first principles and yielding several identities. These equations are not an exhaustive list of all possible identities, but they exemplify the connections that can be found by considering the relation of the level-set movement and the field evolution equations. The equations relating the shape evolution of the level-sets to the time derivatives of the field were derived: the level-set curvature evolution and the dilation of the field were considered. Additionally, for conserved quantity, the relation of continuity equation and flux to the level-set movement was obtained, and the speed of level-set movement within volume was related to the flux over the bounding surface through the divergence theorem.

The change of orientation of the level-set normal is related to the time derivatives describing the evolution of the field in theorem 2. The relation of level-set propagation speed and the spatial derivatives of the first order partial differential equation defining the field evolution is described in corollary 3.1. For the conserved quantity, the propagation speed was related to the flux (theorem 4) and curvature (corollary 4.2). The second order time derivative describing the field evolution was related to the speed and acceleration of the field level-set in theorem 5. Dilation, rotation and curvature evolution were related to the level-set propagation speed and orientation in theorem 9, corollary 9.1, theorem 10 and remark 12. The time derivative of the level-set propagation speed and the conditions for the level-set to propagate at constant speed are described in theorem 11 and remark 15.
# 3 Data availability and conflict of interest

No additional data is associated with the current study. The author has no competing interests to declare that are relevant to the content of this article.

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