In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). These numerical simulations indicate that DNNs seem

This paper is concerned with the Cauchy problem of a bipolar hydrodynamic model for semiconductor device, a system of one dimensional Euler–Poisson equations with time-dependent damping effect in the critical case. The global existence and uniqueness of the solutions to the Cauchy problem are proved by the technical time-weighted energy method, when the initial perturbation around the constant states

This paper is concerned with large-time behavior of solutions to the outflow problem of full compressible Navier–Stokes equations in the half line. This is one out of the series of papers by the authors on the stability of nonlinear waves to the outflow problem. We show the time-asymptotic stability of degenerate (transonic) stationary solution for the general gas including ideal polytropic gas. The

In this paper, an optimal control problem governed by elliptic equations with $L^2$‑norm constraint for state variable is developed. Firstly, the optimality conditions of the optimal control problem are derived, and the optimal control problem is approximated by the Galerkin spectral methods. Similarly, the optimality conditions of the discrete problem are also obtained. Then, some important lemmas

We consider the compressible Ericksen–Leslie system of liquid crystal flows in one dimension. A global weak solution is constructed with initial density $\rho_0 \geq 0$ and $\rho_0 \in L^\gamma$ for $\gamma \gt 1$.

The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We introduce the concept of complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the $\mathcal{CM}$ property as the continuous equations. We prove that $\mathcal{CM}$-preserving schemes

In this paper, we construct a family of finite energy classical solutions to the 3D tropical climate model equations. The approach is by studying the steady-state Beltrami flows and using the standard cut-off technique to establish the global regularity of the systems.

The main purpose of this paper is to show the well-posedness theory of two-dimensional symmetric compressible subsonic impinging jet flow. More precisely, for any given atmospheric pressure, we show that there exists a critical value, such that if the incoming mass flux is less than the critical value, then there exists a smooth symmetric compressible subsonic impinging jet flow, and the free boundary

The Riemann problem of one dimensional shallow water equations with discontinuous topography has been constructed recently. The elementary waves include shock waves, rarefaction waves, and the stationary wave. The stationary wave appears when the water depth changes, especially when there exists a bottom step. In this paper, we are mainly concerned with the interaction between a stationary wave with

Many technologically useful materials are polycrystals composed of a myriad of small monocrystalline grains separated by grain boundaries. Dynamics of grain boundaries play an essential role in defining the material’s properties across multiple scales. In this work, we study the largetime asymptotic behavior of the model for the motion of grain boundaries with the dynamic lattice misorientations and

We discuss the asymptotic frequency synchronization for the non-identical Kuramoto oscillators with time delayed interactions. We provide explicit lower bound on the coupling strength and upper bound on the time delay in terms of initial configurations ensuring exponential synchronization. This generalizes not only the frequency synchronization estimate by Choi et al. [Phys. D, 241(7):735–754, 2012]

We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $\rho$ ahead. The averaging kernel is of exponential type: $w_\varepsilon (s)=\varepsilon^{-1} e^{-s / \varepsilon}$. For any decreasing velocity function $v$, we prove that, as $\varepsilon \to 0$, the limit of solutions to the nonlocal equation coincides with the

In this work, we study the global existence of one-dimensional isentropic gas dynamics system. We prove a global-in-time existence result of this system in the framework of discontinuous non-decreasing solutions considering bounded initial data. We remark that this result allows us to give a global meaning to the gas dynamics system in distributional sense, considering discontinuous solutions with

By proposing a notion of Radon measure solutions of the compressible Euler equations, we consider in the paper uniform stationary hypersonic-limit flows passing a two-dimensional finite non-symmetric obstacle with static gas downstream behind the obstacle, and construct solutions with mass concentrated on the boundary of the obstacle and then on free layers beyond it. The Newton–Busemann pressure law

