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This paper reinterprets the Freidlin--Wentzell variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. Through a one-dimensional irreversible diffusion process on a torus, we explicitly characterize essential concepts in the weak KAM theory, such as the Peierls barrier and the pro jected Mather/Aubry/ Man\~\'e sets. The weak KAM representation of the Freidlin--Wentzell variational construction of the rate function is discussed based on the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. This rate function gives the maximal Lipschitz continuous viscosity solution to the corresponding stationary Hamilton--Jacobi equation (HJE), satisfying the Freidlin--Wentzell variational formula for the boundary data on the Aubry set. Choosing meaningful self-consistent boundary data at each local attractor is essential to select a unique weak KAM solution to stationary HJE. This selected viscosity solution also serves as the global energy landscape of the original stochastic process. This selection for stationary HJEs can be described by first taking the long time limit and then taking the zero noise limit, which also provides a special construction of vanishing viscosity approximation. [ABSTRACT FROM AUTHOR]

We present a data-driven point of view for rare events, which represent conformational transitions in biochemical reactions modeled by overdamped Langevin dynamics on manifolds in high dimensions. We first reinterpret the transition state theory and the transition path theory from the optimal control viewpoint. Given a point cloud probing the manifold, we construct a discrete Markov chain with a Q-matrix computed from an approximated Voronoi tesselation via the point cloud. We use this Q-matrix to compute a discrete committor function whose level set automatically orders the point cloud. Then based on the committor function, an optimally controlled random walk on point clouds is constructed and utilized to efficiently sample transition paths, which become an almost sure event in O(1) time instead of a rare event in the original reaction dynamics. To compute the mean transition path efficiently, a local averaging algorithm based on the optimally controlled random walk is developed, which adapts the finite temperature string method to the controlled Monte Carlo samples. Numerical examples on sphere/torus including a conformational transition for the alanine dipeptide in vacuum are conducted to illustrate the data-driven solver for the transition path theory on point clouds. The mean transition path obtained via the controlled Monte Carlo simulations highly coincides with the computed dominant transition path in the transition path theory. [ABSTRACT FROM AUTHOR]

The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to compute the contact line dynamics with unavoidable merging and splitting of droplets due to the impermeable obstacle. To efficiently solve the parabolic variational inequality, we propose an unconditional stable explicit boundary updating scheme coupled with a projection method. The explicit boundary updating efficiently decouples the computation of the motion by mean curvature of the capillary surface and the moving contact lines. Meanwhile, the projection step efficiently splits the difficulties brought by the obstacle and the motion by mean curvature of the capillary surface. Furthermore, we prove the unconditional stability of the scheme and present an accuracy check. Convergence of the proposed scheme is also proved using a nonlinear Trotter-Kato product formula under the pinning contact line assumption. After incorporating the phase transition information at splitting points, several challenging examples including splitting and merging of droplets are demonstrated. [ABSTRACT FROM AUTHOR]

In this work, the primary goal is to establish a rigorous connection between the Fokker--Planck equation of neural networks and its microscopic model: the diffusion-jump stochastic process that captures the mean-field behavior of collections of neurons in the integrate-and-fire model. The proof is based on a novel iteration scheme: with an auxiliary random variable counting the firing events, both the density function of the stochastic process and the solution of the PDE problem admit series representations, and thus the difficulty in verifying the link between the density function and the PDE solution in each subproblem is greatly mitigated. The iteration approach provides a generic framework for integrating the probability approach with PDE techniques, with which we prove that the density function of the diffusion-jump stochastic process is indeed the classical solution of the Fokker--Planck equation with a unique flux-shift structure. [ABSTRACT FROM AUTHOR]

In this paper we study a cross-diffusion system whose coefficient matrix is non-symmetric and degenerate. The system arises in the study of tissue growth with autophagy. The existence of a weak solution is established. We also investigate the limiting behavior of solutions as the pressure gets stiff. The so-called incompressible limit is a free boundary problem of Hele-Shaw type. Our key new discovery is that the usual energy estimate still holds as long as the time variable stays away from 0. [ABSTRACT FROM AUTHOR]

