Persistence of $C^2$ regularity for aggregation patches

Introduction

Active scalar equations form an important class of nonlocal transport equations arising in fluid dynamics and mathematical biology. In these equations, a scalar quantity is transported by a velocity field that depends nonlocally on the scalar itself. Patch solutions provide a natural setting in which to study the propagation of geometric regularity for this class of equations. In this paper we study this question for the two-dimensional aggregation transport equation and, more generally, for kernels obtained by applying a linear map to the gradient of the Newtonian potential. Our principal concern is the persistence of $C^2$ regularity for patch boundaries.

We consider the Cauchy problem

where $D_0 \subset\mathbb{R}^2$ is a bounded domain and $K$ is the interaction kernel. For sufficiently regular velocity fields, the transport equation preserves the value of the scalar along the trajectories of the flow. Hence, if $\rho(\cdot,0) = \chi_{D_0}$, then $\rho(\cdot,t) = \chi_{D(t)}$, where $D(t)$ is a bounded domain evolving under the flow generated by $v$. Such solutions are called patch solutions.

Two important examples of this equation are the two-dimensional Euler equation and the aggregation equation. For Euler, the scalar $\rho$ is the vorticity and it is usually denoted by $\omega$, and the velocity is given by the Biot–Savart law, whose kernel is

where $N(x) = \frac{1}{2\pi}\log|x|$ is the fundamental solution of the Laplacian in two dimensions. Thus, the Euler velocity satisfies $v_{\mathrm{E}} = \nabla^\perp N * \rho= -\nabla^\perp(-\Delta)^{-1}\rho$, $\operatorname{div} v_{\mathrm{E}} = 0$.

The aggregation equation considered here corresponds to the kernel

Consequently, the aggregation velocity satisfies $v_{\mathrm{Ag}} = \nabla(-\Delta)^{-1}\rho$, and $\operatorname{div} v_{\mathrm{Ag}} = -\rho$ in the sense of distributions.

The literature on vortex patches for the two-dimensional Euler equation is extensive, and we highlight here the results most relevant to our study.

At the level of weak solutions, Yudovich’s theorem (see [20]) ensures global existence and uniqueness for initial vorticity data in $L^1 \cap L^\infty$. In particular, this provides global well-posedness for vortex patches. However, this result does not provide control of the regularity of the boundary of the patch. This question was resolved in the Hölder setting by Chemin [9], who proved global persistence of $C^{1,\alpha}$ regularity ($0 < \alpha< 1$) using paradifferential calculus. Shortly thereafter, Bertozzi and Constantin [1] obtained an alternative proof based on geometric methods in classical analysis. Their arguments can be extended to show persistence of $C^{k,\alpha}$ regularity, for $0 < \alpha< 1$ for any $k \in\mathbb{N}$, $k \ge1$. The endpoint regularity question for $C^2$ and $C^{1,1}$ was settled by Kiselev and Luo in [16]. They constructed $C^2$ initial patches whose boundary curvature becomes unbounded immediately for every $t > 0$, thereby establishing ill-posedness in both $C^2$ and $C^{1,1}$.

For the specific case of rotating vortex patches, several questions concerning their existence and regularity have been addressed in the literature. The Rankine vortex [18], corresponding to a circular patch, and Kirchhoff’s rotating ellipse [15] are classical examples of such solutions, with explicitly known shapes. Deem and Zabusky [11] numerically computed the first families of nontrivial rotating patches, the so-called V-states, emanating from the disk, and Burbea [4] later gave a rigorous construction of these branches by bifurcation from the disk. Hmidi, Mateu and Verdera [14] proved smoothness of these patches, while Hmidi and Mateu [13] studied bifurcation from Kirchhoff ellipses. Later on, spatial analyticity was established for such solutions. Castro, Córdoba and Gómez-Serrano [7] constructed uniformly rotating analytic patches. More recently, Castro-López and Gómez-Serrano [8] established the existence of uniformly rotating analytic patches with non-convex boundaries.

