Sodium channel. $(n_i(t), m_i(t), h_i(t))$ solve uncoupled ODE's that can be represented by adding some self-interaction in our model, for example

$$ egin{cases} 
\frac{d}{dt} n_i(t) = \alpha_n(V_i(t))(1-n_i(t)) - \beta_n(V_i(t)) n_i(t), \\ 
\frac{d}{dt} m_i(t) = \alpha_m(V_i(t))(1-m_i(t)) - \beta_m(V_i(t)) m_i(t), \\ 
\frac{d}{dt} h_i(t) = \alpha_h(V_i(t))(1-h_i(t)) - \beta_h(V_i(t)) h_i(t), 
\end{cases} 
\quad (78) $$

for some given functions $\alpha_{n,m,h}$ and $\beta_{n,m,h}$.

The neurons are coupled through their membrane potentials solving

$$ C_m \frac{dV_i}{dt} + g_K n_i^4 (V_i - V_K) + g_{Na} m_i^3 h_i (V_i - V_{Na}) + g_l (V_i - V_l) + \sum_{j \neq i} w_{ij} (V_i - V_j) = 0, \quad (79) $$

for given constants $C_m, g_K, g_{Na}, g_l, V_K, V_{Na}, V_l$.

The coupled system composed of (78)-(79) obviously fits within the general framework of systems like (1). Provided all coefficients are smooth, it also satisfies the assumptions of Theorem 1.1 so that all our results are immediately applicable.

The simulation of such large systems is challenging, and mean-field limits are again an attractive alternative; see [59, 65, 68, 76]. For identical weights $w_{ij} = 1/N$, the mean-field limit can be rigorously derived by classical methods and has proved useful in understanding some of the large scale behavior of such systems, see for example [27, 26, 29, 33, 35] on the stochastic side or [15, 16, 17, 18, 33, 66, 69] on the deterministic side.

Those studies do not account for the structure of connectivity in the neural network, whose effect has been shown to be critical, as in [71, 72, 73, 83, 84, 85]. Of course, a first practical modeling issue, when considering more complex networks, simply is that the connectivity map $w_{ij}$ has long remain mysterious. Random weights $w_{ij}$ are popular for this reason and naturally lead to random networks, see [4]. The corresponding graph of interactions is typically dense and some notion of mean-field limit, based on graphons, has been derived for example in [21, 19, 20] for smooth interaction kernels $K$, and actually for the so-called Kuramoto [51, 52] model without learning (see (84) below with $\eta = 0$).

Those follow an interesting approach to the problem when the interacting network is described as a random graph, usually based on symmetric Erdős-Rényi type graphs. The family of Erdős-Rényi graphs is one example of a convergent family of random graphs. The limiting behavior of such families is determined by a symmetric measurable function on the unit square: a classical graphon. In the case of the Erdős-Rényi graphs, the limiting graphon is even a constant function. This also leads to investigating bifurcations in the Kuramoto models on a variety of graphs ranging from symmetric and random small-world and power law graphs. The graph structure plays a key role in the transition to synchronization in the model. In [21], Kuramoto models on Erdős-Rényi and Paley graphs have been proven to have the same critical value that defines synchronization, an the authors explain the onset of synchronization in the Kuramoto model on a broad class of dense and sparse random graphs by establishing an explicit link between the network structure and the onset of synchronization. Furthermore, the mean-field limits of the Kuramoto models on these graphs can be proved to coincide with that for the Kuramoto models on weighted complete graphs [19, 20].

However, from a scaling point of view, it is expected that the graph of connection maps $w_{ij}$ should be sparse for applications in neurosciences. Mammalian brains contain between $10^8 - 10^{11}$ neurons ($86 \cdot 10^9$ actually in humans). Although, of course, the models generally apply only to subdomains or certain neural clusters in charge of some specific mission, the dimensions of these clusters is very large $N \gg 1$ and even these clusters tend to be correlated with others, which also implies very large dimensions. For example, in the human brain each neuron has on average 7000 synaptic connections to other neurons. This implies that synaptic weights must have a scale according to this number of the order of $\sup_{i,j} w_{ij} \sim 10^{-3} - 10^{-4} \ll 1$.

We also emphasize the importance of not imposing symmetry conditions to keep $w_{ij} \neq w_{ji}$ in general. In the neuronal framework, the breakdown in symmetry of network connectivity can be correlated with a time scale hierarchy in some special situations. Small deviations from symmetric connectivity will cause small deviations from the invariant dynamics and introduce a temporal hierarchy, ensuring the consequence of an emergent separation on the time scale. Exploring network mechanisms due to asymmetric couplings to create more complex processes suggests that asymmetric coupling is not always a variation that needs to be averaged, since complex dynamics can be formed by simpler dynamics through a combination of asymmetric couplings. For example, stimulating a local part of neural tissue can trigger
temporarily extended paths of motion by inciting the spread and sustained activation of motor networks. Asymmetries in connectivity can reshape the emerging dynamics of a neural network, see [72, 85].

