Our focus is the Boltzmann equation in a torus under very soft potentials around equilibrium. We analyze the asymptotics of the equation from angular cutoff to non-cutoff. We first prove a refined decay result of the semi-group stemming from the linearized Boltzmann operator. Then we prove the global well-posedness of the equations near equilibrium, refined decay patterns of the solutions. Finally

In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric $N$-player non-zero-sum stochastic differential games, for which we refer as deep fictitious play, a multi-stage learning process. Specifically at each stage, we propose the strategy of letting individual player optimize

In this paper, we consider numerical approximations for the anisotropic modified phasefield crystal model with a strong nonlinear vacancy potential, which describes microscopic phenomena involving atomic hopping and vacancy diffusion. The model is a nonlinear damped wave equation that includes an anisotropic Laplacian and a strong nonlinear vacancy term. To develop an easy to implement time marching

In this paper, we study the Cauchy problem for quantum Zakharov system in three space dimensions. We prove that the quantum Zakharov system scatters in low regularity space $L^2$ with small radial initial data basing on some radial improved Strichartz estimates with wider range and the normal form transformation technique.

We show that the nonlinear Schrödinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in $L^2$ once the free deterministic Schrödinger group satisfies a natural $L^1 - L^\infty$ decay, which is verified in many examples. Also, we investigate the well-posedness in the energy domain in general and in concrete situations, as well as the fact that the solution with

We establish two geometric inequalities, respectively, for harmonic functions in exterior Dirichlet problems, and for Green’s functions in interior Dirichlet problems, where the boundary surfaces are smooth and convex. Both inequalities involve integrals over the mean curvature and the Gaussian curvature on an equipotential surface, and the normal derivative of the harmonic potential thereupon. These

Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the harmonic map heat flow equation. The Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M‑H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation

We study the large-time behavior of the sublinear viscosity solution to a singular Hamilton–Jacobi equation that appears in a critical coagulation-fragmentation model with multiplicative coagulation and constant fragmentation kernels. Our results include complete characterizations of stationary solutions and optimal conditions to guarantee large-time convergence. In particular, we obtain convergence

In this paper, we will show the blowup phenomenon of solutions to the compressible Euler equations with time-dependent damping. Firstly, under the assumptions that the radially symmetric initial data and initial density contains vacuum states, the singularity of the classical solutions will formed in finite time in $\mathbb{R}^n (n \geq 2)$. Furthermore, we can also find a sufficient condition for

We propose a central scheme framework for the approximation of hyperbolic systems of conservation laws in any space dimension. The new central schemes are defined so that any convex invariant set containing the initial data can be an invariant domain for the numerical method. The underlying first-order central scheme is the analog of the guaranteed maximum speed method of [J.‑L. Guermond and B. Popov

We construct singular solutions to a semilinear elliptic equation with exponential nonlinearity on $\Omega \subset \mathbb{R}^2$ by a shrinking hole argument, which we call Born–Infeld approximation scheme. With some natural assumptions on the nonlinearity $f (e^u)$, we classify all these singular solutions by the order of $u$ near each singularity. To show the existence of singular solutions, we introduce

The primitive equations (PEs) are a basic model in the study of large scale oceanic and atmospheric dynamics. Its high non-linearity and anisotropic structure attract much attention from mathematicians. In the present article, we consider the corresponding stochastic model. As studies from climate sciences show that the complex multi-scale nature of the earth’s climate system results in many uncertainties

We provide a general framework to extend the relative entropy method to a class of diffusive relaxation systems with discrete velocities. The methodology is detailed in the toy case of the 1D Jin–Xin model under the diffusive scaling, and provides a direct proof of convergence to the limit parabolic equation in any interval of time, in the regime where the solutions are smooth. Recently, the same approach

We consider an initial boundary value problem of two-dimensional (2D) compressible nematic liquid crystal flows. Under a geometric condition for the initial orientation field, we show that the strong solution exists globally if the density is bounded from above. Our proof relies on elementary energy estimates and critical Sobolev inequalities of logarithmic type.