Despite its importance in applications such as biological swarming and the evolution of vortex densities in superconductors (see [3] and the references therein), the mathematical literature on the aggregation equation is considerably more limited. The Newtonian aggregation equation is usually written in divergence form,

In a framework analogous to Yudovich’s theory for the Euler equation, Bertozzi, Laurent and Léger [3] proved that, for compactly supported $\rho_0\in L^1\cap L^\infty$, (3) has a unique distributional solution in $C([0,T];L^1)\cap L^\infty$ for every $T<(\sup\rho_0)^{-1}$. For $\rho_0=\chi_{D_0}$ this solution is

where $\chi_{D_t}$ is the solution of (1) with $K=K_{\mathrm{Ag}}$ considered in Theorem 1.1 (see [3 Section 4]). In the divergence formulation the patch collapses at the finite time $s=1$ onto a set of Lebesgue measure zero, observed numerically to be a set of codimension one called the skeleton [3] (see [12, 6] for rigorous results in particular cases). By (4), $|D_t|=e^{-t}|D_0|$ and the collapse corresponds to $t\to\infty$, so the transport formulation is defined for all $t\ge0$. Bertozzi, Garnett, Laurent and Verdera [2] proved, in every dimension, that $C^{1,\alpha}$ regularity of the boundary of aggregation patches persists for all times.

Thus, both the two-dimensional Euler equation and the aggregation equation preserve $C^{1,\alpha}$ regularity of patch boundaries. A natural question was whether they should have the same behavior for $C^2$ boundaries. The answer is negative. We prove that the two-dimensional aggregation transport equation preserves $C^2$ boundaries. This contrast motivates our main result.

Both Euler and aggregation kernels belong to a broader family of kernels of the form $K_L=L(\nabla N)$, where $L:\mathbb{R}^2\to\mathbb{R}^2$ is a linear map. Another notable member of this family corresponds to $L=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$, and is given in complex notation by the Cauchy kernel $K(z)=\frac{1}{2\pi z}$. Although the associated velocity field is not divergence free, the existence of Yudovich-type solutions has been established for this kernel by Clop and Sengupta [10] via quasiconformal map techniques.

Cantero, Mateu, Orobitg and Verdera [5] established a more general patch persistence result in arbitrary dimension $d$, replacing the Biot-Savart and aggregation kernels with a general $K$ that is odd, homogeneous of degree $-(d-1)$ and $C^2$ away from the origin. Under these assumptions, they proved global persistence of $C^{1,\alpha}$ patch boundaries, together with uniqueness in the corresponding class of regular patches. Their theorem includes Euler and aggregation, but also kernels which are neither divergence free nor gradients of scalar potentials. Later, Verdera [19] gave a different planar proof for their family of kernels. In particular, both results apply in the plane to kernels of the form $K_L(x)=L(\nabla N(x))$.

This raises the question of whether $C^2$ regularity is also preserved for the family of kernels $K_L$. Our next theorem answers this question by characterizing precisely the linear maps $L$ for which persistence of $C^2$ regularity holds. In particular, it shows that the aggregation kernel is exceptional within this family: persistence for every $C^2$ initial patch holds precisely when $L$ is a scalar multiple of the identity map.

The persistence of $C^2$ regularity for aggregation patches is particularly striking in view of the contrasting behavior of Euler patches established by Kiselev and Luo. Indeed, within the family $K_L$, the Euler kernel, up to scalar multiples, is the only nontrivial one that generates a divergence-free velocity field, yet $C^2$ regularity may fail to persist for Euler patches. By contrast, the aggregation velocity satisfies $\operatorname{div}v_{\mathrm{Ag}}=-\rho$, which is discontinuous across the patch boundary. Thus, the persistence of $C^2$ regularity for aggregation cannot be attributed to incompressibility, but rather stems from the geometric structure of the curvature equation, as we will see.

