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We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^\alpha$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),\alpha_t)dt+\sigma(t,X_t,\mathcal L(X_t),\alpha_t)dW_t,$ $X_0\sim \mu$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $\alpha_t$ is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipshitz condition, and that the dynamics is subject to a (convex) level set constraint of the form $\pi(X_t)\leq0$. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipshitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle and then numerically investigate a gradient algorithm for the approximation of the optimal control., Comment: 32 pages; 11 figures

We study the mean field equation on the flat torus T σ : = C / ( Z + Z σ ) Δ u + ρ ( e u ∫ T σ e u − 1 | T σ | ) = 0 , where ρ is a real parameter. For a general flat torus, we obtain the existence of two-dimensional solutions bifurcating from the trivial solution at each eigenvalue (up to a multiplicative constant | T σ | ) of Laplace operator on the torus in the space of even symmetric functions. We further characterize the subset of all eigenvalues through which only one bifurcating curve passes. Finally local convexity near bifurcating points of the solution curves are obtained.

We investigate the dynamics of a limit of interacting FitzHugh--Nagumo neurons in the regime of large interaction coefficients. We consider the dynamics described by a mean-field model given by a n...

We consider the extinction regime in the spatial stochastic logistic model in R d (a.k.a. Bolker–Pacala–Dieckmann–Law model of spatial populations) using the first-order perturbation beyond the mea...

In this paper, the second in a series, we continue to study the generalized Lame equation with the Treibich-Verdier potential y ″ ( z ) = [ ∑ k = 0 3 n k ( n k + 1 ) ℘ ( z + ω k 2 | τ ) + B ] y ( z ) , n k ∈ Z ≥ 0 from the monodromy aspect. We prove the existence of a pre-modular form Z r , s n ( τ ) of weight 1 2 ∑ n k ( n k + 1 ) such that the monodromy data ( r , s ) is characterized by Z r , s n ( τ ) = 0 . This generalizes the result in [17] , where the Lame case (i.e. n 1 = n 2 = n 3 = 0 ) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δ u + e u = 16 π δ 0 and Δ u + e u = 8 π ∑ k = 1 3 δ ω k 2 on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

This paper deals with radial and non-radial solutions for local and nonlocal Liouville type equations. At first non-degenerate and degenerate mean field equations are studied and radially symmetric solutions to the Dirichlet problem for them are written into explicit form. Non-radial solution is constructed in the case of Blaschke type nonlinearity. The Cauchy boundary value problem for nonlinear Laplace equation with several exponential nonlinearities is considered and C^2 smooth monotonically decreasing radial solution u ( r ) is found. Moreover, u ( r ) has logarithmic growth at ∞. Our results are applied to the differential geometry, more precisely, minimal non-superconformal degenerate two dimensional surfaces are constructed in R^4 and their Gaussian, respectively normal curvatures are written into explicit form. At the end of the paper several examples of local Liouville type PDE with radial coefficients which do not have radial solutions are given.

Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schr\"odinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global., Comment: 19 pages

This work deals with the aggregation diffusion equation \[\partial_t \rho = \Delta_p\rho + \lambda div((K_a*\rho)\rho),\] where $K_a(x)=\frac{x}{|x|^a}$ is an attraction kernel and $\Delta_p$ is the so called $p$-Laplacian. We show that the domain $a < p(d+1)-2d$ is subcritical with respect to the competition between the aggregation and diffusion by proving that there is existence unconditionally with respect to the mass. In the critical case we show existence of solution in a small mass regime for an $L\ln L$ initial condition., Comment: 7 pages, 1 figure

This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equationΔu−βu+λeu∫Ωeu=0 in Ω, (*)where Ω is a bounded domain in R2 and β,λ are positive param...

We study the robustness in the nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck (KS-FP for short) equation in the presence of frustrations. For this, we construct a new unstable mode for the corresponding linear part of the perturbation around the incoherent state, and we show that the nonlinear perturbation stays close to the unstable mode in some small time interval which depends on the initial size of the perturbations. Our instability results improve the previous results on the KS-FP with zero frustration [J. Stat. Phys. 160 (2015), pp. 477–496] by providing a new linear unstable mode and detailed energy estimates.

A multiscale meshfree particle method for macroscopic mean field approximations of generalized interacting particle models is developed and investigated. The method is working in a uniform way for ...

We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

In this paper we study the following mean field type equation \begin{equation*} (MF) \qquad -\D_g u \, = \varrho ( \frac{K e^{u}}{\int_{\Sig} K e^{u} dV_g} \, - \, 1) \, \mbox{ in } \Sigma, \end{equation*} where $(\Sigma, g)$ is a closed oriented surface of unit volume $Vol_g(\Sigma)$ = 1, $K$ positive smooth function and $\varrho= 8 \pi m$, $ m \in \N$. Building on the critical points at infinity approach initiated in \cite{ABL17} we develop, under generic condition on the function $K$ and the metric $g$, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters $B_m(\Sigma)$.\\ We derive from these \emph{Morse inequalities at infinity} various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. $\varrho \in 8 \pi \N$., Comment: 31 pages. More details have been added

The critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non-Markovian biased random walks with reactions. From the Hamilton-Jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation. If the reaction rate is larger than the jump frequency, invasion always succeeds, even in the case of maximal bias. Numerical simulations support our analytical predictions.