The derivation of mean-field limits for neuron models with sparse, non-symmetric connection maps had remained fully open and is critical for biological applications. The question was made even more pressing by recent progress that have lead to the first fully detailed representation of the connections between neurons, for some simple animals such as the *Drosophila*, see for example [42]. The present article is the first to allow handling of arbitrary connection maps. However our investigations still leave a certain number of open questions which we briefly discuss below.

## A.2. Open problem: Integrate and fire models.
Many variants of models (78)-(79) have been studied to make the individual dynamics of a neuron more accurate. Those include adding noise to the system in various forms that often still fully compatible with our results.

However, so-called integrate and fire models are considered to better capture the dynamics of large networks; see for example [13, 14] or [1] for a historical perspective. Among many elaborate variants, a basic model is described here for simplicity. The state of each neuron is again represented by its membrane potential $V_i(t)$ which follows a simple SDE at almost all time,

$$dV_i(t) = b(V_i(t)) dt + \sigma_N dW_i, \qquad (80)$$

but with a probability to spike

$$\mathbb{P}(\text{spike between } t \text{ and } t+dt) \approx f(V(t)) dt. \qquad (81)$$

In the case of a spike at time $t$, $V_i$ is reset at 0 while all connected neurons are activated or inhibited

$$V_i(t_+) = 0, \quad V_j(t_+) = V_j(t_-) + \alpha w_{ij} \quad \forall j \neq i. \qquad (82)$$

There exist many variations around those models, for example to include more complex equations instead of (80), or deterministic spiking when $V_i$ reaches a given threshold. While (80)-(82) would formally fit in a modified version of (1), it includes singular interactions that prevent our results to apply. The nature of this singularity is however different from many of the other applications in this proposal, as it comes from the jumps in the $V_i$ at spiking times, instead of unbounded interactions at $X_i = X_j$. The derivation of integrate and fire models for general connectivities hence remains a major, challenging open question

## A.3. Open problem: Learning mechanisms in neuron dynamics.
Long-term perspectives also include understanding how to incorporate learning mechanisms into system (1). This is a key question for biological and artificial neuron dynamics as learning is of course a critical function of the network.

Learning can simply translate in having time dependent connection weights $w_{ij}(t)$ that evolve according to the dynamics of the agents themselves, see [2] for instance; this corresponds for example to the well-known Hebb rule that neurons wire together if they fire together. Considering again $N$ neurons with activation $X_i$ and time-dependent connectivity or synaptic weights $w_{ij}(t)$ between neuron $i$ and $j$, a straightforward extension of (1) simply reads

$$\left\{
\begin{array}{l}
\displaystyle \frac{dX_i}{dt} = \sum_{j=1}^{N} w_{ij}(t) K(X_i, X_j) \quad (+\text{noise}), \\
\\
\displaystyle \frac{dw_{ij}}{dt} = \eta (\Gamma(X_i, X_j) - w_{ij}) \quad (+\text{noise}), \\
\\
X_i(0) = X_i^0, \quad w_{ij}(0) = w_{ij}^0,
\end{array}
\right.
\qquad (83)$$

for some plasticity function $\Gamma$, where $\eta$ is regarded as the learning rate parameter.

How the mechanisms of learning can be reinforced or relaxed was specified in the neuronal context by Hebb [40]. Hebb proposed a theory of the adaptation of neurons in the brain during the learning process. The basic idea of his rule is based on the fact that neurons wire together if they fire together. Such a mechanism is based on the hypothesis that synchronous activation of cells (firing of neurons) leads to selectively pronounced increases in synaptic strength between these cells. From this process emerges the self-organized patterns. These postulates provide the neural basis for unsupervised complex learning of cognitive function in neural networks and may explain some processes that occur in the development of the nervous system.

The modeling is however made especially difficult in the present setting: changes in the $w_{ij}(t)$ need to be able to altering the structure of sparse networks while preserving bounds such as (3). If all $w_{ij}$ are of the same order, which is typical of dense graphs, then system (83) likely has a similar behavior to (1). However for sparse graphs, we expect the values of the $w_{ij}$ to vary wildly and a straightforward system like (83) just cannot correctly capture very different orders of magnitude in connectivities.
On the analytical side, weights $w_{ij}$ that adapt in time can lead to stronger synchronization between agents, strengthening correlations between them and putting in doubt the propagation of independence. Useful hints can be derived from studies on the Kuramoto model [51, 52] of coupled oscillators that was briefly mentioned above. The system is posed on the frequencies $\theta_i$ and can read, with learning,

$$ egin{cases} 
\frac{d\theta_i}{dt} = \Omega_i + \frac{1}{N} \sum_{j=1}^{N} w_{i,j} \sin(\theta_j - \theta_i), \\ 
\frac{dw_{ij}}{dt} = \eta (\Gamma(\theta_i, \theta_j) - w_{ij}) (+\text{noise}), \\ 
\theta_i(0) = \theta_i^0, \quad w_{ij}(0) = w_{ij}^0, 
\end{cases} 
\quad (84) $$

where the $\Omega_i$ are the natural frequencies of oscillators.