Outline of the paper.

In Section 2 we prove the $W^{2,p}$ regularity. In Section 3 we derive the curvature equation for aggregation. Section 4 contains the main theorem: persistence of $C^2$ regularity for the aggregation model. Finally, Section 5 deals with general kernels of the form $L(\nabla N)$.

Notation, conventions, useful relations and function spaces

To make the paper easier to read, we begin by introducing the notation and conventions that will be used throughout the paper. For the reader’s convenience, we also recall some useful geometric identities and define the function spaces in which our arguments will be carried out.

For a vector $v=(v_1,v_2)\in\mathbb{R}^2$, we define its perpendicular by $v^\perp=(-v_2,v_1)$. We identify $\mathbb{T}$ with $\mathbb{R}/(2\pi\mathbb{Z})$. For a curve $\mathbb{T}\ni\xi\mapsto\gamma(\xi)$, we define the arc-length metric $g(\xi)=|\partial_\xi\gamma(\xi)|$, the unit tangent vector $\tau(\xi)=\frac{\partial_\xi\gamma(\xi)}{g(\xi)}$, and the normal vector $n(\xi)=-\tau^\perp(\xi)$. If a closed simple curve is parametrized counterclockwise, then $n(\xi)$ is the outer normal vector with this convention.

The arc-length parameter is defined by $s=\int_0^\xi g(\eta)\,\mathrm{d}\eta$. The chain rule gives the relationship $\partial_\xi=g\partial_s$. Moreover, the tangent and normal vectors satisfy the Frenet formulas

where $\kappa$ is the signed curvature of $\gamma$.

Regarding the function spaces we will work with, we set the quantities

where $|\xi-\eta|$ denotes the periodic distance on $\mathbb{T}$. The quantity $\Gamma(\gamma)$ is the inverse arc-chord ratio and controls self-intersections and degeneration of the parametrization, while $|g|_*$ controls the size of the parametrization speed and its reciprocal. Note that $|g|_*$ is not a norm.

Then we will work with the Banach space

and a closed subset of it,

$W^{2,p}$ regularity

To study the persistence of $W^{2,p}$ regularity of the boundary of an aggregation patch, we use a contour dynamics formulation. This formulation describes the evolution of the boundary through an equation for its parametrization, reducing the patch evolution problem to a nonlocal evolution equation for a curve. The contour dynamics formulation is classical for the two-dimensional Euler equations; see, for instance, Deem and Zabusky [11] and Majda and Bertozzi [17] for background and a detailed treatment. For the aggregation equation, and for the kernels of § 5, the contour dynamics equation is obtained analogously; see [2], § 2 and [5], Section 2.1.

Let $\rho(\cdot,t)=\chi_{D(t)}$ be the aggregation patch, and let $\gamma(\cdot,t)$ be a counterclockwise parametrization of $\partial D(t)$ whose points move with the velocity field, $\partial_t\gamma(\xi,t)=v_{\mathrm{Ag}}(\gamma(\xi,t),t)$. Since $v_{\mathrm{Ag}}=-\nabla N*\rho$, where $N(x)=\frac{1}{2\pi}\log|x|$, Green’s formula yields the contour dynamics equation (CDE), up to a tangential reparametrization:

Local existence and uniqueness in $W^{2,p}$ follow from a standard fixed-point argument, which yields a unique solution on a finite time interval. We omit the technical details and refer the interested reader to [16 Section 2], where the same argument is developed for vortex patches. This is possible due to the fact that the velocity field considered here is obtained by rotating the Euler velocity field, as we argued before. Nevertheless, we include the formulas for the velocity derivatives, which are needed throughout. More precisely, the velocities at a point in $\gamma(\cdot,t)$ for Euler and for Aggregation are

we observe that the only difference is the factor $n(\eta)$ instead of the factor $\tau(\eta)$ in the integrand.