For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [28] and/or with symmetric domains [23]. Toward our goal we parametrize the branch not by the $L^{\infty}(\Omega)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysis of mean field type equations., Comment: Intro has been expanded. References has been added. Minor expository improvements

Inspired by works of Cast\'eras (Pacific J. Math., 2015), Li-Zhu (Calc. Var., 2019) and Sun-Zhu (Calc. Var., 2020), we propose a heat flow for the mean field equation on a connected finite graph $G=(V,E)$. Namely $$ \left\{\begin{array}{lll} \partial_t\phi(u)=\Delta u-Q+\rho \frac{e^u}{\int_Ve^ud\mu}\\[1.5ex] u(\cdot,0)=u_0, \end{array}\right. $$ where $\Delta$ is the standard graph Laplacian, $\rho$ is a real number, $Q:V\rightarrow\mathbb{R}$ is a function satisfying $\int_VQd\mu=\rho$, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is one of certain smooth functions including $\phi(s)=e^s$. We prove that for any initial data $u_0$ and any $\rho\in\mathbb{R}$, there exists a unique solution $u:V\times[0,+\infty)\rightarrow\mathbb{R}$ of the above heat flow; moreover, $u(x,t)$ converges to some function $u_\infty:V\rightarrow\mathbb{R}$ uniformly in $x\in V$ as $t\rightarrow+\infty$, and $u_\infty$ is a solution of the mean field equation $$\Delta u_\infty-Q+\rho\frac{e^{u_\infty}}{\int_Ve^{u_\infty}d\mu}=0.$$ Though $G$ is a finite graph, this result is still unexpected, even in the special case $Q\equiv 0$. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz-Simon type inequality and use it to conclude the convergence of the heat flow., Comment: 15 pages

We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$\left\{ \begin{array}{ll} -\Delta u=\lambda_1\dfrac{V_1 e^{u}}{ \int_{\Omega_{\boldsymbol\epsilon}} V_1 e^{u} dx } - \lambda_2\tau \dfrac{ V_2 e^{-\tau u}}{ \int_{\Omega_{\boldsymbol\epsilon}}V_2 e^{ - \tau u} dx}&\text{in $\Omega_{\boldsymbol\epsilon}=\Omega\setminus \displaystyle \bigcup_{i=1}^m \overline{B(\xi_i,\epsilon_i)}$}\\ \ \ u=0 &\text{on $\partial \Omega_{\boldsymbol\epsilon}$}, \end{array} \right. $$ where $B(\xi_i,\epsilon_i)$ is a ball centered at $\xi_i\in\Omega$ with radius $\epsilon_i$, $\tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $\lambda_1>8\pi m_1$ and $\lambda_2 \tau^2>8\pi (m-m_1)$ with $m_1 \in \{0,1,\dots,m\}$, there exist radii $\epsilon_1,\dots,\epsilon_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $\xi_1,\dots,\xi_{m_1}$ and $\xi_{m_1+1},\dots,\xi_{m}$, respectively, as the radii approach zero., Comment: 23 pages

Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in Crisan & Xiong (2010) and Clark & Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive It\^{o} representations of their limiting forms as the approximation parameter $\delta \rightarrow 0$. All filters require the solution of a Poisson equation defined on $\mathbb{R}^{d}$, for which existence and uniqueness of solutions can be a non-trivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.

We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

In this paper we are interested in understanding the structure of domains of first and second kind, a concept motivated by problems in statistical mechanics. We prove some openness property for domains of first kind with respect to a suitable topology, as well as some sufficient condition for a simply connected domain to be of first kind in terms of the Fourier coefficients of the Riemann map. Finally, we show that the set of simply connected domains of first kind is contractible., Comment: 15 pages

We derive a singular version of the Sphere Covering Inequality which was recently introduced in Gui and Moradifam (Invent Math. https://doi.org/10.1007/s00222-018-0820-2 , 2018) suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in Luo and Tian (Proc Am Math Soc 116(4):1119–1129, 1992). Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.

In this paper, we give an algebraic construction of the solution to the following mean field equation $$ \Delta \psi+e^{\psi}=4\pi\sum_{i=1}^{2g+2}\delta_{P_{i}}, $$ on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$ Furthermore, we study the rescaled equation $$ \Delta \psi+\gamma e^{\psi}=4\pi\gamma \sum_{i=1}^{2g+2}\delta_{P_{i}} $$ and its adiabatic limit at $\gamma=0$., Comment: 14 pages

Let X be a complex hyperelliptic curve of genus two equipped with the canonical metric d s 2 . We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of ( X , d s 2 ) determines an explicit solution to a mean field equation.