For large $\eta$, the other variable in the model that is the choice of $\Gamma$ takes on great relevance since, at the limit $\eta \to \infty$, we have that $w_{ij}(t) = \Gamma(\theta_i, \theta_j)$. If for example $\Gamma$ is assumed to be $\Gamma(\theta_i, \theta_j) = \cos(\theta_i - \theta_j)$, the attraction between near oscillators is reinforced whereas repulsive interaction arises between apart phases. On the other hand if one takes $\Gamma(\theta_i, \theta_j) = C|\sin(\theta_i - \theta_j)|$, synchronization only slowly emerges due to the reduction in connectivities for nearby oscillators.

Hebbian-type dynamics can be considered for the Kuramoto model. This learning law ensures that the weight of the adaptive coupling increases if the phases of the oscillators are close. One possibility discussed in [67] is to assume that the Hebbian-like plasticity function $\Gamma$ is given by

$$ \Gamma(\theta, \theta') := \frac{1}{c_{\alpha, \zeta}^{\alpha} |\theta - \theta'|_{0}^{2\alpha}}, \quad (85) $$

where $|\theta - \theta'|_0$ is the orthodromic distance (to zero) over the unit circle and the parameter $c_{\alpha, \zeta} := 1 - \zeta^{-1/\alpha}$, $\zeta \in (0, 1]$, has been chosen so that whenever two phases $\theta_i$ and $\theta_j$ stay at orthodromic distance $\sigma$ or larger, then the adaptive function $\Gamma$ predicts a maximum degree of connectedness not larger than $\zeta$ between such oscillators. This is another way to create and amplify correlations between neurons.

## APPENDIX B. TECHNICAL PROOFS FROM SUBSECTION 3.2

For the sake of completeness, we collect here the more technical proofs from subsection 3.2, starting with the proof of Lemma 3.7.

*Proof of Lemma 3.7.* Set any $w \in L_{\zeta}^{\infty} \mathcal{M}_{\zeta} \cap L_{\zeta}^{\infty} \mathcal{M}_{\xi}$, any $\phi \in L^{\infty}([0, 1])$ and any couple of sequences $\{w_n\}_{n \in \mathbb{N}}$ and $\{\phi_n\}_{n \in \mathbb{N}}$ as in the statement of the lemma. On the one hand, since $\phi \in L^1([0, 1])$, then

$$ \int_0^1 \psi(\xi) \left( \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right) d\xi = \int_0^1 \phi(\zeta) \left( \int_0^1 \psi(\xi) w(d\xi, \zeta) \right) d\zeta \le \|w\|_{L_{\zeta}^{\infty} \mathcal{M}_{\xi}} \|\phi\|_{L^1} \|\psi\|_{L^{\infty}}, $$

for any $\psi \in C([0, 1])$. This implies that $\int_0^1 \phi(\zeta) w(\cdot, d\zeta) \in \mathcal{M}([0, 1])$ and

$$ \left\| \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right\|_{\mathcal{M}_{\xi}} \le \|w\|_{L_{\zeta}^{\infty} \mathcal{M}_{\xi}} \|\phi\|_{L^1}. \quad (86) $$

Moreover, given any $\psi \in C([0, 1])$ we have that $\psi \otimes \phi_n \to \psi \otimes \phi$ strongly in $L_{\zeta}^1 C_{\xi}$. Since in addition $w_n \stackrel{*}{\to} w$ weakly-star in $L_{\zeta}^{\infty} \mathcal{M}_{\xi} = (L_{\zeta}^1 C_{\xi})^*$, then we also obtain

$$ \int_0^1 \psi(\xi) \left( \int_0^1 \phi_n(\zeta) w_n(\xi, d\zeta) \right) d\xi \to \int_0^1 \psi(\xi) \left( \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right) d\zeta. $$

Due to the arbitrariness of $\psi$, the above amounts to

$$ \int_0^1 \phi_n(\zeta) w_n(\cdot, d\zeta) \stackrel{*}{\to} \int_0^1 \phi(\zeta) w(\cdot, d\zeta), \quad (87) $$

weakly-star in $\mathcal{M}([0, 1])$. On the other hand, set any $\psi \in C([0, 1])$ and another sequence $\{\tilde{\phi}_n\}_{n \in \mathbb{N}}$ as above consisting of continuous functions. It clearly exists by density of $C([0, 1])$ in $L^1([0, 1])$ and we can indeed assume that $\{\tilde{\phi}_n\}_{n \in \mathbb{N}} \in C([0, 1])$, $\tilde{\phi}_n \to \phi$ strongly in $L^1$ and $\|\tilde{\phi}_n\|_{L^\infty} \le \|\phi\|_{L^\infty}$. Then,

$$ \int_0^1 \psi(\xi) \left( \int_0^1 \tilde{\phi}_n(\zeta) w(\xi, d\zeta) \right) d\xi \le \|w\|_{L_{\zeta}^{\infty} \mathcal{M}_{\xi}} \|\psi\|_{L^1} \|\tilde{\phi}_n\|_{L^{\infty}} \le \|w\|_{L_{\zeta}^{\infty} \mathcal{M}_{\zeta}} \|\psi\|_{L^1} \|\phi\|_{L^{\infty}}. $$