We introduce a principled method for the signed clustering problem, where the goal is to partition a weighted undirected graph whose edge weights take both positive and negative values, such that edges within the same cluster are mostly positive, while edges spanning across clusters are mostly negative. Our method relies on a graph-based diffuse interface model formulation utilizing the Ginzburg–Landau

In this paper we present an analytical formula for pricing discretely-sampled variance swaps with the realized variance being defined in terms of squared log return of the underlying asset. The dynamics of the underlying asset price follows the Schwartz’s two-factor model which can be used to describe commodity prices and allows the convenience yields to be stochastic. A partial differential equation

In this paper, we propose a normalized Goldstein-type local minimax method (NG-LMM) to seek for multiple minimax-type solutions. Inspired by the classical Goldstein line search rule in the optimization theory in $\mathbb{R}^m$, which is aimed to guarantee the global convergence of some descent algorithms, we introduce a normalized Goldstein-type search rule and combine it with the local minimax method

We study dissipative solutions to a 3D simplified hyperbolic Ericksen–Leslie system for liquid crystals with Ginzburg–Landau approximation. First, we establish a weak-strong stability principle, which leads to a suitable notion of dissipative solutions to the hyperbolic Ericksen–Leslie system. Then, we introduce a regularized system to approximate the original system, for which we can prove the existence

Magnetic materials possess the intrinsic spin order, whose disturbance leads to spin waves. From the mathematical perspective, a spin wave is known as a traveling wave, which is often seen in wave and transport equations. The dynamics of intrinsic spin order is modeled by the Landau–Lifshitz–Gilbert equation, a nonlinear parabolic system of equations with a pointwise length constraint. In this paper

We study emergent dynamics of the discrete Cucker–Smale (in short, DCS) model with randomly switching network topologies. For this, we provide a sufficient framework leading to the stochastic flocking with probability one. Our sufficient framework is formulated in terms of an admissible set of network topologies realized by digraphs and probability density function for random switching times. As examples

We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can

In this paper, we analyze the behavior of viscous shock profiles of one-dimensional compressible Navier–Stokes equations with a singular pressure law which encodes the effects of congestion. As the intensity of the singular pressure tends to $0$, we show the convergence of these profiles towards free-congested traveling front solutions of a two-phase compressible-incompressible Navier–Stokes system

In this paper we study the asymptotic behavior of Brownian motion in both comb-shaped planar domains, and comb-shaped graphs. We show convergence to a limiting process when both the spacing between the teeth and the width of the teeth vanish at the same rate. The limiting process exhibits an anomalous diffusive behavior and can be described as a Brownian motion time-changed by the local time of an

In this paper we derive a Bismut–Elworthy–Li–type formula with respect to strong solutions to singular stochastic differential equations (SDE’s) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H \lt 1/2$. “Singular” here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this

In this paper, the local well-posedness of strong solutions to the Cauchy problem of the isentropic compressible Navier–Stokes equations is proved with the initial data being allowed to have vacuum. The main contribution of this paper is that the well-posedness is established without assuming any compatibility condition on the initial data, which was widely used before in many works concerning the

We study a class of fourth-order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type\[\int_{\Omega} u^{2 \gamma - \alpha - \beta} \Delta u^\alpha \Delta u^\beta dx \geq c \int_{\Omega} {\lvert \Delta u^\gamma \rvert}^2 dx \; \textrm{,}\]which

In this work, we investigate a variational formulation for a time-fractional Fokke–Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [R. Jordan, D. Kinderlehrer, and F. Otto, SIAM

In this paper, we study the long-time behavior of the solutions for the initial-boundary value problem to a one-dimensional Navier–Stokes equations for a reacting mixture in a half line $\mathbb{R}_{+} := (0, \infty)$. We give the asymptotic stability of not only stationary solution for the impermeability problem but also the composite waves consisting of the subsonic BL-solution, the contact wave

In this paper, we investigate the well-posedness/ill-posedness of the stationary solutions to the isothermal bipolar hydrodynamic model of semiconductors driven by Euler–Poisson equations. Here, the density of electrons is proposed with sonic boundary and considered in interiorly subsonic case or interiorly supersonic case, while the density of holes is considered in fully subsonic case or fully supersonic

The free boundary problem of one-dimensional heat conducting compressible Navier–Stokes equations with large initial data is investigated. We obtain the global existence of strong solution under stress-free boundary condition along the free surface, where the heat conductivity depends on temperature $(\kappa = \overline{\kappa} \theta^b , b \in (0, \infty))$ and the viscosity coefficient depends on