Indeed, both kernels are of the form $L(\nabla N)$, with $L=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ for Euler and $L=-\operatorname{Id}$ for Aggregation, and the velocity at $\gamma(\xi)$ is then $-\frac{1}{2\pi}\int_{\mathbb{T}}Ln(\eta)\log|\gamma(\xi)-\gamma(\eta)|g(\eta)\,\mathrm{d}\eta$. Since $L$ is constant, it commutes with $\partial_{\xi}$ and $\partial_s$, so all the computations involving derivatives in $\xi$ hold for every constant matrix $L$. Since the Aggregation velocity is obtained by applying an orthogonal rotation to the Euler velocity, the corresponding formulas are obtained by applying the same rotation to the vector-valued factors in the formulas for Euler. In particular, whenever they appear outside the scalar kernels, $\tau(\eta)$ is replaced by $-n(\eta)$, while $n(\eta)$ is replaced by $\tau(\eta)$. Therefore, [16 Proposition 2.4] yields the following result for Aggregation patches.

Now, since the kernels involved in (5) and (6) are of the same type as those for the Euler case, we can recover (see [16 Propositions 2.7 and 2.8]) the following bounds for $\gamma_1,\gamma_2\in B_p^M$,

Hence for the equation $\partial_t\gamma(t)=v(\gamma(t))$ with $\gamma(0)=\gamma_0$ the arc-chord parametrization of $\partial D_0$ the fixed point argument of [16 Proposition 2.9] gives the local-in-time existence and uniqueness in $W^{2,p}$. Consequently, we have the following theorem.

The scheme to prove global-in-time regularity of solutions is not that classical. We follow the clever argument in [16 Section 3]. The idea is to embed $W^{2,p}$ in a space where global solutions exist and then get an improved $W^{2,p}$ estimate taking advantage of the fact that the solution is $C^{1,\alpha}$. More specifically, since $W^{2,p}(\mathbb{T}) \hookrightarrow C^{1,\alpha}(\mathbb{T})$ provided $\alpha= 1 - 1/p$, the global $C^{1,\alpha}$ theory controls all the quantities defining $B_{p}^{M}$ except $\|\gamma\|_{W^{2,p}}$, which is then controlled by an improved estimate. The only ingredient that depends on the equation is [16 Proposition 3.1]: the solution of (CDE) continues for all times in $C^{1,\alpha}$ and, for every $T > 0$,

For Aggregation patches, by the uniqueness in [2, Theorem 2.1] the local solution in Theorem 2.2 coincides with the global $C^{1,\alpha}$ solution of [2 Sections 2 and 7], and (7) follows from the a priori bounds in [2 (7.15)–(7.16)], which control $\|\nabla v_{\mathrm{Ag}}(\cdot,t)\|_{L^\infty}$ and the $C^{1,\alpha}$ character of $\partial D_t$ on bounded time intervals. Indeed, $|\partial_s v_{\mathrm{Ag}}| \leq\|\nabla v_{\mathrm{Ag}}\|_{L^\infty}$; by (CDE), $|\partial_t g| \leq\|\nabla v_{\mathrm{Ag}}\|_{L^\infty} g$ and $\partial_t|\gamma(\xi,t)-\gamma(\eta,t)| \leq\|\nabla v_{\mathrm{Ag}}\|_{L^\infty}|\gamma(\xi,t)-\gamma(\eta,t)|$, which bound $|g|_{*}$ and $\Gamma(\gamma)$ by Gronwall’s lemma; and $\|\tau(t)\|_{C^\alpha}$ is controlled by the $C^{1,\alpha}$ bound for $\partial D_t$ together with the bound for $g$. Thus, we can follow the computations in [16 Proposition 3.3] to obtain the improved bound

Since solutions $\gamma$ of equation (CDE) satisfy $\gamma(t)=\gamma_0+\int_0^t v_{\mathrm{Ag}}(\gamma(s))\,\mathrm{d}s$, using the bound in (8) and Gronwall’s lemma we get a $W^{2,p}$ bound for $\gamma(t)$ that does not blow-up in finite time,

Thus, the maximum time of existence cannot be finite and the solution is global. Moreover, $v_{\mathrm{Ag}}(\gamma)\in C([0,\infty);W^{2,p}(\mathbb{T}))$ by the Lipschitz bounds above, so $\gamma\in C^1([0,\infty);W^{2,p}(\mathbb{T}))$, as in [16, Theorem 3.4].