By the above weak-star convergence, we obtain

$$ \int_0^1 \psi(\xi) \left( \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right) d\xi \le \|w\|_{L_{\zeta}^{\infty} \mathcal{M}_{\zeta}} \|\psi\|_{L^1} \|\phi\|_{L^{\infty}}, $$
for every $\psi \in C([0, 1])$. Noting that $C([0, 1])$ is dense in $L^1([0, 1])$, we find that the above Radon measure $\int_0^1 \phi(\zeta) w(\cdot, d\zeta)$ actually belongs to $L^1([0, 1])^* = L^\infty([0, 1])$ and

$$ \left\| \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right\|_{L^\infty_\xi} \le \|w\|_{L^\infty_\xi} \mathcal{M}_\zeta \|\phi\|_{L^\infty}, $$

which implies $(32)_2$. Using (86) we also infer (32)$_1$ again because the Radon measure $\int_0^1 \phi(\zeta) w(\cdot, d\zeta)$ is absolutely continuous with respect to the Lebesgue measure and we already had an estimate in the total variation norm. To end the proof of (33) note that $\int_0^1 \phi_n(\zeta) w_n(\cdot, d\zeta)$ are uniformly bounded in $L^1$ and $L^\infty$ thanks to the already proven estimates (32). Since we already had that they converge weakly-star in $\mathcal{M}([0, 1])$ to $\int_0^1 \phi(\zeta) w(\cdot, d\zeta)$ by (87), we conclude that they also converge weakly-star in $L^\infty$. $\square$

We now turn to the proof of Proposition 3.11.

*Proof of Proposition 3.11.* The argument relies on a direct Picard fixed point. Specifically, let us define

$$ E := L^\infty([0, t_*] \times [0, 1], W^{1,1} \cap W^{1,\infty}(\mathbb{R}^d)), \quad E_R := \{f \in E : \|f\|_{L^\infty_{t,\xi} W_x^{1,1} \cap W_x^{1,\infty}} < R\}, $$

for any $R > 0$ to be determined later. It is straightforward to check that $E_R$ is a closed subset of $L^1([0, t_*] \times [0, 1], W^{1,\infty}(\mathbb{R}^d))$ under the $L^1$ norm in all variables $t, \xi$ and $x$. Then, endowed with the $L^1$ norm, $E_R$ becomes a complete and convex metric space, which we shall use in the fixed point argument below.

Notice that the nonlinear PDE (35) can be reformulated as a fixed point equation in the usual way. Specifically, for any $f \in E$ let us set $\mathcal{L}f \in E$ to be the unique solution of the linear Cauchy problem

$$ \left\{ \begin{array}{l} \partial_t \mathcal{L}f(t, x, \xi) + \operatorname{div}_x \left( \mathcal{L}f(t, x, \xi) \int_0^1 w(\xi, d\zeta) \int_{\mathbb{R}^d} K(x-y) f(t, y, \zeta) dy \right) = \nu \Delta_x \mathcal{L}f(t, x, \xi), \\ \mathcal{L}f(0, \cdot, \cdot) = f^0. \end{array} \right. \qquad (88) $$

Hence, the fixed points of the operator $\mathcal{L}$ amount to weak solutions of (35). The rest of the proof will focus on proving that the operator $\mathcal{L}: E \to E$ is well defined, and we have that $\mathcal{L}(E_R) \subset E_R$ and it is contractive in the $L^1$ norm for appropriate $R > 0$.

◇ Step 1: Well-posedness of (88).