At the continuous level, we consider two types of tumor growth models: the cell density model, based on the uid mechanical construction, is more favorable for scientific interpretation and numerical simulations, and the free boundary model, as the incompressible limit of the former, is more tractable when investigating the boundary propagation. In this work, we aim to investigate the boundary propagation speeds in those models based on asymptotic analysis of the free boundary model and efficient numerical simulations of the cell density model. We derive, for the first time, some analytical solutions for the free boundary model with pressure jumps across the tumor boundary in multidimensions with finite tumor sizes. We further show that in the large radius limit, the analytical solutions to the free boundary model in one and multiple spatial dimensions converge to traveling wave solutions. The convergence rate in the propagation speeds are algebraic in multidimensions as opposed to the exponential convergence in one dimension. We also propose an accurate front capturing numerical scheme for the cell density model, and extensive numerical tests are provided to illustrate the analytical findings. [ABSTRACT FROM AUTHOR]

We consider in this work the convergence of the random batch method proposed in our previous work [Jin et al., J. Comput. Phys., 400(2020), 108877] for interacting particles to the case of disparate species and weights. We show that the strong error is of O(√τ) while the weak error is of O(τ) where r is the time step between two random divisions of batches. Both types of convergence are uniform in N, the number of particles. The proof of strong convergence follows closely the proof in [Jin et al., J. Comput. Phys., 400(2020), 108877] for indistinguishable particles, but there are still some differences: Since there is no exchangeability now, we have to use a certain weighted average of the errors; some refined auxiliary lemmas have to be proved compared with our previous work. To show that the weak convergence of empirical measure is uniform in N, certain sharp estimates for the derivatives of the backward equations have been used. The weak convergence analysis is also illustrating for the convergence of the Random Batch Method for N-body Liouville equations. [ABSTRACT FROM AUTHOR]

We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of the corresponding implicit schemes, are proved by its linkage to Volterra integrals with completely monotone kernels. We then apply the backward scheme corresponding to this discretization to two time fractional dissipative problems, and these implicit schemes are helpful for the analysis of the corresponding problems. In particular, we show that the overdamped generalized Langevin equation with fractional noise has a unique limiting measure for strongly convex potentials and we establish the convergence of numerical solutions to the strong solutions of time fractional gradient flows. The proposed scheme and schemes derived using the same philosophy can be useful for many other applications as well. [ABSTRACT FROM AUTHOR]

The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin-Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order γ∊ (0,1) for functions valued in general Banach spaces, consistent with the traditional definition if the space is Rd and functions are absolutely continuous. Based on a Volterra-type integral form, we establish some time regularity estimates of the functions provided that the weak Caputo derivatives are in certain spaces. The compactness criteria are then established using the time regularity estimates. The existence of weak solutions for a special case of time fractional compressible Navier-Stokes equations with constant density and time fractional Keller-Segel equations in R2 are then proved as model problems. This work provides a framework for studying weak solutions of nonlinear time fractional PDEs. [ABSTRACT FROM AUTHOR]

We propose a generalized definition of Caputo derivatives from t = 0 of order γ∈in (0, 1) using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. Our strategy is to define a modified Riemann-Liouville fractional calculus which agrees with the traditional Riemann-Liouville definition for t > 0 but includes some singularities at t = 0 so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to deconvolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general comparison principle with the most general conditions. This then allows for a priori energy estimates of fractional PDEs. Since the new definition is valid for locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs. Many fundamental results for fractional ODEs are revisited within this framework under very weak conditions. [ABSTRACT FROM AUTHOR]

In this paper, we present a dispersive regularization approach to construct a global N -peakon weak solution to the modified Camassa-Holm equation (mCH) in one dimension. In particular, we perform a double mollification for the system of ODEs describing trajectories of N - peakon solutions and obtain N smoothed peakons without collisions. Though the smoothed peakons do not give a solution to the mCH equation, the weak consistency allows us to take the smoothing parameter to zero and the limiting function is a global N -peakon weak solution. The trajectories of the peakons in the constructed solution are globally Lipschitz continuous and do not cross each other. When N = 2, the solution is a sticky peakon weak solution. At last, using the N -peakon solutions and through a mean field limit process, we obtain global weak solutions for general initial data m0 in Radon measure space. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation ∂u/ ∂t = Δ(Δu) -3. This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution u with Δu ≥ 0 is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation ∂tρ+ρ2Δ2ρ3 = 0, another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare [Phys. D, 240 (2011), pp. 1771-1784]. [ABSTRACT FROM AUTHOR]