In this paper, we study some properties of positive solutions of nonlinear integral equations with axis-symmetric kernels, which arise from weak-type convolution-Young’s inequality and the stationary magnetic compressible fluid stars. With the help of the method of moving planes and regularity lifting lemma, we show that all of the positive solutions in certain functional spaces are symmetric and monotonically

The derivation of different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and the nonlinear progressive wave equation (NPE) from an isentropic Navier–Stokes/Euler system is systematized using the Hilbert-type expansion in the corresponding perturbative and (for the KZK and NPE equations) paraxial ansatz. The use of

Cahn–Hilliard/Navier–Stokes system is the combination of the Cahn–Hilliard equation with the Navier–Stokes equations. It describes the motion of unsteady mixing fluids and has a wide range of applications ranging from turbulent two-phase flows to microfluidics. In this paper we consider the non-local Cahn–Hilliard equation (the gradient term of the order parameter in the free energy is replaced with

In this paper, we develop a robust and efficient numerical method for shallow water equations with moving bottom topography. The model consists of the Saint-Venant system governing the water flow coupled with the Exner equation for the sediment transport. One of the main difficulties in designing good numerical methods for such models is related to the fact that the speed of water surface gravity waves

The present paper is dedicated to the study of the Cauchy problem for compressible Navier–Stokes–Poisson–Korteweg model in any dimension $d \geq 2$, which simultaneously involves the lower order potential term and the higher order capillarity term. The unique global solvability of the system is obtained when the initial data are close to a stable equilibrium state in a functional setting invariant

The Cauchy problem of the 1D compressible radiative and reactive gas without viscosity is studied in this paper. When the radiation effect is under consideration, the equations present high nonlinearity, together with the lack of viscosity, which result in many more difficulties. When the solution to the corresponding Riemann problem of the Euler equation consists of a contact discontinuity and rarefaction

The paper proves existence of renormalized solutions for a class of velocity-discrete coplanar stationary Boltzmann equations with given indata. The proof is based on the construction of a sequence of approximations with $L^1$ compactness for the integrated collision frequency and gain term. $L^1$ compactness of a sequence of approximations is obtained using the Kolmogorov–Riesz theorem and replaces

This paper is devoted to studying the inflow problem for an ideal polytropic model with non-viscous gas in one-dimensional half space. We show the existence of the boundary layer in different areas. By employing the energy method, we also prove the unique global-in-time existence of the solution and the asymptotic stability of both the boundary layers, the $3$‑rarefaction wave and their superposition

In this paper, we derive some new lower bounds of possible blow up solutions in $\dot{H}^1_p (\mathbb{R}^3)$ with $3 / 2 \lt p \lt \infty$ to the 3D Navier–Stokes equations, which provides a new proof of the corresponding recent results involving blow up rates in $\dot{H}^s$ with $1 \leq s \lt 5/2$ in [A. Cheskidov and M. Zaya, J. Math. Phys., 57:023101, 2016; J.C. Cortissoz and J.A. Montero, J. Math

We examine challenges to sampling from Boltzmann distributions associated with multiscale energy landscapes. The multiscale features—or “roughness”—correspond to highly oscillatory, but bounded, perturbations of a smooth landscape. Through a combination of numerical experiments and analysis we demonstrate that the performance of Metropolis adjusted Langevin algorithm can be severely attenuated as the

In this work, we introduce a new Monte Carlo method for solving the Boltzmann model of rarefied gas dynamics. The method works by reformulating the original problem through a micro-macro decomposition and successively by solving a suitable equation for the perturbation from the local thermodynamic equilibrium. This equation is then discretized by using unconditionally stable exponential schemes in

We propose a novel problem formulation of continuous-time information propagation on heterogeneous networks based on jump stochastic differential equations (JSDE). The structure of the network and activation rates between nodes are naturally taken into account in the JSDE. This new formulation allows for efficient and stable algorithms for a variety of challenging information propagation problems,

We show the vanishing viscosity limit to entropy shocks for the fractal Burgers equation in one space dimension. More precisely, we quantify the rate of convergence of the inviscid limit in $L^2$ for large initial perturbations around the entropy shock on any bounded time interval. This is the first result on the inviscid limit to entropy shock for the fractal Burgers equation with the quantified convergence

We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas in particular. We assume the discontinuity in the piecewise-smooth solution is an extremal shock. We use only mild hypotheses on the system. Our techniques and result