The curvature equation for aggregation

In [16], a differential equation for the curvature $\kappa$ is derived from geometric identities. This is equation (4.7) in their article and hence it is completely independent to the partial differential equation we are dealing with.

We start with the first term in the right hand side of the equation. We need to give an expression for $\partial_s\partial_t\gamma\cdot\tau$.

every $\xi\in\mathbb{T}$,

Consequently, for the aggregation velocity,

Proof. Fix $\xi\in\mathbb{T}$ and set $x=\gamma(\xi)$ and $y=\gamma(\eta)$. Also, set $B_\varepsilon(x)$ the ball centered at $x$ with radius $\varepsilon$. It is easy to check that $\frac{x-y}{|x-y|^2}$ is a divergence free vector field. Then, by the divergence theorem applied to the domain $D\backslash B_\varepsilon(x)$ we have

where we have used that the normal in the circular part is $(x-y)/\varepsilon$. Taking the limit when $\varepsilon\to0$ and putting the labels $x$ and $y$ as $\gamma(\xi)$ and $\gamma(\eta)$ we obtain the identity (10).

Projecting onto the normal and tangential directions, we have

The first integral is constantly equal to $-1/2$ as we have just seen. The second one is equal to $\partial_s\partial_t\gamma(\xi)\cdot\tau(\xi)$ because of identity (5) and thus we have proved (11). □

Remark 1. Although the geometric computations remain valid, rotating the velocity changes the projections entering the curvature equation. Then, the equivalent version of the previous lemma for Euler equation is not valid here.

We take care now of the second term in the curvature equation.

Proof. Taking the normal projection of (6), note that there is a term containing $\kappa(\xi)$, which is

By (11) this is equal to

The rest of the terms of the normal projection of (6) are exactly $-F(\xi)$, which finishes the proof. □

Remark 2. Note that the integral with principal value in (6) is integrable after taking the scalar product with $n(\xi)$ because

This inequality is true for $\gamma\in C^{1,\alpha}(\mathbb{T})$. See, for instance, [16], Lemma 3.2.

Finally, by substituting identities (11) and (12) into (9) we obtain the desired equation for the curvature

where $F$ is defined in (13) and $G(\xi):=\partial_s\partial_t\gamma(\xi)\cdot\tau(\xi)$.

Finding an equation describing the evolution of the metric $g$ is simpler. Since $g=|\partial_\xi\gamma|$, differentiating $g^2$ gives $\partial_tg=g\partial_sv_Ag\cdot\tau$. Again, the term $\partial_sv_Ag$ in (5) contains a singular integral but turns integrable when multiplied by $\tau$. In fact,

Equations (9) and (15) establish a system of differential equations

for $F$ defined in (13) and $G$ defined in (15). This is the curvature-metric system. Solving it is equivalent to solving (CDE). The following statement is exactly [16 Proposition 4.2] and it states this equivalence. We repeat it here for reader’s convenience, but omit the proof.

Persistence of C$^{2}$ regularity

In the previous section, we derived a system of differential equations involving the curvature and the metric (CM). Now, we solve this system and verify that the solution provides a $C^{2}$ regular Jordan curve.

Proof. We can integrate the system (CM). The term $G$ is well-defined and uniformly bounded in $L^{\infty}$ for every time interval. We can integrate the metric $g$ in terms of $G$ as

Since $|G(\xi,t)|\leq C_{T}$ for every $t\in[0,T]$ we have

Thus, the metric remains uniformly bounded above and below from zero and so the parametrization remains regular for every time.