The Kuramoto model (KM) of coupled phase oscillators on graphs provides the most influential framework for studying collective dynamics and synchronization. It exhibits a rich repertoire of dynamical regimes. Since the work of Strogatz and Mirollo, the mean field equation derived in the limit as the number of oscillators in the KM goes to infinity, has been the key to understanding a number of interesting effects, including the onset of synchronization and chimera states. In this work, we study the mathematical basis of the mean field equation as an approximation of the discrete KM. Specifically, we extend the Neunzert's method of rigorous justification of the mean field equation to cover interacting dynamical systems on graphs. We then apply it to the KM on convergent graph sequences with non-Lipschitz limit. This family of graphs includes many graphs that are of interest in applications, e.g., nearest-neighbor and small-world graphs.

In this paper, we prove uniqueness of solutions of mean field equations with general boundary conditions for the critical and subcritical total mass regime, extending the earlier results for null Dirichlet boundary condition. The proof is based on new Bol’s inequalities for weak radial solutions obtained from rearrangement of the solutions.

An exact low-dimensional system of mean-field equations for an infinite-size network of pulse coupled integrate-and-fire neurons with a bimodal distribution of an excitability parameter is derived. Bifurcation analysis of these equations shows a rich variety of dynamic modes that do not exist with a unimodal distribution of the excitability parameter. New modes include multistable equilibrium states with different levels of the spiking rate, collective oscillations and chaos. All oscillatory modes coexist with stable equilibrium states. The mean field equations are a good approximation to the solutions of a microscopic model consisting of several thousand neurons.

We study symmetry of solutions of the mean field equation \[ \Delta u +\rho(\frac{Ke^u}{\int_{T_\epsilon} Ke^u} -\frac{1}{|T_\epsilon|} )=0\] on the flat torus $T_\epsilon=[-\frac{1}{2\epsilon}, \frac{1}{2\epsilon}] \times [-\frac{1}{2}, \frac{1}{2}]$ with $0

We prove the existence or non-existence of solutions of the mean field equation (1.1) with n = 1 n=1 or n = 2 n=2 for a rectangular torus E E . It is interesting to see that the results depend on the location of the half period ω k 2 \frac {\omega _{k}}{2} .

We consider an interacting system of one-dimensional structures modelling fibers with fiber-fiber interaction in a fiber lay-down process. The resulting microscopic system is investigated by looking at different asymptotic limits of the corresponding stochastic model. Equations arising from mean-field and diffusion limits are considered. Furthermore, numerical methods for the stochastic system and its mean-field counterpart are discussed. A numerical comparison of solutions corresponding to the different scales (microscopic, mesoscopic and macroscopic) is included., 23 pages, 16 figures

This article is aimed at proposing a generalization of the Ross-Macdonald model for the transmission of Vector-borne diseases in which human-to-human contagions are also considered. We first present this generalized model by formulating a mean field theory, checking its validity by comparing to numerical simulations. To make the premises of our model more realistic, we adapt the mean field equations to the case in which human contacts are described by a complex network. In this case we are also able to derive an analytical expression for the epidemic threshold. In both the mean-field and network-based models, we estimate the value of the epidemic threshold which corresponds to the boundary between the disease-free and epidemic regimes. The expression of this threshold allows us to discuss the impact that human-to-human contagions have on the spread of vector-borne diseases., 7 pages, 5 figures

We study a double mean field-type PDE related to a prescribed curvature problem on compacts surfaces with boundary. We provide a general blow-up analysis, then a Moser-Trudinger inequality, which gives energy-minimizing solutions for some range of parameters. Finally, we provide existence of min-max solutions for a wider range of parameters, which is dense in the plane if $��$ is not simply connected., 29 pages, 4 figures

We are concerned with the mean field equation with singular data on bounded domains. By assuming a singular point to be a critical point of the 1-vortex Kirchhoff-Routh function, we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at a singular point. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and a suitably defined Pohozaev-type identity.

In this paper, we prove two existence results of solutions to mean field equations $$\begin{aligned} \Delta u+e^u=\rho \delta _0 \end{aligned}$$and $$\begin{aligned} \Delta u=\lambda e^u(e^u-1)+4 \pi \sum _{j=1}^{M}{\delta _{p_j}} \end{aligned}$$on an arbitrary connected finite graph, where $$\rho >0$$ and $$\lambda >0$$ are constants, M is a positive integer, and $$p_1,\ldots ,p_M$$ are arbitrarily chosen distinct vertices on the graph.

In this paper, we prove that the even solution of the mean field equation $\Delta u=\lambda(1-e^u) $ on $S^2$ must be axially symmetric when $4, Comment: 15 pages

We prove uniqueness of blow up solutions of the mean field equation as $\rho_n \rightarrow 8\pi m$, $m\in\mathbb{N}$. If $u_{n,1}$ and $u_{n,2}$ are two sequences of bubbling solutions with the same $\rho_n$ and the same (non degenerate) blow up set, then $u_{n,1}=u_{n,2}$ for sufficiently large $n$. The proof of the uniqueness requires a careful use of some sharp estimates for bubbling solutions of mean field equations [24] and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in [51] in the context of the Chern-Simons-Higgs equations. Moreover, motivated by the Onsager statistical description of two dimensional turbulence, we are bound to obtain a refined version of an estimate about $\rho_n-8\pi m$ in case the first order evaluated in [24] vanishes., Comment: Accepted for JMPA