Multi-agent systems can be successfully described by kinetic models, which allow one to explore the large scale aggregate trends resulting from elementary microscopic interactions. The latter may be formalised as collision-like rules, in the spirit of the classical kinetic approach in gas dynamics, but also as Markov jump processes, which assume that every agent is stimulated by the other agents to

We introduce a model dealing with conservation laws on networks and coupled boundary conditions at the junctions. In particular, we introduce buffers of fixed arbitrary size and time-dependent split ratios at the junctions, which represent how traffic is routed through the network, while guaranteeing spill-back phenomena at nodes. Having defined the dynamics at the level of conservation laws, we lift

We present three examples of chemical reaction networks whose ordinary differential equation scaling limits are almost identical and in all cases stable. Nevertheless, the Markov jump processes associated to these reaction networks display the full range of behaviors: one is stable (positive recurrent), one is unstable (transient) and one is marginally stable (null recurrent). We study these differences

In this work we study the nonlocal transport equation derived recently by Steinerberger [Proc. Amer. Math. Soc., 147(11):4733–4744, 2019]. When this equation is considered on the real line, it describes how the distribution of roots of a polynomial behaves under iterated differentiation of the function. This equation can also be seen as a nonlocal fast diffusion equation. In particular, we study the

This paper studies the asymptotic behavior of the solution to the initial boundary value problem of a one-dimensional compressible viscous heat-conducting gas with radiation. We consider an outflow problem, where the gas blows out the region through the boundary, of the general gases including ideal polytropic gas. First, we give the necessary and sufficient conditions for an existence of the non-degenerate

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0 \lt \alpha \leq 2$. Special cases are the two-dimensional incompressible Euler equations $(\alpha = 2)$ and the surface quasi-geostrophic (SQG) equations $(\alpha = 1)$. We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when

In this note, we study the Nernst–Planck–Navier–Stokes system for the transport and diffusion of ions in electrolyte solutions. The key feature is to establish three energy-dissipation equalities. As their direct consequence, we obtain global existence for two-ionic species case in $\mathbb{R}^n , n \geq 2$, and multi-ionic species case in $\mathbb{R}^n , n=2,3$.

Sharp bounds for the Rademacher complexity and the generalization error are derived for the residual network model. The Rademacher complexity bound has no explicit dependency on the depth of the network, while the generalization bounds are comparable to the Monte Carlo error rates, suggesting that they are nearly optimal in the high dimensional setting. These estimates are achieved by constraining

It is well known that conjugate gradient methods are suitable for large-scale nonlinear optimization problems, due to their simple calculation and low storage. In this paper, we present a three-term conjugate gradient method using subspace technique for large-scale unconstrained optimization, in which the search direction is determined by minimizing the quadratic approximation of the objective function

In this paper, we consider the steady irrotational Euler flows in multidimensional nozzles. The first rigorous proof on the existence and uniqueness of the incompressible flow is provided. Then, we justify the corresponding low Mach number limit, which is the first result of the low Mach number limit on the steady Euler flows. We establish several uniform estimates, which does not depend on the Mach

In this paper, we study the propagation of the mono-kinetic distribution in the Cucker–Smale-type kinetic equations. More precisely, if the initial distribution is a Dirac mass for the variables other than the spatial variable, then we prove that this “mono-kinetic” structure propagates in time. For that, we first obtain the stability estimate of measure-valued solutions to the kinetic equation, by

This paper concerns with the stability of the plane Couette flow resulting from the motions of boundaries such that the top boundary $\Sigma_1$ and the bottom one $\Sigma_0$ move with constant velocities $(a,0)$ and $(b,0)$, respectively. If one imposes Dirichlet boundary condition on the top boundary and Navier boundary condition on the bottom boundary with Navier coefficient $\alpha$ , there always

We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker–Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties

It is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_n = 2 \cos ((\frac{n}{2}) \omega + ny + x)$ with $\omega$ an irrational number and $x, y \in [0, 1]$. Recently, Han, Schlag, and the second author

Under consideration in the present paper is a mathematical model proposed as an equation of long-crested shallow-water waves propagating in one direction with the effect of Earth’s rotation. The system is called Rotation-Camassa-Holm system (RCH2). The local well-posedness of the periodic Cauchy problem is then established by the linear transport theory. Then, wave-breaking phenomena is investigated