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Let (Σ, g) be a compact Riemann surface with smooth boundary ∂E, ∆g be the Laplace-Beltrami operator, and h be a positive smooth function. Using a min-max scheme introduced by Djadli and Malchiodi (2008) and Djadli (2008), we prove that if Σ is non-contractible, then for any ρ Σ (8kπ, 8(k +1)π) with k Σ ℕ*, the mean field equation { Δ g u = ρ h e u ∫ Σ h e u d v g in Σ , u = 0 on ∂ Σ has a solution. This generalizes earlier existence results of Ding et al. (Ann Inst H Poincaré Anal Non Linéaire, 1999) and Chen and Lin (2003) in the Euclidean domain. Also we consider the corresponding Neumann boundary value problem. If h is a positive smooth function, then for any ρ ∈ (4kπ, 4(k + 1)π) with k ∈ ℕ*, the mean field equation { Δ g u = ρ ( h e u ∫ Σ h e u d v g − 1 | Σ | ) in Σ , ∂ u / ∂ v = 0 on ∂ Σ has a solution, where v denotes the unit normal outward vector on ∂Σ. Note that in this case we do not require the surface to be non-contractible. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum σ(L) of the complex Hill operator with Darboux-Treibich-Verdier potential L =d²/dx² - 6P(x + z0; σ) - 2P(x +1/2 + z0; σ) in L²(R,C), where P(z; σ) is the Weierstraß elliptic function with periods 1 and σ, and z0 㼰 C is chosen such that L has no singularities on R. We give a complete picture of the deformation of the spectrum with σ = 1/2 + ib as b > 0 varies. A new idea of the proof is to apply the result of the mean field equation and its connection with this operator. [ABSTRACT FROM AUTHOR]

In this paper we study the blow-up analysis for some mean field equation on point vortices which is derived by C. Neri under a stochastic assumption. In particular, we derive some estimate which is the asymptotic behavior of blow-up solutions near the blow-up points. To obtain this result we shall employ the new scaling argument for blow-up solutions of Neri's mean field equation. Moreover, we also study the power type elliptic equation in higher dimension case related to mean field equation and derive such estimate. [ABSTRACT FROM AUTHOR]

We prove the existence of a family of blow-up solutions of a mean field equation on the sphere. The solutions blow up at four points where the minimum value of a potential energy function (involving the Green's function) is attained. The four blow-up points form a regular tetrahedron. Moreover, the solutions we build have a group of symmetry Td which is isomorphic to the symmetric group S4. Other families of solutions can be similarly constructed with blow-up points at the vertices of equilateral triangles on a great circle or other inscribed platonic solids (cubes, octahedrons, icosahedrons and dodecahedrons). All of these solutions have the symmetries of the corresponding configuration, while they are non-axially symmetric. [ABSTRACT FROM AUTHOR]

We prove the existence of a family of blow-up solutions of a mean field equation on the sphere. The solutions blow up at four points where the minimum value of a potential energy function (involving the Green's function) is attained. The four blow-up points form a regular tetrahedron. Moreover, the solutions we build have a group of symmetry Td which is isomorphic to the symmetric group S4. Other families of solutions can be similarly constructed with blow-up points at the vertices of equilateral triangles on a great circle or other inscribed platonic solids (cubes, octahedrons, icosahedrons and dodecahedrons). All of these solutions have the symmetries of the corresponding configuration, while they are non-axially symmetric. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let (Σ , g) be a closed Riemannian surface, G = { σ 1 , ... , σ N } be an isometric group acting on it. Denote a positive integer ℓ = inf x ∈ Σ I (x) , where I (x) is the number of all distinct points of the set { σ 1 (x) , ... , σ N (x) }. A sufficient condition for existence of solutions to the mean field equation Δ g u = 8 π ℓ h e u ∫ Σ h e u d v g − 1 Vol g (Σ) is given. This recovers results of Ding–Jost–Li–Wang, Asian J. Math. (1997) 230–248 when ℓ = 1 or equivalently G = { Id } , where Id is the identity map. [ABSTRACT FROM AUTHOR]

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on an arbitrary flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus. [ABSTRACT FROM AUTHOR]

The seminal work [7] by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in [30] found that the "bubbling implies mass concentration" phenomena might not hold if there is a collapse of singularities. Furthermore, a sharp estimate [23] for the bubbling solutions has been obtained. In this paper, we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs. The main difficulty comes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearized operator. It is well-known that the kernel space is three dimensional. So the main technical ingredient of the proof is to show that the limit after re-scaling is orthogonal to the kernel space. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The authors deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that they allow heavily mean field dependent dynamics. This in particular leads to a system of PDE’s with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, they introduce a structural assumptions that cover many cases in stochastic differential games with mean field dependent dynamics for which they are able to establish the existence of a weak solution. In addition, the authors present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis. [ABSTRACT FROM AUTHOR]

We consider a multi-group microscopic model for pedestrian flow describing the behaviour of large groups. It is based on an interacting particle system coupled to an eikonal equation. Hydrodynamic multi-group models are derived from the underlying particle system as well as scalar multi-group models. The eikonal equation is used to compute optimal paths for the pedestrians. Particle methods are used to solve the equations on all levels of the hierarchy. Numerical test cases are investigated and the models and, in particular, the resulting evacuation times are compared for a wide range of different parameters. [ABSTRACT FROM AUTHOR]

