Interface Dynamics in a Two-phase Tumor Growth Model

Consider two species of cells in $\mathbb{R}^{2}$, one being actively growing tumor cell and the other being inactive normal cell. Spatial densities of tumor and normal cells, each denoted by $m$ and $n$, satisfy

(1.1)		$\displaystyle\partial_{t}m-\mathrm{div}\,(\mu m\nabla p)=$	$\displaystyle\;mG(p),$
(1.2)		$\displaystyle\partial_{t}n-\mathrm{div}\,(\nu n\nabla p)=$	$\displaystyle\;0,$
(1.3)		$\displaystyle m+n\leq$	$\displaystyle\;1.$

Here $\mu,\nu>0$ denote mobilities of the tumor and normal cells. $p$ is the pressure generated by the cells, serving as a Lagrange multiplier for the constraint $m+n\leq 1$. It satisfies

(1.4)		$\displaystyle-\mathrm{div}\,((\mu m+\nu n)\nabla p)=mG(p)$	$\displaystyle\quad\mbox{ if }m+n=1,$
(1.5)		$\displaystyle p=0$	$\displaystyle\quad\mbox{ if }m+n<1.$

In (1.1) and (1.4), $G(p)$ represents pressure-dependent proliferation rate of the tumor cell. In the spirit of [1], we assume that

1. (1)

   $G\in C^{1}[0,+\infty)$.

2. (2)

   $G(p)$ is decreasing.

3. (3)

   $G(0)>0$ and $G(p_{M})=0$ for some $p_{M}>0$.

In short, (1.1)-(1.5) models the scenario where the tumor keeps growing and where two species of cells migrate with different mobilities, according to the Darcy’s law [2], under the pressure they generate together.

Mathematical analysis of strongly-coupled competitive systems such as (1.1)-(1.5) can be challenging [3, 4, 5, 6, 7, 8]. To the best of our knowledge, existing analyses of such problems are carried out either in one space dimension or with equal mobility of the two species. In contrast, it is suggested in [1] that the cells moving with different mobilities is an important feature of the model (1.1)-(1.5). Indeed, the numerical results in [1] show that when $\mu<\nu$, certain radially symmetric solution is stable, while when $\mu>\nu$ a Saffman-Taylor type instability [9] can occur.

In this paper, we study (1.1)-(1.5) with the restriction that $m$ and $n$ are segregated and fully saturated in their regions. Namely, we assume that $m=\chi_{\Omega}$ and $n=\chi_{\tilde{\Omega}\backslash\Omega}$, where $\Omega\subset\subset\tilde{\Omega}$ are two time-varying bounded domains. This gives rise to a free boundary problem that concerns dynamics of both $\gamma:=\partial\Omega$ and $\tilde{\gamma}:=\partial\tilde{\Omega}$.

First, the equation for $p$ reduces to

(1.6)		$\displaystyle-\mathrm{div}\,((\mu\chi_{\Omega}+\nu\chi_{\tilde{\Omega}\backslash\Omega})\nabla p)=\chi_{\Omega}G(p)$	$\displaystyle\quad\mbox{in }\tilde{\Omega},\quad p|_{\partial\tilde{\Omega}}=0,$
(1.7)		$\displaystyle p=0$	$\displaystyle\quad\mbox{in }\tilde{\Omega}^{c}.$

Then the motion law of the free boundaries are given as follows. From (1.1), we may derive the normal velocity for $\gamma$:

(1.8)

$$V_{n,\gamma}=-\mu\frac{\partial p}{\partial\sigma_{\Omega}}.$$

Here $\sigma_{\Omega}$ denotes the outward normal of $\gamma$ with respect to $\Omega$. Similarly, the normal speed of $\tilde{\gamma}$ is given by (1.2):

(1.9)

$$V_{n,\tilde{\gamma}}=-\nu\frac{\partial p}{\partial\sigma_{\tilde{\Omega}}},$$

where $\sigma_{\tilde{\Omega}}$ denotes outward normal of $\tilde{\gamma}$ with respect to $\tilde{\Omega}$.

Our main result is the local-in-time well-posedness of the free boundary problem (1.6)-(1.9). Inspired by the numerical results in [1], we assume $\mu<\nu$ for the well-posedness. Interestingly, we will illustrate later that even with this assumption, instabilities may still occur along $\gamma$ without further geometric assumptions on $\Omega$ and $\tilde{\Omega}$ (see Remark 2.5). We thus need to restrict ourselves to the case where the initial configuration is nearly radial (see Figure 1). More precise statement of our main results can be found in Theorem 2.1 and Theorem 2.2 in Section 2.3.