We study in this work a continuum model derived from a one-dimensional attachmentdetachment-limited type step ow on a vicinal surface, ut = -u2(u3)hhhh; where u, considered as a function of step height h, is the step slope of the surface. We formulate a notion of a weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of the weak solution and prove it converges to a constant solution as time goes to infinity. The space-time Hölder continuity of the weak solution is also discussed as a byproduct. [ABSTRACT FROM AUTHOR]

In this paper, we prove convergence of a sticky particle method for the modified Camassa—Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) — uxx(•, t) is bounded by the total variation of the initial data m0. We also obtain W1,1(ℝ)-stability of weak solutions when solutions are in L∞(0,∞;W2,1(ℝ)). (Notice that peakon weak solutions are not in W2,1(ℝ).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation. [ABSTRACT FROM AUTHOR]

In this paper we study the existence assertion of the initial boundary value problem for the equation ... This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in the equation can have a singular part in the sense of the Lebesgue decomposition theorem, and the exponential nonlinearity somehow "cancels" it out. The net result is that we obtain a solution u that satisfies the equation and the initial boundary conditions in the almost everywhere (a.e.) sense. [ABSTRACT FROM AUTHOR]

From a variational perspective, we derive a series of magnetization and quantum spin current systems coupled via an "s-d" potential term, including the Schrödinger-Landau-Lifshitz- Maxwell system, the Pauli-Landau-Lifshitz system, and the Schrödinger-Landau-Lifshitz system with successive simplifications. For the latter two systems, we propose using the time splitting spectral method for the quantum spin current and the Gauss-Seidel projection method for the magnetization. Accuracy of the time splitting spectral method applied to the Pauli equation is analyzed and verified by numerous examples. Moreover, behaviors of the Schrödinger-Landau- Lifshitz system in different "s-d" coupling regimes are explored numerically. [ABSTRACT FROM AUTHOR]

We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the transported quantity in the neighboring cells. A motivation is pedestrian dynamics during panic situations in a small corridor where the propagation of people in a part of the corridor can be either left- or right-going. Under the assumptions of propagation of chaos and mean-field limit, we derive a master equation and the corresponding mean-field kinetic and macroscopic models. Steady-states are computed and analyzed and exhibit the possibility of multiple metastable states and hysteresis. [ABSTRACT FROM AUTHOR]

We propose a simple finite difference scheme for Navier--Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, it can be shown that this spatially compatible discretization leads to uniform inf-sup condition, which plays a crucial role in the pressure approximation of both dynamic and steady state calculations. Numerical experiments demonstrate the robustness and efficiency of our scheme. [ABSTRACT FROM AUTHOR]

We studied coupled systems of the Fokker--Planck equation and the Navier--Stokes equation modeling the Hookean and the finitely extensible nonlinear elastic (FENE)-type polymeric flows. We proved the continuous embedding and compact embedding theorems in weighted spaces that naturally arise from related entropy estimates. These embedding estimates are shown to be sharp. For the Hookean polymeric system with a center-of-mass diffusion and a superlinear spring potential, we proved the existence of a global weak solution. Moreover, we were able to tackle the FENE model with L² initial data for the polymer density instead of the L∞ counterpart in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with a degenerate diffusion Patlak--Keller--Segel system in n ≥ 3 dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent m = 2n/(n + 2), which is smaller than the usual exponent m* = 2 - 2/n used in other studies. With the exponent m = 2n/(n + 2), the associated free energy is conformal invariant, and there is a family of stationary solutions ... For radially symmetric solutions, we prove that if the initial data are strictly below U λ,0(X) for some λ, then the solution vanishes in Lloc¹ as t → → if the initial data are strictly above U λ,0(X) for some λ, then the solution either blows up at a finite time or has a mass concentration at r = 0 as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the Lm norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow-up of the solution if the Lm norm for initial data is larger than the Lm norm of ..., which is constant independent of λ and x0, and the free energy of initial data is smaller than that of ... [ABSTRACT FROM AUTHOR]