We note that (88) is a linear transport equation in $\mathcal{L}f$ parametrized by $\xi$ and whose velocity field $\mathcal{V}_f$ is given in Definition 3.9. Notice that integrating by parts we have

$$ \nabla_x \mathcal{V}_f(t, x, \xi) = \int_0^1 w(\xi, d\zeta) \int_{\mathbb{R}^d} K(x-y) \otimes \nabla_y f(t, y, \zeta) dy, $$

which is bounded uniformly in all variables by the regularity of $f \in E$. Therefore, the velocity field verifies $\mathcal{V}_f \in L^\infty([0, t_*] \times [0, 1], W^{1,\infty}(\mathbb{R}^d, \mathbb{R}^d))$. For $\nu = 0$ and for a.e. value $\xi \in [0, 1]$ we can associate a unique and well-defined flow solving the characteristic system

$$ \left\{ \begin{array}{l} \displaystyle \frac{dX_f}{dt}(t, s, x; \xi) = \mathcal{V}_f(t, X_f(t, s, x; \xi); \xi), \\ X_f(s, s, x; \xi) = x, \end{array} \right. \qquad (89) $$

in the sense of Caratheodory. Hence, the classical results ensure that the unique solution of (88) can be obtained for example by the method of characteristics, namely, $\mathcal{L}f(t, \cdot, \xi) = X_f(t, 0, \cdot; \xi) \# f^0(\cdot, \xi)$ for $t \in [0, t_*]$ and a.e. $\xi \in [0, 1]$. In other words, we have that

$$ \mathcal{L}f(t, x, \xi) = f_0(X_f(0, t, x; \xi)) J_f(0, t, x; \xi), \qquad (90) $$

where $J_f(t, s, x; \xi) := \det(\nabla_x X_f(t, s, x; \xi))$ denotes the Jacobian associated to the flow.

For $\nu > 0$, we may instead use the heat kernel $G_\nu(t, x)$ and we can again classically obtain a solution through the mild formulation

$$ \mathcal{L}f(t, x, \xi) = G_\nu(t, \cdot) \star_x f^0(x, \xi) \\ + \int_0^t \int_{\mathbb{R}^d} G_\nu(t-s, x-y) \operatorname{div}_y \left( \mathcal{L}f(s, y, \xi) \int_0^1 w(\xi, d\zeta) \int_{\mathbb{R}^d} K(y-z) f(s, z, \zeta) dz \right) ds dy. \qquad (91) $$

◇ Step 2: A priori estimates of (88).

We first recall classical estimates for advection and advection-diffusion equations. Consider any $v(t, x) \in L^\infty([0, t^*], W^{1,\infty}(\mathbb{R}^d))$, any weak solution $u$ in $L^1_{\text{loc}}$ to

$$ \left\{ \begin{array}{l} \partial_t u + \operatorname{div}_x(u v) = \nu \Delta_x u + R(t, x), \\ u(0, x) = u^0(x). \end{array} \right. \qquad (92) $$
Then if $R \in L_{t,x}^1$, we have that $u \in L_t^\infty L_x^1$ with

$$ \|u(t, \cdot)\|_{L_x^1} \le \|u^0\|_{L_x^1} + \int_0^t \|R(s, \cdot)\|_{L_x^1} ds. \quad (93) $$

If $R \in L_t^1 L_x^\infty$, then $u \in L_{t,x}^\infty$ with

$$ \begin{aligned} \|u(t, \cdot)\|_{L_x^\infty} \le & \|u^0\|_{L_x^\infty} \exp \left( t \| \operatorname{div}_x v \|_{L_{t,x}^\infty} \right) \\ & + \int_0^t \|R(s, \cdot)\|_{L_x^\infty} \exp \left( (t-s) \| \operatorname{div}_x v \|_{L_{t,x}^\infty} \right) ds. \end{aligned} \quad (94) $$

If additional derivatives are available on $u^0$, $v$ and $R$. Sobolev bounds are also propagated. Provided that $v$ is $C^1$, $u$ is automatically a classical solution to (92) so that

$$ \partial_t \nabla_x u + \operatorname{div}(v \nabla_x u) + \operatorname{div}(\nabla_x v u) = \nu \Delta_x \nabla_x u + \nabla_x R. $$

In particular, we may apply estimates similar to the previous ones as $\nabla_x u$ solves (92) with $\tilde{R} = \nabla_x R - \operatorname{div}(\nabla_x v u)$. We consequently find that, if $u^0 \in W_x^{1,1}$ and $R = 0$ for simplicity, then

$$ \begin{aligned} \|\tilde{R}(t, \cdot)\|_{L_x^1} &\le \|\operatorname{div} v(t, \cdot)\|_{W_x^{1,\infty}} \|u(t, \cdot)\|_{L_x^1} + \|v(t, \cdot)\|_{W_x^{1,\infty}} \|\nabla_x u(t, \cdot)\|_{L_x^1} \\ &\le \|\operatorname{div} v(t, \cdot)\|_{W_x^{1,\infty}} \|u^0\|_{L_x^1} + \|v(t, \cdot)\|_{W_x^{1,\infty}} \|\nabla_x u(t, \cdot)\|_{L_x^1}. \end{aligned} $$

Hence we obtain by Gronwall's lemma that

$$ \begin{aligned} \|\nabla_x u(t, \cdot)\|_{L_x^1} \le & \|\nabla_x u^0\|_{L_x^1} \exp \left( t \| \operatorname{div}_x v \|_{L_{t,x}^\infty} \right) \\ & + \|u^0\|_{L^1} \int_0^t \| \operatorname{div} v(s, \cdot) \|_{W_x^{1,\infty}} \exp \left( (t-s) \| v \|_{L_{t,x}^\infty W_x^{1,\infty}} \right) ds. \end{aligned} \quad (95) $$