In order to achieve a closed formula for the curvature we differentiate with respect to time the function $t\mapsto e^{t/2}g(\xi,t)^{3}\kappa(\xi,t)$, so $e^{t/2}g^{3}$ is an integrating factor. Making use of the equation for $g$, we have

Now integrating with respect to time between $0$ and $t$, we have

The integrals in the solutions $g(\xi,t)$ and $\kappa(\xi,t)$ are both well defined, since $F,G\in L^{1}_{\mathrm{loc}}([0,\infty);C(\mathbb{T}))$ and $g$ is bounded on compact time intervals. Also, since $\kappa_{0}$ is continuous and $g$ is continuous and bounded away from zero, then $\kappa(\xi,t)\in C(\mathbb{T})$ for all time $t>0$, which proves the theorem. □

General kernel

Let $L:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a linear transform. We consider now the transport equation with patch initial data

Thus, (17) becomes 2D-Euler for $L=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and becomes the Aggregation transport equation for $L=-\operatorname{Id}$. Hence, equation (16) is a generalization of both of them.

A first remark is that, since $K_{\mathrm{Ag}}=-\operatorname{Id}(\nabla N)$, then $v_L=-Lv_{\mathrm{Ag}}$ and hence some of the computations done until now can be easily adapted. On the other hand, note that the divergence of the velocity field can be more involved than in the previous scenarios. For Euler we have incompressibility, for Aggregation the divergence is bounded for density patches, since $\operatorname{div}v_{\mathrm{Ag}}=-\rho=-\chi_D$. But in general $\operatorname{div}v_L$ is the action of a Calderón-Zygmund operator on $\rho=\chi_D$ and then it does not have to be even bounded. In fact, one can only conclude $\operatorname{div}v_L$ belongs to the BMO class. In the arguments below this difficulty on the divergence is not relevant, but it is when dealing with the $C^{1,\alpha}$ persistence of regularity, which is an intermediate step of the proof. We use the main theorem in [5] to ensure this regularity.

Since $v_L=-Lv_{\mathrm{Ag}}$, integration by parts gives an associated Contour Dynamics Equation for the problem

(LCDE) $\partial_t\gamma(\xi,t)=-\frac{1}{2\pi}\int_{\mathbb{T}}Ln(\eta,t)\log\lvert\gamma(\xi,t)-\gamma(\eta,t)\rvert g(\eta,t)\,d\eta$.

The first thing we need is global-in-time existence and uniqueness of solutions for (LCDE).

Proof. The velocity in (LCDE) is the one considered in Section 2 for a general constant matrix $L$, so the proofs of Theorems 2.2 and 2.3 apply verbatim, with (7) provided by [5 Section 2]. □

Now we establish the curvature-metric system that will allow us to study the persistence of $C^2$ smoothness of the curve.

and $F_L$ is a sum of absolutely convergent integrals. Explicitly,

Proof. Throughout the proof, $v_{\mathrm{Ag}}$ denotes the aggregation velocity associated with the current domain, evaluated along its boundary. Since $L$ is constant, it commutes with time and spatial derivatives. Hence the identity for the evolution of the curvature yields

We therefore use the formula for $\partial_s^2v_{\mathrm{Ag}}$ obtained in the aggregation case, apply $L$, and project onto $n$. The two terms in that formula which do not explicitly contain curvature become precisely the second and third integrals in (19). It remains to identify the contributions of the terms containing $\kappa(\xi)$ and $\kappa(\eta)$.

The term containing $\kappa(\xi)$ is

To evaluate it, we need both components, in the frame at $\xi$, of the vector integral before applying $L$. Its normal component is the identity (11):

The tangential component simply satisfies

Decomposing $n(\eta)$ in the orthonormal frame $(\tau(\xi),n(\xi))$ and using these two identities, the term containing $\kappa(\xi)$ becomes

Together with the geometric term $-2\kappa\,\partial_s\partial_t\gamma\cdot\tau$, this gives exactly $c_L\kappa$.