We prove the existence of stationary turbulent flows with arbitrary positive vortex circulation on non-simply connected domains. Our construction yields solutions for all real values of the inverse temperature with the exception of a quantized set, for which blow-up phenomena may occur. Our results complete the analysis initiated in Ricciardi and Zecca (2016). [ABSTRACT FROM AUTHOR]

In this article, we consider the evolution model $$ \partial_t{u} -\nabla\cdot(u\nabla Pu)=0,\quad Pu=(-\Delta)^{-s}u, \quad 0< s\leq 1,\; x\in\mathbb{R}^d,\; t>0. $$ We show that when $s\in[1/2,1)$, $\alpha>d+1$, $d\geq 2$, the equation has a unique local in time solution for any initial data in $B^\alpha_{1,\infty}$. Moreover, in the critical case $s=1$, the solution exists in $B^\alpha_{p,\infty}$, $2\leq p\leq\infty$, $\alpha> d/p$, $d\geq3$.

A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with variable intensities. We start by performing a blow-up analysis in order to derive some information on the local blow-up masses. As a consequence we get a compactness property in a supercritical range. We next introduce a variational argument based on improved Moser–Trudinger inequalities which yields existence of solutions for any choice of the underlying surface. [ABSTRACT FROM AUTHOR]

We study the mean field equation derived by Neri in the context of the statistical mechanics description of 2D-turbulence, under a “stochastic” assumption on the vortex circulations. The corresponding mathematical problem is a nonlocal semilinear elliptic equation with exponential type nonlinearity, containing a probability measure P ∈ M ( [ − 1 , 1 ] ) which describes the distribution of the vortex circulations. Unlike the more investigated “deterministic” version, we prove that Neri's equation may be viewed as a perturbation of the widely analyzed standard mean field equation, obtained by taking P = δ 1 . In particular, in the physically relevant case where P is non-negatively supported and P ( { 1 } ) > 0 , we prove the mass quantization for blow-up sequences. We apply this result to construct minimax type solutions on bounded domains in R 2 and on compact 2-manifolds without boundary. [ABSTRACT FROM AUTHOR]

We consider the problem where Ω is a smooth bounded open domain in ℝ² which contains the point ξ. We prove that if λ > 8π, problem (P) has a solutions u_ such that u∈(x) 8π + λ/2 G(x, ξ) uniformly on compact sets of Ω \ {ξ} as∈ goes to zero. Here G denotes Green's function of Dirichlet Laplacian in Ω. If λ ... 8πℕ we will not make any symmetry assumptions on Ω, while if λ ∈ 8πℕ we will assume that Ω is invariant under a rotation through an angle 8π2 λ around the point ξ. [ABSTRACT FROM AUTHOR]

For a general class of elliptic PDE’s in mean field form on compact Riemann surfaces with exponential nonlinearity, we address the question of the existence of solutions with concentrated nonlinear term, which, in view of the applications, are physically of definite interest. In the model, we also include the possible presence of singular sources in the form of Dirac masses, which makes the problem more difficult to attack. [ABSTRACT FROM AUTHOR]

We outline some results obtained in recent years concerning a class of semilinear elliptic equations of the 'mean field' type containing a probability measure, motivated by the statistical mechanics description of 2D turbulence. In particular, we exhibit the optimal Moser-Trudinger constants, we study the blow-up of solution sequences, we prove their mass quantization under suitable conditions and we consequently derive the existence of solutions by variational methods. [ABSTRACT FROM AUTHOR]

Let (Σ , g) be a closed Riemann surface, and G = { σ 1 , ⋯ , σ N } be a finite isometric group acting on it. Denote a positive integer ℓ = min x ∈ Σ ⁡ I (x) , where I (x) is the number of all distinct points of the set { σ 1 (x) , ⋯ , σ N (x) }. In this paper, we consider the following G -invariant mean field type flow { ∂ ∂ t e u = Δ g u + 8 π ℓ (f e u ∫ Σ f e u d v g − 1 | Σ |) u (⋅ , 0) = u 0 , where u 0 belongs to C 2 + α (Σ) for some α ∈ (0 , 1) , f is a sign-changing smooth function such that ∫ Σ f e u 0 d v g ≠ 0 , both u 0 and f are G -invariant, and | Σ | denotes the area of (Σ , g). Such kind of flow was originally proposed by Castéras [6]. Through a priori estimates, we prove that the flow u (x , t) exists for all time t ∈ [ 0 , ∞). Moreover, by employing blow-up procedure, we obtain that under certain geometric conditions, u (x , t) converges to u (x) in H 2 (Σ) as t → ∞ , where u (x) is a solution of the mean field equation − Δ g u = 8 π ℓ (f e u ∫ Σ f e u d v g − 1 | Σ |). This generalizes recent results of Li-Zhu [27] and Sun-Zhu [37]. [ABSTRACT FROM AUTHOR]

We consider the Trudinger-Moser type functional... where Ω is a two-dimensional Riemannian surface without boundary, v 2 H1.Ω/,RΩ v D 0, I = [-1, 1], P is a Borel probability measure on I and λ > 0. The functional Jλ arises in the statistical mechanics description of equilibrium turbulence, under the assumption that the intensity and the orientation of the vortices are determined by P. We formulate a Toland nonconvex duality principle for Jλ and we compute the optimal value of λ for which Jλ is bounded from below. [ABSTRACT FROM AUTHOR]