The evolution of the inner interface $\gamma$ is similar to the 2-D Muskat problem [10, 11] with viscosity jump [12, 13], which is concerned with a close-to-flat interface between two fluids driven by the Darcy’s law. In the case when the more viscous fluid is pushed towards the less viscous one, [12] establishes global well-posedness for small initial data; in the opposite case, ill-posedness is shown. With generalized Rayleigh-Taylor condition [14], [13] formulates similar result on the well-posedness in a more general setup allowing density-viscosity jumps. Note that these rigorous results agree very well with [9] and the aforementioned numerical results in [1]. They are obtained by exploring the inherent parabolicity in the interface motion with complex analysis [12] and functional analytic [13] approaches. However, it is not clear if these approaches can be directly applied here as our model involves a geometry-dependent source term, whose support touches $\gamma$.

Notably there is a lot more literature concerning the Muskat problem with density jump [15, 16, 17, 18, 19] or density-viscosity jumps [20, 21, 22, 23, 13]. In both of these cases, the smoothing mechanism is essentially provided by the fact that a heavier fluid sits below a lighter one, where the gravity naturally damps the oscillation of the interface. In contrast, the smoothing mechanism is much less explicit when there is only jump in the viscosity across the interface [12, 13].

Motion of the outer interface $\tilde{\gamma}$ is reminiscent of the free boundary arising in the one-phase Hele-Shaw problem [24], where a blob of fluid is injected into a Hele-Shaw cell or a porous medium and expands according to the Darcy’s law. Despite its similarity with the Muskat problem in some aspects, it admits a few other treatments. We direct the readers to [25, 26, 27, 28, 29, 30] and the references therein. Once again, in our problem, the presence of the source term depending nonlocally on $\tilde{\gamma}$ and $\gamma$ may hinder direct applications of these approaches.

In this paper, we study the dynamics of both interfaces $\gamma$ and $\tilde{\gamma}$ in a unified framework, adapted from the study of contour equations in the Muskat problem [21, 22]. We first reduce (1.6)-(1.9), which involves an elliptic equation for $p$ in a time-varying domain, partially into contour equations for the interface configurations and quantities along them; see (2.16), (2.17), (2.33) and (2.34). A key step in this reduction is to represent the transporting velocity over $\tilde{\Omega}$ as a sum of three parts, which arise from the discontinuity of the cell mobilities across $\gamma$, the zero Dirichlet boundary condition of $p$ along $\tilde{\gamma}$, and the source term in $\Omega$, respectively; see (2.12) and also (2.3). Then by linearizing these contour equations around radially symmetric configurations, we show their parabolic nature under suitable conditions (c.f., Section 2.4). In particular, the interfaces can smooth themselves according to a fractional-heat-type equation with source terms. After deriving good estimates for these source terms, we prove well-posedness of the interface motion by a fixed-point argument. Smallness of the geometric deviation of $\gamma$ and $\tilde{\gamma}$ from radially symmetric configurations helps close the estimates needed in this argument. See Section 2 for more details.

This problem features a geometry-dependent source term $\chi_{\Omega}G(p)$ in (1.6) that is supported up to the inner interface. It may be tempting to think of it as an innocuous regular term, but in fact, it changes the dynamics in a crucial way compared to the related problems discussed above.

Firstly, on the technical level, the source term seems to prevent the complex analysis approach in [12] from being applied here. Secondly, the parabolicity of $\gamma$ relies on the fact that the cell with lower mobility is displacing the other species, i.e., $(\mu-\nu)\frac{\partial p}{\partial\sigma_{\Omega}}|_{\gamma}>0$ (c.f., (1.8) and Remark 2.4), in line with the classic Saffman-Taylor condition [9]. Since $\chi_{\Omega}G(p)$ depends on the domain geometry, one can manufacture such $\Omega$ and $\tilde{\Omega}$, so that the tumor is pushed by the normal tissue along some part of $\gamma$ under the assumption $\mu<\nu$. This is possible even and both $\gamma$ and $\tilde{\gamma}$ are required to be graphs of functions over $\mathbb{T}$ in the polar coordinate; see Remark 2.5. In this sense, simply assuming $\mu<\nu$ is not enough for proving well-posedness, and it is reasonable to additionally require that $\gamma$ and $\tilde{\gamma}$ are close to concentric circles (see Figure 1). Then characterization of the parabolicity of $\gamma$ is based on a good understanding of $p$. In Section 3, we apply elliptic regularity theory to justify that given the domain geometry close to a radially symmetric one, the corresponding $p$ should not be far away from a radially solution. That would be sufficient to guarantee parabolicity in the motion of $\gamma$ as $\mu<\nu$. Furthermore, these elliptic estimates together with the results in Section 4 and Section 5 will help justify that such parabolicity can be characterized by a fractional heat operator with exponent $\frac{1}{2}$, which plays a central role in our analysis. See Section 2.4 and Section 8.