We then focus on the term containing $\kappa(\eta)$

This will be compared with a Hilbert transform as in Kiselev-Luo. In fact, we will follow exactly the arguments in [16 Section 5.1]. Specifically, in the singular integral containing $\kappa(\eta)$ we add and subtract $q_L(\xi)H\kappa(\xi)$, where $q_L(\xi)=\frac{1}{2}L\tau(\xi)\cdot n(\xi)$. At the level of the kernel, this corresponds to adding and subtracting

By the same diagonal comparison and periodization used in [16 Proposition 5.2], this kernel matches the leading singularity of the original one. The added term contributes exactly $q_LH\kappa$, while the difference between (20) and (21) is the first integral in (19) and is absorbed into $F_L$. The cancellation makes this difference absolutely integrable for $p>2$. The other two integrals in (19) are the remaining terms of the curvature equation. □

Remark 3. Note that in the previous proposition, if $L=c\operatorname{Id}$ the Hilbert transform term vanishes. In particular if $L=-\operatorname{Id}$ we recover the curvature-metric system (CM).

Now we are prepared to distinguish for which linear transforms $L$ we recover the persistence of the $C^2$ regularity of the patch.

Proof. If we apply the Hilbert transform $\mathcal{H}$ to the first equation in (18), we get

where $[T_1,T_2]f=T_1(T_2f)-T_2(T_1f)$ is a commutator and where we have defined

Also, we have used $\mathcal{H}^2=-\operatorname{Id}$. Hence we can write a system of differential equations for $\kappa$ and $\mathcal{H}\kappa$ as

Although $\widetilde{F}_L$ contains a term involving $\mathcal{H}\kappa$, it is regular even when $\kappa$ is merely continuous. This is due to an extra cancellation of the commutator. In fact, we can verify that $\widetilde{F}_L$ belongs to Hölder class $C^{1/2}$, using [16 Lemma 5.5] for $p=4$ and $\sigma=3/4$ for instance.

We can solve the system of differential equations (22) by solving the homogeneous part and then by variation of constants. Suppressing the common material label $\xi$, we indeed have

where

and

The remainders $R$ and $\widetilde{R}$ are uniformly bounded in $C^{1/2}(\mathbb{T})$ as before by Lemma 5.5 in [16]. In particular, for a given time $t>0$

If $L=c\operatorname{Id}$ then $q_L=0$ so there is no Hilbert contribution in $\kappa(t)$ and we can proceed as in the Aggregation case to show persistence of $C^2$ regularity of the boundary.

On the other hand, if $L\ne c\operatorname{Id}$, we choose an initial curve $\gamma_0$ as constructed in [16 Equation (5.25)] orientated such that $q_L(0,0)\ne0$. We can do such a construction since there exists a direction $\tau$ such that

provided $L$ is not a multiple of the identity. There is a neighborhood $U$ of $(\xi=0,t=0)$ such that $\operatorname{sign}(q_L(\xi,t))=\operatorname{sign}(q_L(0,0))$. Consequently, for $(\xi,t)\in U$ we have ensured $|\sin(b(\xi,t))|>0$.

Then, for the bounded curvature $\kappa_0$ associated to $\gamma_0$ we have that $\kappa(t)\notin L^\infty(\mathbb{R}^2)$ for short time, which proves the theorem. □

Declaration on the use of AI

All mathematical computations and proofs in this paper were initially done by the authors without the use of artificial intelligence (AI) tools. Only after the mathematical statements were done, we used GPT-6 Astra (OpenAI) and Claude Opus 5.5 (Anthropic) for writing assistance and to help check the manuscript and verify the steps of the arguments. The authors have reviewed and edited all the content and take full responsibility for this publication.

References

Departament de Matemàtiques – Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Email address: [email protected]

Departament de Matemàtiques – Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Email address: [email protected]