Similarly if $u^0 \in W_x^{1,\infty}$ and $R = 0$, then

$$ \begin{aligned} \|\tilde{R}(t, \cdot)\|_{L_x^\infty} &\le \|\operatorname{div} v(t, \cdot)\|_{W_x^{1,\infty}} \|u(t, \cdot)\|_{L_x^\infty} + \|v(t, \cdot)\|_{W_x^{1,\infty}} \|\nabla_x u(t, \cdot)\|_{L_x^\infty} \\ &\le \|\operatorname{div} v(t, \cdot)\|_{W_x^{1,\infty}} \|u^0\|_{L_x^\infty} \exp \left( t \| \operatorname{div}_x v \|_{L_{t,x}^\infty} \right) + \|v(t, \cdot)\|_{W_x^{1,\infty}} \|\nabla_x u(t, \cdot)\|_{L_x^\infty}, \end{aligned} $$

and one has that, still by Gronwall's lemma,

$$ \begin{aligned} \|\nabla_x u(t, \cdot)\|_{L_x^\infty} \le & \|\nabla_x u^0\|_{L_x^\infty} \exp \left( t \| \operatorname{div}_x v \|_{L_{t,x}^\infty} \right) \\ & + \|u^0\|_{L^\infty} \int_0^t \| \operatorname{div} v(s, \cdot) \|_{W_x^{1,\infty}} \exp \left( (t-s) \left( \|v\|_{L_{t,x}^\infty W_x^{1,\infty}} + \|\operatorname{div}_x v\|_{L_{t,x}^\infty} \right) \right) ds. \end{aligned} \quad (96) $$

We may now propagate estimates to show that the solution $\mathcal{L}f$ lies in $E$, and indeed $\mathcal{L}(E_R) \subset E_R$ for an appropriate choice of $R > 0$. We note that Equation (88) is immediately under the form (92) where $\xi$ is only a parameter and

$$ v_{\xi}(t, x) = V_f(t, x, \xi) = \int_{0}^{1} w(\xi, d\zeta) \int_{\mathbb{R}^d} K(x-y) f(t, y, \zeta) dy. $$

From here, since $f^0 \in L_\xi^\infty L_x^1$ we obtain that

$$ \begin{aligned} \|\mathcal{L}f(t, \cdot, \xi)\|_{L_x^1} &= \|f^0(\cdot, \xi)\|_{L_x^1}, \\ \|\mathcal{L}f(t, \cdot, \xi)\|_{L_x^\infty} &\le \|f^0(\cdot, \xi)\|_{L^\infty} \exp \left( t_* \| \operatorname{div}_x V_f \|_{L_{t,x}^\infty} \right), \end{aligned} \quad (97) $$

for $t \in [0, t_*]$ and a.e. $\xi \in [0, 1]$.

From the definition of $V_f$, we also directly obtain that

$$ \begin{aligned} \| \operatorname{div}_x V_f \|_{L_{t,x,\xi}^\infty} &\le \| K \|_{L^\infty} \| w \|_{L_\xi^\infty} \mathcal{M}_\zeta \cap L_\zeta^\infty \mathcal{M}_\xi \| \nabla_x f \|_{L_{t,\xi}^\infty} L_x^1 \\ \| V_f \|_{L_{t,\xi}^\infty} \| W_x^{1,\infty} \| &\le \| K \|_{L^\infty} \| w \|_{L_\xi^\infty} \mathcal{M}_\zeta \cap L_\zeta^\infty \mathcal{M}_\xi \| \nabla_x f \|_{L_{t,\xi}^\infty} L_x^1 \\ \| \operatorname{div} V_f \|_{L_{t,\xi}^\infty} \| W_x^{1,\infty} \| &\le \| \operatorname{div} K \|_{L^1} \| w \|_{L_\xi^\infty} \mathcal{M}_\zeta \cap L_\zeta^\infty \mathcal{M}_\xi \| \nabla_x f \|_{L_{t,\xi}^\infty} L_x^\infty. \end{aligned} \quad (98) $$

Putting (97), and (98) together into (95) and (96) implies that

$$ \begin{aligned} \|\mathcal{L}f\|_{L_{t,\xi}^\infty W_x^{1,1} \cap W_x^{1,\infty}} &\le C_1 \|f^0\|_{L_\xi^\infty W_x^{1,1} \cap W^{1,\infty}} \\ &\quad \times \exp \left( C_2 t_* (\|K\|_{L^1} + \|\operatorname{div} K\|_{L^1}) \|w\|_{L_\xi^\infty} \mathcal{M}_\zeta \cap L_\zeta^\infty \mathcal{M}_\xi \|f\|_{L_{t,\xi}^\infty} W_x^{1,1} \cap W_x^{1,\infty} \right), \end{aligned} \quad (99) $$

for some universal $C_1, C_2 \ge 1$. In particular, the above estimate show that $\mathcal{L}f \in E$ for every $f \in E$.
◇ Step 3: Invariance of $E_R$ and contractivity of $\mathcal{L}$.