The source term also poses new difficulty in studying global well-posedness and stability properties near the radially symmteric solutions. Indeed, as the tumor grows larger, the pressure becomes more sensitive to the interface geometry. We demonstrate this by a scaling argument. Suppose at given time $T>0$, $\Omega$ and $\tilde{\Omega}$ are close to two concentric discs, and $\tilde{\Omega}$ has radius of order $R\gg 1$. Define $p_{R}(x,t):=p(Rx,R(t-T))$ and let $\tilde{\Omega}_{R}$ and $\Omega_{R}$ denote the corresponding dilated version of $\tilde{\Omega}$ and $\Omega$ according to the scaling. Then (1.6) becomes

(1.10)

$$-\mathrm{div}\,((\mu\chi_{\Omega_{R}}+\nu\chi_{\tilde{\Omega}_{R}\backslash\Omega_{R}})\nabla p_{R})=\chi_{\Omega_{R}}R^{2}G(p_{R}),$$

with zero boundary data on $\partial\tilde{\Omega}_{R}$, while the boundary motion laws (1.8) and (1.9) remain the same. In this new problem, the proliferation rate $R^{2}G(\cdot)$ can have a large magnitude where $p_{R}$ is small and it is sensitive to the pressure. This results in concentration of the source term near the inner interface and a steep growth of $p_{R}$ there. On the other hand, the total mass of the normal tissue is preserved due to (1.2), and thus $\tilde{\Omega}_{R}\backslash\Omega_{R}$ is extremely thin as $R\gg 1$. So in the rescaled problem the source term is close to both the inner and outer interfaces. It is then conceivable that $p_{R}$ will be highly sensitive to the domain geometry in the sense that even when the domain is pretty close to being radial, $p_{R}$ may be highly oscillatory and far from being radially symmetric. Therefore, because of the source term, nonlinear stability of the interface configurations around radially symmetric ones becomes a much more subtle issue when it comes to long time asymptotics.

This work is partially supported by National Science Foundation under Award DMS-1900804.

Define a potential $\varphi$ to be

(2.1)

$$\varphi:=\mu p\mbox{ in }\Omega,\quad\varphi:=\nu p\mbox{ in }\Omega^{c}.$$

So $\varphi$ solves

$$-\Delta\varphi=G(p)\chi_{\Omega}\quad\mbox{in }\tilde{\Omega}\backslash\gamma,\quad\varphi|_{\tilde{\gamma}}=0,$$

and $\varphi\equiv 0$ on $\tilde{\Omega}^{c}$. When $\mu\not=\nu$, $\varphi$ has discontinuity across $\gamma$, denoted by

$$[\varphi]_{\gamma}(x):=\varphi|_{\gamma,\Omega}(x)-\varphi|_{\gamma,\Omega^{c}}(x),\quad x\in\gamma.$$

(1.8) and (1.9) yield that each cell phase is transported by the velocity field $u=-\nabla\varphi$. It has discontinuity across $\gamma$ in the tangential component, but not in its normal component.

Let $\Gamma$ denote the fundamental solution of the Laplace’s equation in $\mathbb{R}^{2}$,

$$\Gamma(x):=-\frac{1}{2\pi}\ln|x|.$$

Let $\mathcal{D}_{\gamma}$ denote the double layer potential operator associated with $\gamma$. Namely, with a boundary potential $\psi$ defined on $\gamma$, we define $\mathcal{D}_{\gamma}\psi$ on $\mathbb{R}^{2}$ to be

(2.2)

$$\mathcal{D}_{\gamma}\psi(x):=\int_{\gamma}\sigma_{y}\cdot\nabla_{y}(\Gamma(x-y))\psi(y)\,dy.$$

Note that here the gradient is taken with respect to $y$. It is well-known that for $\gamma$ and $\psi$ sufficiently smooth, say $C^{1,\alpha}(\mathbb{T})$, $[\mathcal{D}_{\gamma}\psi]_{\gamma}=-\psi$. Then $\varphi$ admits the following representation

(2.3)

$$\varphi=-\mathcal{D}_{\gamma}[\varphi]-\mathcal{D}_{\tilde{\gamma}}\phi+\Gamma*g\quad\mbox{in }\tilde{\Omega}\backslash\gamma.$$

where $\phi$ is some boundary potential defined along $\tilde{\gamma}$ to be determined in order for the boundary condition $\varphi|_{\tilde{\gamma}}=0$, and where

(2.4)

$$g=G(p)\chi_{\Omega}=G(\mu^{-1}\varphi)\chi_{\Omega}\geq 0.$$

Assume $C^{1,\alpha}(\mathbb{T})$-regularity of $\gamma$ and $[\varphi]$. Then the representation (2.3) along $\gamma$ takes the average of $\varphi$ on two sides of $\gamma$, i.e.,

$$\left.\left(-\mathcal{D}_{\gamma}[\varphi]-\mathcal{D}_{\tilde{\gamma}}\phi+\Gamma*g\right)\right|_{\gamma}=\frac{1}{2}(\varphi|_{\gamma,\Omega}+\varphi|_{\gamma,\Omega^{c}})=\frac{\mu+\nu}{2}p=\frac{\mu+\nu}{2(\mu-\nu)}[\varphi].$$