In this paper, we study the non-isentropic Euler–Poisson system and the non-uniqueness of transonic shock solutions is obtained. More precisely, prescribing a class of physical boundary conditions on the boundary of a flat nozzle with finite length, we prove that there exist two and only two transonic shocks. This is motivated by the result of existence of multiple transonic shock solutions for isentropic

Moving bottlenecks in road traffic represent an interesting mathematical problem, which can be modeled via coupled PDE-ODE systems. We consider the case of a scalar conservation law modeling the evolution of vehicular traffic and an ODE with discontinuous right-hand side for the bottleneck introduced in [M.L. Delle Monache and P. Goatin, J. Diff. Eqs., 257(11):4015–4029, 2014]. The bottleneck usually

It is said that classical theories are sometimes inappropriate to describe very efficient biological processes in nature, which seem to be better understood via quantum mechanical models. We are however still very far from understanding how quantum features can survive in open quantum systems. In this paper the author shall present a simple mathematical model for the illustration of the excitation

In this paper we consider the well-posedness of the compressible Prandtl boundary layer for the Navier–Stokes–Poisson equations with nonslip boundary condition and obtain the linear stability of the approximate solution constructed by boundary layer expansion.

We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schrödinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the variational characterization of the

In this paper, we shall study the initial-boundary value problem of a mathematical model describing the branching of capillary sprouts during angiogenesis in one dimensional space. Under homogeneous Neumann boundary conditions, we show the existence of a unique global classical solution with uniform-in-time bound for all suitably regular initial data. Moreover, we show that the unique solution will

We study a Lie splitting scheme for a nonlinear Schrödinger-type equation with random dispersion. The main result is an approximation of the local error. Then we can deduce sharp order estimates, for instance in the case of a white noise dispersion.

We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson–Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space $W^{1,\infty} (\mathbb{R}^d)$ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity

We study the conditional regularity of solutions to the Navier-Stokes equations in the three dimensional space. Let $u=(u_1,u_2,u_3)$ denote the velocity. We impose an additional condition only on one diagonal entry of the velocity gradient, namely $\partial_3 u_3$, and show, using a technique based on the mixed multiplier theorem and an anisotropic version of the Troisi inequality, that if $\partial_3

This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil $(A,B)$. Based on Zolotarev’s best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval via function compositions with

In this paper, we study the long-time existence of smooth solutions to the Cauchy problem for the Boussinesq equation with large initial data in $\mathbb{R}^n$. Due to the strong dispersive effect in the Boussinesq equation, the method of combining the blowup criterion and Strichartz estimate are used to show that the lifespan of the solutions can be taken arbitrarily large provided that the dispersive

The interaction of elementary waves for isentropic flow in a variable cross-section duct is obtained [W.C. Sheng and Q.L. Zhang, Commun. Math. Sci., 16:1659–1684, 2018]. The authors have discussed rarefaction wave or shock wave interactions with stationary wave. In this paper, we extend their results to the nonisentropic flow. It can be proved that if one changes $(u,\rho)$ plane in isentropic flow

Frozen Gaussian approximation (FGA) has been applied and numerically verified as an efficient tool to compute high-frequency wave propagation modeled by non-strictly hyperbolic systems, such as the elastic wave equations [J.C. Hateley, L. Chai, P. Tong and X. Yang, Geophys. J. Int., 216:1394–1412, 2019] and the Dirac system [L. Chai, E. Lorin and X. Yang, SIAM J. Numer. Anal., 57:2383–2412, 2019].

We establish the existence of global martingale solutions of the stochastic Camassa–Holm equation in $H^1(\mathbb{R})$. The construction of the solution is based on the regularization method and the stochastic compactness method. Furthermore, we use Borel–Cantelli Lemma to prove the global existence of mild solution of the stochastic Camassa–Holm equation with small noise in $L^2(\mathbb{R})$.