Set now any $R > C_1 \|f^0\|_{L^\infty_W^{1,1}W_x^{1,\infty}}$ and note that the above estimates also show that $\mathcal{L}(E_R) \subset E_R$ if the maximal time $t_*$ is taken sufficiently small, namely,

$$0 < t_* \le t_1(R) := \frac{1}{C_2(\|K\|_{L^\infty} + \|\operatorname{div} K\|_{L^1})\|w\|_{L^\infty_W \mathcal{M}_\zeta \cap L^\infty_W \mathcal{M}_\xi} R} \log \left( \frac{R}{C_1 \|f^0\|_{L^\infty_W^{1,1} \cap W_x^{1,\infty}}} \right).$$

Our next goal is to show that $\mathcal{L}$ is contractive under the $L^1$ norm. To do so, let us consider $f, g \in E_R$ and note that $f - g$ follows the following linear transport equation with a source term

$$\left\{ \begin{array}{l} \partial_t(\mathcal{L}f - \mathcal{L}g) + \operatorname{div}_x(\mathcal{V}_g(\mathcal{L}f - \mathcal{L}g)) + \operatorname{div}_x((\mathcal{V}_f - \mathcal{V}_g)\mathcal{L}f) = \nu\Delta_x(\mathcal{L}f - \mathcal{L}g), \\ (\mathcal{L}f - \mathcal{L}g)(0, \cdot, \cdot) = 0. \end{array} \right.$$

Taking $L^1$ norms with respect to $x$ and $\xi$ yields

$$\| (\mathcal{L}f - \mathcal{L}g)(t, \cdot, \cdot) \|_{L^1_{t,x,\xi}} \le \| \operatorname{div}_x((\mathcal{V}_f - \mathcal{V}_g)\mathcal{L}f) \|_{L^1_{t,x,\xi}} \\ \le \| \operatorname{div}_x(\mathcal{V}_f - \mathcal{V}_g) \|_{L^\infty_{t,\xi} L^1_x} \| \mathcal{L}f \|_{L^\infty_{t,x,\xi}} + \| \mathcal{V}_f - \mathcal{V}_g \|_{L^\infty_{t,\xi} L^1_x} \| \nabla_x \mathcal{L}f \|_{L^\infty_{t,x,\xi}}, \quad (100)$$

for $t \in [0, t_*]$. By the estimate (34)$_1$ in Lemma 3.8 we have

$$\begin{aligned} \|\mathcal{V}_f - \mathcal{V}_g\|_{L^1_{t,x,\xi}} &\le \|K\|_{L^1} \|w\|_{L^\infty_W \mathcal{M}_\zeta \cap L^\infty_W \mathcal{M}_\xi} \|f - g\|_{L^1_{t,x,\xi}}, \\ \|\operatorname{div}_x(\mathcal{V}_f - \mathcal{V}_g)\|_{L^1_{t,x,\xi}} &\le \|\operatorname{div} K\|_{L^1} \|w\|_{L^\infty_W \mathcal{M}_\zeta \cap L^\infty_W \mathcal{M}_\xi} \|f - g\|_{L^1_{t,x,\xi}}. \end{aligned} \quad (101)$$

Then, putting (100) and (101) together, and using that $\mathcal{L}f \in E_R$ by the above step, we have that

$$\|\mathcal{L}f - \mathcal{L}g\|_{L^1_{t,x,\xi}} \le R t_* (\|K\|_{L^1} + \|\operatorname{div} K\|_{L^1}) \|w\|_{L^\infty_W \mathcal{M}_\zeta \cap L^\infty_W \mathcal{M}_\xi} \|f - g\|_{L^1_{t,x,\xi}}. \quad (102)$$

Take $t_*$ sufficiently small so that $t_* \le t_1(R)$ as above and, in addition

$$0 < t_* < t_2(R) := \frac{1}{R (\|K\|_{L^1} + \|\operatorname{div} K\|_{L^1}) \|w\|_{L^\infty_W \mathcal{M}_\zeta \cap L^\infty_W \mathcal{M}_\xi}}.$$

Hence $\mathcal{L}$ is contractive on $E_R$ under the $L^1$ norm by the stability estimate (102). By the Banach fixed point theorem there must exist a (unique) fixed point $f \in E_R$ of $\mathcal{L}$, leading to a weak solution of (35) in $[0, t_*]$. Note that $t_*$ must be taken small enough so that $0 < t_* < \min\{t_1(R), t_2(R)\}$. However, restarting the argument from $t_*$ allows extending this weak solution to any finite time interval as usual, since the above a priori estimate (99) above does not blow up in time. □

We finish this technical appendix with the proof of Lemma 3.12.