This implies

(2.5)

$$[\varphi]=2A(-\mathcal{D}_{\gamma}[\varphi]-\mathcal{D}_{\tilde{\gamma}}\phi+\Gamma*g)|_{\gamma},$$

where

$$A=\frac{\mu-\nu}{\mu+\nu}.$$

On the other hand, the zero Dirichlet boundary condition of $\varphi$ along $\tilde{\gamma}$ requires that

$$\lim_{x\to\tilde{\gamma}(\theta)\atop x\in\tilde{\Omega}}(-\mathcal{D}_{\tilde{\gamma}}\phi)(x)=(\mathcal{D}_{\gamma}[\varphi]-\Gamma*g)|_{\tilde{\gamma}(\theta)}.$$

Assuming $C^{1,\alpha}(\mathbb{T})$-regularity of $\tilde{\gamma}$ and $\phi$, by the property of the double layer potential, $\phi$ should solve

$$-(\mathcal{D}_{\tilde{\gamma}}\phi)|_{\tilde{\gamma}}+\frac{1}{2}\phi=(\mathcal{D}_{\gamma}[\varphi]-\Gamma*g)|_{\tilde{\gamma}}$$

along $\tilde{\gamma}$, i.e.,

(2.6)

$$\phi=2(\mathcal{D}_{\tilde{\gamma}}\phi+\mathcal{D}_{\gamma}[\varphi]-\Gamma*g)|_{\tilde{\gamma}}.$$

Finally, (1.8) and (1.9) become

(2.7)

$$V_{n,\gamma}=-\frac{\partial\varphi}{\partial\sigma_{\Omega}},\quad V_{n,\tilde{\gamma}}=-\frac{\partial\varphi}{\partial\sigma_{\tilde{\Omega}}}.$$

(2.3)-(2.7) readily form a closed system.

We consider the case when $\gamma$ and $\tilde{\gamma}$ are close to two concentric circles centered at the origin, with some radii $r<R$, respectively. See Figure 1. We parameterize $\gamma$ and $\tilde{\gamma}$ using the polar coordinate,

(2.8)		$\displaystyle\gamma(\theta,t)=$	$\displaystyle\;f(\theta,t)(\cos\theta,\sin\theta),$
(2.9)		$\displaystyle\tilde{\gamma}(\theta,t)=$	$\displaystyle\;F(\theta,t)(\cos\theta,\sin\theta),$

where $\theta\in\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})=[-\pi,\pi)$. Then $[\varphi]$ and $\phi$ can be naturally understood as functions of $\theta\in\mathbb{T}$. Next we shall derive equations for $\gamma$ and $\tilde{\gamma}$ (or equivalently, for $f$ and $F$).

Note that $\sigma_{\Omega}(\theta)=-\gamma^{\prime}(\theta)^{\perp}/|\gamma^{\prime}(\theta)|$, where $v^{\perp}$ denote a vector $v\in\mathbb{R}^{2}$ rotated counter-clockwise by $\pi/2$. By (2.2), all $x\in\tilde{\Omega}\backslash\gamma$,

(2.10)

$$\mathcal{D}_{\gamma}[\varphi]=\frac{1}{2\pi}\int_{\mathbb{T}}\frac{(x-\gamma(\theta^{\prime}))\cdot(-\gamma^{\prime}(\theta^{\prime}))^{\perp}}{|x-\gamma(\theta^{\prime})|^{2}}[\varphi](\theta^{\prime})\,d\theta^{\prime}.$$

By assuming $[\varphi]\in C^{1}(\mathbb{T})$,

(2.11)

$$\begin{split}\nabla\mathcal{D}_{\gamma}[\varphi]=&\;\frac{1}{2\pi}\int_{\mathbb{T}}\frac{\partial}{\partial\theta^{\prime}}\left(-\frac{(x-\gamma(\theta^{\prime}))^{\perp}}{|x-\gamma(\theta^{\prime})|^{2}}\right)[\varphi](\theta^{\prime})\,d\theta^{\prime}\\ =&\;\frac{1}{2\pi}\int_{\mathbb{T}}\frac{(x-\gamma(\theta^{\prime}))^{\perp}}{|x-\gamma(\theta^{\prime})|^{2}}[\varphi]^{\prime}(\theta^{\prime})\,d\theta^{\prime}.\end{split}$$

which is a Birkhoff-Rott-type integral [31]. Hence, by (2.3), for $x\in\tilde{\Omega}\backslash\gamma$,

(2.12)

$$u(x)=-\nabla\varphi(x)=\frac{1}{2\pi}\int_{\mathbb{T}}\frac{(x-\gamma(\theta^{\prime}))^{\perp}}{|x-\gamma(\theta^{\prime})|^{2}}[\varphi]^{\prime}(\theta^{\prime})\,d\theta^{\prime}+\frac{1}{2\pi}\int_{\mathbb{T}}\frac{(x-\tilde{\gamma}(\theta^{\prime}))^{\perp}}{|x-\tilde{\gamma}(\theta^{\prime})|^{2}}\phi^{\prime}(\theta^{\prime})\,d\theta^{\prime}-\nabla(\Gamma*g).$$