Motivated by a phenomenon of phase transition in a model of alignment of selfpropelled particles, we obtain a kinetic mean-field equation which is nothing more than the Smoluchowski equation on the sphere with dipolar potential. In this self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global wellposedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, the other equilibria are the Fisher--von Mises distributions with arbitrary direction and a concentration parameter determined by the intensity of the noise. For any initial condition, we give a rigorous proof of convergence of the solution to a steady state as time goes to infinity. In particular, when the noise is under the threshold and with nonzero initial mean velocity, the solution converges exponentially fast to a unique Fisher--von Mises distribution. We also found a new conservation relation, which can be viewed as a convex quadratic entropy when the noise is above the threshold. This provides a uniform exponential rate of convergence to the uniform distribution. At the threshold, we show algebraic decay to the uniform distribution. [ABSTRACT FROM AUTHOR]

We consider the vorticity-stream formulation of axisymmetric incompressible flows and its equivalence with the primitive formulation. It is shown that, to characterize the regularity of a divergence free axisymmetric vector field in terms of the swirling components, an extra set of pole conditions is necessary to give a full description of the regularity. In addition, smooth solutions up to the axis of rotation give rise to smooth solutions of primitive formulation in the case of the Navier-Stokes equation, but not the Euler equation. We also establish a proper weak formulation and show its equivalence to Leray's formulation. [ABSTRACT FROM AUTHOR]

The sequential regularization method is a reformulation of the unsteady Navier-Stokes equations from the viewpoint of constrained dynamical systems or the approximate Helmholtz-Hodge projection. In this paper we study the long time behavior of the sequential regularization formulation. We give a uniform-in-time estimate between the solution of the reformulated system and that of the Navier-Stokes equations. We also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time. A couple of long time flow examples are computed to demonstrate this method. [ABSTRACT FROM AUTHOR]

In a previous work [P. Crispel, P. Degond, and M.-H. Vignal, J. Comput. Phys., 223 (2007), pp. 208-234], a new numerical discretization of the Euler-Poisson system was proposed. This scheme is "asymptotic preserving" in the quasineutral limit (i.e., when the Debye length ε tends to zero), which means that it becomes consistent with the limit model when ε →→? 0. In the present work, we show that the stability domain of the present scheme is independent of ε. This stability analysis is performed on the Fourier transformed (with respect to the space variable) linearized system. We show that the stability property is more robust when a space-decentered scheme is used (which brings in some numerical dissipation) rather than a space-centered scheme. The linearization is first performed about a zero mean velocity and then about a nonzero mean velocity. At the various stages of the analysis, our scheme is compared with more classical schemes and its improved stability property is outlined. The analysis of a fully discrete (in space and time) version of the scheme is also given. Finally, some considerations about a model nonlinear problem, the Burgers-Poisson problem, are also discussed. [ABSTRACT FROM AUTHOR]

We give an error estimate for the energy and helicity preserving scheme (EHPS) in second order finite difference setting on axisymmetric incompressible flows with swirling velocity. This is accomplished by a weighted energy estimate, along with careful and nonstandard local truncation error analysis near the geometric singularity and a far field decay estimate for the stream function. A key ingredient in our a priori estimate is the permutation identities associated with the Jacobians, which are also a unique feature that distinguishes EHPS from standard finite difference schemes. [ABSTRACT FROM AUTHOR]

This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the nonequilibrium part of the distribution function. This equation is solved only locally and is related to the fluid equation through a downscaling effect. The method does not need to find an interface condition as do usual domain decomposition methods to match fluid and kinetic representations. We show our approach applies to problems that have a hydrodynamic time scale as well as to problems with diffusion time scale. Simple numerical schemes are proposed to discretize our models, and several numerical examples are used to validate the method. [ABSTRACT FROM AUTHOR]