**Proof of Lemma 3.12.** As usual, one uses an approximation by convolution. Because $w$ is defined on $[0, 1]^2$, we first extend $w$ as a periodic function on with period 1 in both variables. Similarly we naturally consider any function on $[0, 1]$ as periodic, allowing to naturally use convolutions. Then, choose any convolution kernel $L_n \ge 0$ with $L_n \in C_c^\infty(\mathbb{R})$ for any fixed $n$ and define $w_n = L_n \otimes L_n \star_{\xi, \zeta} w$. Since $w \in \mathcal{M}_{\xi, \zeta}$, we immediately note that $w_n \in L_{\xi, \zeta}^\infty$.

Next, for any $f \in L_\xi^1 C_\zeta$, we trivially have that $\|L_n \otimes L_n \star f\|_{L_\xi^1 L_\zeta^\infty} \le \|f\|_{L_\xi^1 L_\zeta^\infty}$. By duality, we deduce that $w_n$ is uniformly bounded in $L_\xi^\infty \mathcal{M}_\zeta \cap L_\zeta^\infty \mathcal{M}_\xi$ and, hence, in $L_\xi^\infty L_\zeta^1 \cap L_\zeta^\infty L_\xi^1$, since $w_n \in L_{\xi, \zeta}^\infty$.

Consider now any $\phi_n$ converging weakly to $\phi$ in $H^1([0, 1])$, we have that

$$\int_0^1 \phi_n(\zeta) L_n \star_{\zeta} w(\xi, \zeta) d\zeta = \int_0^1 \tilde{L}_n \star_{\zeta} \phi_n(\zeta) w(\xi, \zeta) d\zeta,$$

with $\tilde{L}_n(\xi) = L_n(-\xi)$. By Lemma 3.7, for any $\psi \in L^\infty([0, 1])$, we deduce that

$$\int_0^1 \psi(\xi) \left( \int_0^1 \phi_n(\zeta) L_n \star_{\zeta} w(\xi, \zeta) d\zeta - \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right) d\xi \le \|\psi\|_{L^\infty} \|w\|_{L^\infty_W \mathcal{M}_\zeta \cap L^\infty_W \mathcal{M}_\xi} \|\phi - \tilde{L}_n \star_{\zeta} \phi_n\|_{L^1}.$$

Since $H^1$ is compactly embedded in $L^1$, this shows that

$$\left\| \int_0^1 \phi_n(\zeta) L_n \star_{\zeta} w(\xi, \zeta) d\zeta - \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right\|_{L^1_{\xi}} \to 0, \quad \text{as } n \to \infty.$$
Still by Lemma 3.7 since $H^1 \subset L^\infty$, $\int_0^1 \phi(\zeta) w(\xi, d\zeta)$ is a fixed function in $L^\infty$ and consequently its convolution by $L_n$ converges in $L^1$ (and in all $L^p, p < \infty$). Therefore

$$ \left\| \int_0^1 \phi_n(\zeta) w_n(\xi, \zeta) d\zeta - \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right\|_{L_\xi^1} \\ \le \left\| L_n \star_\xi \left( \int_0^1 \phi_n(\zeta) L_n \star_\zeta w(\xi, \zeta) d\zeta - \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right) \right\|_{L_\xi^1} \\ + \left\| L_n \star_\xi \int_0^1 \phi_n(\zeta) w(\xi, \zeta) d\zeta - \int_0^1 \phi(\zeta) w(\xi, d\zeta) \right\|_{L_\xi^1} \to 0, $$

proving the convergence of $w_n$ in $L_\xi^1 H_\zeta^{-1}$.

<figure><img src="image_5.png" /></figure>

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(Pierre-Emmanuel Jabin)

DEPT. OF MATHEMATICS, HUCK INSTITUTES, AND EXCELLENCE RESEARCH UNIT “MODELING NATURE”, PENNSYLVANIA STATE UNIVERSITY, UNIVERSITY PARK, PA 16803, USA

Email address: pejabin@psu.edu

(David Poyato)

INSTITUT CAMILLE JORDAN (ICJ), UMR 5208 CNRS & UNIVERSITÉ CLAUDE BERNARD LYON 1, 69100 VILLEURBANNE, FRANCE, AND RESEARCH UNIT “MODELING NATURE” (MNAT), UNIVERSIDAD DE GRANADA, GRANADA, 18071, SPAIN

Email address: poyato@math.univ-lyon1.fr davidpoyato@ugr.es

(Juan Soler)

DEPARTAMENTO DE MATEMÁTICA APLICADA AND RESEARCH UNIT “MODELING NATURE” (MNAT), FACULTAD DE CIENCIAS, UNIVERSIDAD DE GRANADA, 18071 GRANADA, SPAIN

Email address: jsoler@ugr.es