On the other hand, by (2.7) and (2.8),

(2.13)

$$\partial_{t}f=u(\gamma(\theta))\cdot\sigma_{\Omega}(\theta)\cdot\frac{|\gamma^{\prime}(\theta)|}{f(\theta)}=-\frac{1}{f}\cdot u(\gamma(\theta))\cdot\gamma^{\prime}(\theta)^{\perp}.$$

Although $u(\gamma(\theta))$ here should be understood as the limit of (2.12) when letting $x\rightarrow\gamma(\theta)$ from the inside of $\gamma$, it is safe to simply take $x=\gamma(\theta)$ since the normal component of $u$ does not have discontinuity across $\gamma$. Define

(2.14)		$\displaystyle\mathcal{K}_{\gamma}\psi:=$	$\displaystyle\;\frac{1}{2\pi}\mathrm{p.v.}\int_{\mathbb{T}}\frac{\gamma(\theta)-\gamma(\theta^{\prime})}{|\gamma(\theta)-\gamma(\theta^{\prime})|^{2}}\cdot\psi(\theta^{\prime})\,d\theta^{\prime},$
(2.15)		$\displaystyle\mathcal{K}_{\gamma,\tilde{\gamma}}\psi(\theta):=$	$\displaystyle\;\frac{1}{2\pi}\int_{\mathbb{T}}\frac{\gamma(\theta)-\tilde{\gamma}(\theta^{\prime})}{|\gamma(\theta)-\tilde{\gamma}(\theta^{\prime})|^{2}}\cdot\psi(\theta^{\prime})\,d\theta^{\prime}.$

Let $\mathcal{K}_{\tilde{\gamma},\gamma}\psi(\theta)$ be defined symmetrically by interchanging $\gamma$ and $\tilde{\gamma}$ in (2.15). Thanks to (2.11) and (2.12), (2.13) can be rewritten as

(2.16)

$$\partial_{t}f=-\frac{1}{f}\gamma^{\prime}(\theta)\cdot\mathcal{K}_{\gamma}[\varphi]^{\prime}-\frac{1}{f}\gamma^{\prime}(\theta)\cdot\mathcal{K}_{\gamma,\tilde{\gamma}}\phi^{\prime}+\frac{1}{f}\nabla(\Gamma*g)|_{\gamma}\cdot\gamma^{\prime}(\theta)^{\perp}.$$

Similarly,

(2.17)

$$\partial_{t}F=-\frac{1}{F}\tilde{\gamma}^{\prime}(\theta)\cdot\mathcal{K}_{\tilde{\gamma}}\phi^{\prime}-\frac{1}{F}\tilde{\gamma}^{\prime}(\theta)\cdot\mathcal{K}_{\tilde{\gamma},\gamma}[\varphi]^{\prime}+\frac{1}{F}\nabla(\Gamma*g)|_{\tilde{\gamma}}\cdot\tilde{\gamma}^{\prime}(\theta)^{\perp}.$$

These equations are coupled with initial conditions

(2.18)

$$f(t=0)=f_{0}(\theta),\quad F(t=0)=F_{0}(\theta).$$

For future use, we introduce

(2.19)

$$h(\theta,t)=\frac{f(\theta,t)}{r}-1,\quad H(\theta,t)=\frac{F(\theta,t)}{R}-1.$$

They are relative deviations of $\gamma$ and $\tilde{\gamma}$ from radially symmetric configurations.

We first introduce $W^{k-\frac{1}{p},p}(\mathbb{T})$-space for $k\in\mathbb{Z}_{+}$ and $p\in(1,\infty)$ [32, § 2.12.2]. Let $\{e^{-t(-\Delta)^{1/2}}\}_{t\geq 0}$ denote the Poisson semi-group on $\mathbb{T}$ with generator $-(-\Delta)^{1/2}$. For $f\in L^{p}(\mathbb{T})$, let

(2.20)

$$\|f\|_{\dot{W}^{k-\frac{1}{p},p}(\mathbb{T})}:=\left\|e^{-t(-\Delta)^{1/2}}f\right\|_{L^{p}_{[0,\infty)}\dot{W}^{k,p}(\mathbb{T})}.$$

We say $f\in W^{k-\frac{1}{p},p}(\mathbb{T})$ if and only if $f\in L^{p}(\mathbb{T})$ such that $\|f\|_{\dot{W}^{k-\frac{1}{p},p}(\mathbb{T})}<+\infty$.

Our main results are as follows.

Thanks to the estimates for the local solution, one can apply Theorem 2.1 iteratively and show that local solutions exist for an arbitrary time period $\tilde{T}>0$ as long as $h_{0}$ and $H_{0}$ are correspondingly sufficiently small.

Uniqueness of local solutions can be shown if $G$ is more regular.

To elucidate the hidden parabolicity of (2.16)-(2.18), we linearize it around the radially symmetric configurations.