In [T.C. Sideris, Comm. Part. Diff. Eqs., 8:1291–1323, 1983], the author proves the solution of the nonlinear wave equations breaks down in finite time, if the initial data is radially symmetric and arbitrarily small. The present article is devoted to the study of the lower bound of blow-up rate of blow-up solution and the global solution to a class of nonlinear wave equations in $\mathbb{R}^d , d

This paper studies the pricing problem for callable-puttable convertible bonds with callput protections and call notice period as a Dynkin game via the backward stochastic differential equation (BSDE) with two reflecting barriers. By virtue of reflected BSDEs, we first reduce such a Dynkin game to an optimal stopping time problem and establish the formulae for the fair price of puttable convertible

We consider the numerical approximations of the Cahn–Hilliard equation with dynamic boundary conditions [C. Liu et al., Arch. Ration. Mech. Anal., 2019]. We propose a first-order in time, linear and energy-stable numerical scheme, which is based on the stabilized linearly implicit approach. The energy stability of the scheme is proved and the semi-discrete-in-time error estimates are carried out. Numerical

Recently, energetic variational approach was employed to derive models for nonisothermal electrokinetics by Liu et al. [P. Liu, S. Wu, and C. Liu, Commun. Math. Sci., 16(5):1451–1463, 2018]. In particular, the Poisson–Nernst–Planck–Fourier (PNPF) system for the dynamics of $N$-ionic species in a solvent was derived. In this paper we first reformulate PNPF ($4N +6$ equations) into an evolutional system

A collisionless plasma is modeled by the Vlasov–Poisson system. Smooth solutions are considered in three spatial dimensions with compactly supported initial data. The main theorem of this work is a small data result that improves an earlier theorem of Bardos and Degond in that it does not require the derivatives of the initial data to be small. Another theorem is presented here that gives a sufficient

This paper is concerned with a mathematical model of competition for resource where species consume noninteracting resources. This system of differential equations is formally obtained by renormalizing the MacArthur’s competition model at equilibrium, and agrees with the trait-continuous model studied by [S. Mirrahimi, B. Perthame and J.Y. Wakano, J. Math. Biol., 64(7):1189–1223, 2012]. As a dynamical

A set-pair Lovász extension is established to construct equivalent continuous optimization problems for graph $k$-cut problems.

We consider the stabilization of linear kinetic equations with a random relaxation term. The well-known framework of hypocoercivity by J. Dolbeault, C. Mouhot and C. Schmeiser (2015) ensures the stability in the deterministic case. This framework, however, cannot be applied directly for arbitrarily small random relaxation parameters. Therefore, we introduce a Galerkin formulation, which reformulates

In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels $\phi (x) := {\lvert x \rvert}^{- (n + \alpha)}$ for $\alpha \in (0,2)$. The solutions describe unidirectional parallel motion of agents governing multi-dimensional collective behavior of flocks. Here, we consider the range $1\lt \alpha \lt 2$ and establish

We consider the spatially two-dimensional version of a cross-diffusion system, as originally proposed by Short et al. in [M.B. Short, M.R. D’Orsogna, V.B. Pasour, G.E. Tita, P.J. Brantingham, A.L. Bertozzi, and L.B. Chayes, Math. Mod. Meth. Appl. Sci., 18:1249–1267, 2008] to describe the evolution of urban crime. Although sharing some basic structure elements with the well-studied classical Keller–Segel

We give a sharp convergence rate for the asynchronous stochastic gradient descent (ASGD) algorithms when the loss function is a perturbed quadratic function based on the stochastic modified equations introduced in [An et al. “Stochastic modified equations for the asynchronous stochastic gradient descent”, arXiv:1805.08244]. We prove that when the number of local workers is larger than the expected

This note shows that the matrix forms of several one-parameter distribution families satisfy a hierarchical low-rank structure. Such families of distributions include binomial, Poisson, and $\chi^2$ distributions. The proof is based on a uniform relative bound of a related divergence function. Numerical results are provided to confirm the theoretical findings.

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