It is convenient to first derive equations for $[\varphi]^{\prime}$ and $\phi^{\prime}$ by taking derivative in (2.5) and (2.6). Assuming $\gamma,[\varphi]\in C^{1}(\mathbb{T})$, we have

(2.29)

$$\frac{d}{d\theta}(\mathcal{D}_{\gamma}[\varphi])|_{\gamma}=-\gamma^{\prime}(\theta)^{\perp}\cdot\mathcal{K}_{\gamma}[\varphi]^{\prime}.$$

Indeed, by integration by parts,

(2.30)

$$\begin{split}(\mathcal{D}_{\gamma}[\varphi])|_{\gamma}=&\;-\frac{1}{2\pi}\mathrm{p.v.}\int_{\mathbb{T}}\partial_{\theta^{\prime}}[\mathrm{arg}((\gamma(\theta)-\gamma(\theta^{\prime}))_{1}+i(\gamma(\theta)-\gamma(\theta^{\prime}))_{2})]\cdot[\varphi](\theta^{\prime})\,d\theta^{\prime}\\ =&\;-\frac{1}{2\pi}\cdot\pi[\varphi](\theta)+\frac{1}{2\pi}\int_{\mathbb{T}}\mathrm{arg}((\gamma(\theta)-\gamma(\theta^{\prime}))_{1}+i(\gamma(\theta)-\gamma(\theta^{\prime}))_{2})\cdot[\varphi]^{\prime}(\theta^{\prime})\,d\theta^{\prime}.\end{split}$$

Here the argument is defined such that its values at $\theta=\pm\pi$ coincide. In the last equality, we need the assumption $\gamma\in C^{1}(\mathbb{T})$. Hence, using the fact that $[\varphi]\in C^{1}(\mathbb{T})$,

(2.31)

$$\begin{split}\frac{d}{d\theta}(\mathcal{D}_{\gamma}[\varphi])|_{\gamma}=&\;-\frac{1}{2}[\varphi]^{\prime}+\frac{1}{2\pi}\frac{d}{d\theta}\int_{\mathbb{T}}\mathrm{arg}((\gamma(\theta)-\gamma(\theta^{\prime}))_{1}+i(\gamma(\theta)-\gamma(\theta^{\prime}))_{2})\cdot[\varphi]^{\prime}(\theta^{\prime})\,d\theta^{\prime}\\ =&\;\frac{1}{2\pi}\mathrm{p.v.}\int_{\mathbb{T}}\frac{d}{d\theta}\left(\mathrm{arg}((\gamma(\theta)-\gamma(\theta^{\prime}))_{1}+i(\gamma(\theta)-\gamma(\theta^{\prime}))_{2})\right)\cdot[\varphi]^{\prime}(\theta^{\prime})\,d\theta^{\prime},\end{split}$$

This justifies (2.29). Next let

(2.32)

$$e_{r}:=(\cos\theta,\sin\theta),\quad e_{\theta}:=(-\sin\theta,\cos\theta).$$

Then $[\varphi]^{\prime}$ and $\phi^{\prime}$ satisfy

(2.33)		$\displaystyle[\varphi]^{\prime}=$	$\displaystyle\;2A\left((f^{\prime}(\theta)e_{r}+f(\theta)e_{\theta})\cdot\nabla(\Gamma*g)|_{\gamma}+\gamma^{\prime}(\theta)^{\perp}\cdot\mathcal{K}_{\gamma}[\varphi]^{\prime}+\gamma^{\prime}(\theta)^{\perp}\cdot\mathcal{K}_{\gamma,\tilde{\gamma}}\phi^{\prime}\right),$
(2.34)		$\displaystyle\phi^{\prime}=$	$\displaystyle\;-2\left((F^{\prime}(\theta)e_{r}+F(\theta)e_{\theta})\cdot\nabla(\Gamma*g)|_{\tilde{\gamma}}+\tilde{\gamma}^{\prime}(\theta)^{\perp}\cdot\mathcal{K}_{\tilde{\gamma}}\phi^{\prime}+\tilde{\gamma}^{\prime}(\theta)^{\perp}\cdot\mathcal{K}_{\tilde{\gamma},\gamma}[\varphi]^{\prime}\right).$

Now we shall linearize the equations (2.16), (2.17), (2.33) and (2.34) around the radially symmetric configurations, i.e., $f\equiv r$, $F\equiv R$, and $[\varphi]^{\prime}=\phi^{\prime}\equiv 0$. The following discussion is only formal and gives an overview of the analysis carried out in the rest of the paper. Let us begin by collecting a few facts that will be justified in later sections.

• •

  It will be clear in Section 4 and Section 7 that

  (2.35)

  $$e_{r}\cdot\nabla(\Gamma*g)|_{\gamma}\approx c_{*}\quad\mbox{ and }\quad e_{r}\cdot\nabla(\Gamma*g)|_{\tilde{\gamma}}\approx\tilde{c}_{*}:=\frac{c_{*}r}{R}.$$

  Here $c_{*}$ and $\tilde{c}_{*}$ are constants defined in (2.22).

• •

  Let $\mathcal{H}$ be the Hilbert transform on $\mathbb{T}$ [33], i.e.,

  (2.36)

  $$\mathcal{H}f(\theta):=\frac{1}{2\pi}\mathrm{p.v.}\int_{\mathbb{T}}\cot\left(\frac{\theta-\theta^{\prime}}{2}\right)f(\theta^{\prime})\,d\theta^{\prime}.$$

  Then in Section 5 we shall show

  (2.37)

  $$\gamma^{\prime}\cdot\mathcal{K}_{\gamma}\approx\frac{1}{2}\mathcal{H}\quad\mbox{ and }\quad\tilde{\gamma}^{\prime}\cdot\mathcal{K}_{\tilde{\gamma}}\approx\frac{1}{2}\mathcal{H}.$$

• •

  Define $\mathcal{S}$ to be a smoothing operator on $\mathbb{T}$ with a Poisson kernel,

  (2.38)

  $$\mathcal{S}\psi(\theta)=\frac{1}{2\pi}P_{\frac{r}{R}}*\psi(\theta)=\frac{1}{2\pi}\int_{\mathbb{T}}\frac{1-\left(\frac{r}{R}\right)^{2}}{1+\left(\frac{r}{R}\right)^{2}-2\left(\frac{r}{R}\right)\cos\xi}\psi(\theta-\xi)\,d\xi.$$

  The notation $P_{\frac{r}{R}}$ will be introduced in Section 6. Then in Section 6 we shall see

  (2.39)		$\displaystyle\gamma^{\prime}\cdot\mathcal{K}_{\gamma,\tilde{\gamma}}\phi^{\prime}\approx\frac{1}{2}\mathcal{H}\mathcal{S}\phi^{\prime},$	$\displaystyle\gamma^{\prime\perp}\cdot\mathcal{K}_{\gamma,\tilde{\gamma}}\phi^{\prime}\approx\frac{1}{2}\mathcal{S}\phi^{\prime},$
  (2.40)		$\displaystyle\tilde{\gamma}^{\prime}\cdot\mathcal{K}_{\tilde{\gamma},\gamma}[\varphi]^{\prime}\approx\frac{1}{2}\mathcal{H}\mathcal{S}[\varphi]^{\prime},$	$\displaystyle\tilde{\gamma}^{\prime\perp}\cdot\mathcal{K}_{\tilde{\gamma},\gamma}[\varphi]^{\prime}\approx-\frac{1}{2}\mathcal{S}[\varphi]^{\prime}.$

• •

  The remaining terms in (2.16), (2.17), (2.33) and (2.34) and the error made above are considered to be smaller or more regular, which will be omitted at this moment.

Putting these facts together, the linearized system can be written as

(2.41)		$\displaystyle\partial_{t}f+c_{*}=$	$\displaystyle\;-\frac{1}{2r}\mathcal{H}([\varphi]^{\prime}+\mathcal{S}\phi^{\prime}),$
(2.42)		$\displaystyle\partial_{t}F+\frac{c_{*}r}{R}=$	$\displaystyle\;-\frac{1}{2R}\mathcal{H}(\phi^{\prime}+\mathcal{S}[\varphi]^{\prime}).$
(2.43)		$\displaystyle[\varphi]^{\prime}=$	$\displaystyle\;2Ac_{*}f^{\prime}+A\mathcal{S}\phi^{\prime},$
(2.44)		$\displaystyle\phi^{\prime}=$	$\displaystyle\;-\frac{2c_{*}r}{R}F^{\prime}+\mathcal{S}[\varphi]^{\prime}.$

See Section 7 and Section 8 for the complete equations.

Combining (2.43) and (2.41), we obtain

(2.45)

$$\partial_{t}f+c_{*}=-\frac{Ac_{*}}{r}(-\Delta)^{1/2}f-\frac{1+A}{2r}\mathcal{H}\mathcal{S}\phi^{\prime}.$$

(2.45) is a fractional heat equation only when $Ac_{*}>0$. Note that the last term in (2.45) and all those omitted ones are supposed to be small or regular source terms. Since $c_{*}<0$, it is natural to believe that the motion of $\gamma$ can be well-posed only when $A<0$, i.e, $\mu<\nu$.

Similarly, by combining (2.42) with (2.44),

(2.46)

$$\partial_{t}F+\frac{c_{*}r}{R}=\frac{c_{*}r}{R^{2}}(-\Delta)^{1/2}F-\frac{1}{R}\mathcal{H}\mathcal{S}[\varphi]^{\prime}.$$

Note that it shows the smoothing of the outer interface not to depend on $A$, but only on the fact that $\frac{c_{*}r^{2}}{R}<0$.

The parabolicity of (2.45) and (2.46) is sufficient to prove existence of local solutions in Section 8, and then uniqueness in Section 9. The proof of the local existence uses two layers of fixed-point arguments. We sketch it as follows.

1. (1)

   Fix a pair of interface dynamics $f$ and $F$.

2. (2)

   First we need to solve for $[\varphi]^{\prime}$ and $\phi^{\prime}$ associated with the domain defined by $f$ and $F$. To do that, in Section 7, we apply a fixed-point argument to static equations (7.1) and (7.2) (or equivalently, (2.33) and (2.34)) with the variable $([\varphi]^{\prime},\phi^{\prime})$. In this argument, we need estimates for the remainder terms that are omitted in (2.43) and (2.44), which turn out to be small.

3. (3)

   Once $[\varphi]^{\prime}$ and $\phi^{\prime}$ are well-defined and their estimates are derived, we use them to bound $\mathcal{H}\mathcal{S}\phi^{\prime}$ and $\mathcal{H}\mathcal{S}[\varphi]^{\prime}$ in (2.45) and (2.46) as well as all the remainder terms omitted there (see (8.1) and (8.2) for the complete equations). They altogether will be put as the source terms in some fractional heat equations similar to (2.45) and (2.46) in order to construct a new pair of interface dynamics, $\tilde{f}$ and $\tilde{F}$. See (8.32)-(8.34). We then show in Section 8 that the map $(f,F)\mapsto(\tilde{f},\tilde{F})$ has a fixed-point, which is a local solution.

4. (4)

   In this process, bounds for all the remainder terms will rely on estimates derived in Sections 3-6. See Section 2.5 for what are exactly covered in them.

The proof of the uniqueness boils down to showing that $[\varphi]^{\prime}$, $\phi^{\prime}$ and all the remainder terms above depend in a Lipschitz manner on the interface configurations. Indeed, what we prove is a stability-type estimate for $f$ and $F$ based on that of the fractional heat equation. We carry out this idea in Section 9 with a twist in order to slightly reduce complexity of the proof.

In Section 3, we first study the pressure $p$ in an almost radially symmetric geometry by elliptic regularity theory. In Section 4, we derive estimates concerning gradients of the growth potential $\Gamma*g$ (c.f., (2.3) and (2.4)) restricted to inner and outer interfaces. Section 5 is devoted to proving estimates for singular integral operators $\mathcal{K}_{\gamma}$ and $\mathcal{K}_{\tilde{\gamma}}$, while Section 6 establishes estimates for integral operators $\mathcal{K}_{\gamma,\tilde{\gamma}}$ and $\mathcal{K}_{\tilde{\gamma},\gamma}$. Section 7 shows well-definedness of $[\varphi]$ and $\phi$ as well as their estimates. Finally, we prove existence of the local solution in Section 8, and uniqueness in Section 9. Some auxiliary estimates and non-essential lengthy proofs are collected in Appendices.

First we introduce a diffeomorphism to transform the physical domain into a reference domain that is perfectly radially symmetric. Given $\delta$ satisfying (2.23), define a cut-off function $\eta_{\delta}\in C_{0}^{\infty}([0,+\infty))$, such that $\eta_{\delta}\in[0,1]$ is only supported on $[1-2\delta,1+2\delta]$, $\eta_{\delta}=1$ on $[1-\delta,1+\delta]$, and for some universal constant $C$,

(3.1)

$$\delta|\eta_{\delta}^{\prime}|+\delta^{2}|\eta_{\delta}^{\prime\prime}|\leq C.$$

Let $X=(\rho\cos\omega,\rho\sin\omega)\in\mathbb{R}^{2}$ be a point in the reference coordinate, with $\rho=|X|$. Define

(3.2)

$$x(X)=\left[1+h(\omega)\eta_{\delta}\left(\frac{\rho}{r}\right)+H(\omega)\eta_{\delta}\left(\frac{\rho}{R}\right)\right]X=:\zeta(X)X.$$

where $h$ and $H$ are given in (2.19). In other words, $x$ deforms the reference domain in the radial direction only in annuli around $\partial B_{r}$ and $\partial B_{R}$. It depends only on $\gamma$ in the annulus $B_{r(1+2\delta)}\backslash B_{r(1-2\delta)}$, and only on $\tilde{\gamma}$ in $B_{R}\backslash B_{R(1-2\delta)}$; $x(X)=X$ elsewhere. We may also write $\zeta(X)$ as $\zeta(\rho,\omega)$. We know that $x(X)$ is a diffeomorphism from $\mathbb{R}^{2}$ to itself provided that $\zeta(\rho,\omega)\rho$ is strictly increasing in $\rho$ for all $\omega\in\mathbb{T}$. This is true if oscillations of $\gamma$ and $\tilde{\gamma}$ in the radial direction are small with respect to the gap between them, i.e.,

(3.3)

$$\delta^{-1}(\|h\|_{L^{\infty}(\mathbb{T})}+\|H\|_{L^{\infty}(\mathbb{T})})\ll 1.$$

Under this assumption, it is clear that $x(X)$ maps $B_{r}$, $B_{R}$, $\partial B_{r}$ and $\partial B_{R}$ to $\Omega$, $\tilde{\Omega}$, $\gamma$ and $\tilde{\gamma}$, respectively. We denote its inverse to be $X(x)$.