# Forthcoming Papers

• Sam G. Krupa
Criteria for the a-contraction and stability for the piecewise-smooth solutions to hyperbolic balance laws

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 hold without smallness assumptions on the solutions. We can handle shocks of any size. We work in the class of bounded, measurable solutions satisfying a single entropy condition. We also assume a strong trace condition on the solutions, but this is weaker than $BV_{\text{loc}}$. We use the theory of a-contraction (see Kang and Vasseur [Arch. Ration. Mech. Anal., 222(1):343--391, 2016]) developed for the stability of pure shocks in the case without source.

Markov jump processes and collision-like models in the kinetic description of multi-agent systems

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 change state according to a certain transition probability distribution. In this paper we establish a parallelism between these two descriptions, whereby we show how the understanding of the kinetic jump process models may be improved taking advantage of techniques typical of the collisional approach.

• Nicolas Laurent-Brouty, Alexander Keimer, Paola Goatin
A macroscopic traffic flow model with finite buffers on networks: Well-posedness by means of Hamilton-Jacobi equations

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 it up to the Hamilton-Jacobi (H-J) formulation and write boundary datum of incoming and outgoing junctions as functions of the queue sizes and vice-versa. The Hamilton-Jacobi formulation provides the necessary regularity estimates to derive a fixed-point problem in a proper Banach space setting, which is used to prove well-posedness of the model. Finally, we detail how to apply our framework to a non-trivial road network, with several intersections and finite-length links.

• Andrea Agazzi, Jonathan C. Mattingly
Seemingly stable chemical kinetics can be stable, marginally stable, or unstable

We present three examples of chemical reaction networks whose ordinary differential equation scaling limit 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 and characterize the invariant measures by Lyapunov function techniques. In particular, we design a natural set of such functions which scale homogeneously to infinity, taking advantage of the same scaling behavior of the reaction rates.

• Anne-Laure Dalibard, Charlotte Perrin
Existence and stability of partially congested propagation fronts in a one-dimensional Navier-Stokes model

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 and we provide a refined description of the profiles in the vicinity of the transition between the free domain and the congested domain. In the second part of the paper, we prove that the profiles are asymptotically nonlinearly stable under small perturbations with zero integral, and we quantify the size of the admissible perturbations in terms of the intensity of the singular pressure.

• John Kelso Hunter, Jingyang Shu, Qingtian Zhang
Two-Front Solutions of the SQG Equation and its Generalizations

We derive contour-dynamics equations for two-front solutions of the Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (GSQG) equations when the fronts are a graph, as well as scalar reductions of these equations, including ones that describe one-front solutions in the presence of a rigid, flat boundary. We also prove the local-in-time existence and uniqueness of smooth solutions of the front equations in different parameter regimes.

• Samuel Cohn, Gautam Iyer, James H Nolen, Robert Pego
Anomalous diffusion in comb-shaped domains and graphs

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 independent sticky Brownian motion. In the two dimensional setting the main technical step is an oscillation estimate for a Neumann problem, which we prove here using a probabilistic argument. In the one dimensional setting we provide both a direct SDE proof, and a proof using the trapped Brownian motion framework in Ben Arous, et.al. (Ann. Probab. 2015).

• Kwang-Il Choe, Hakho Hong, Jongsung Kim
Non-degenerate stationary solution for outflow problem on the 1-D viscous heat-conducting gas with radiation

This paper studied the asymptotic behavior of the solution to the initial boundary value problem of an 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 gave the necessary conditions and suﬃcient conditions for an existence of the non-degenerate stationary solution. In addition, using the energy method, it proves the asymptotic stability of the solutions under the assumption that the initial perturbation and the boundary data in the Sobolev space is small. We also demonstrated the convergence rate for the exponential and logarithmic decay of the solver. Note that it is the result of the outflow problem of the viscous heat-conducting gas with radiation in the half line.

• Rafael Granero-Belinchon
On a nonlocal differential equation describing roots of polynomials under differentiation

In this work we study the nonlocal transport equation derived recently by Steinerberger. When this equation is consireded on the real line, it describes how the distribution of roots of a polynomial behaves under iterated differentation of the function. This equation can also be seen as a nonlocal fast diffusion equation. In particular, we study the well-posedness of the equation, establish some qualitative properties of the solution and give conditions ensuring the global existence of both weak and strong solutions. Finally, we present a link between the equation obtained by Steinerberger and a one-dimensional model of the surface quasi-geostrophic equation used by Chae, Cordoba, Codoba & Fontelos.

• Zhong Tan, Yong Wang, Xu Jiankai
On the integral equation with the axis-symmetric kernel

In this paper, we investigate some properties of positive solutions for a nonlinear integral equation with axis-symmetric kernel functions, which arises from weak type convolution-Young's inequality and the stationary magnetic compressible fluid stars. With the help of the method of moving plane and regularity lifting lemma, we show that all of positive solutions in certain functional spaces is symmetric and monotonic decreasing on the symmetric axis, and the integrable intervals of positive solutions are also obtained. Furthermore, by analyzing the decay rate of positive solutions in different directions, we show that the kind of integral equation does not admit radial solutions in a weighted functional space.

• Meichen Hou, Lili Fan
Nonlinear stability of the boundary layer and rarefaction wave for the inflow problem governed by the heat-conductive ideal gas without viscosity

This paper is devoted to studying the inflow problem for an ideal polytropic model with non-viscous gas in one-dimensional half space. We showed the existence of the boundary layer in different areas. By employing the energy method, we also proved the unique global-in-time solution existed and the asymptotic stability of both the boundary layer, the $3-$rarefaction wave and their superposition wave under some smallness conditions. Series of simple but tricky operation on boundary need to be carefully done by taking good advantages of construction on the system and domain properties.

• Mihai Cucuringu, Andrea Pizzoferrato, Yves van Gennip
An MBO scheme for clustering and semi-supervised clustering of signed networks

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 functional, based on an adaptation of the classic numerical Merriman-Bence-Osher (MBO) scheme for minimizing such graph-based functionals. The proposed objective function aims to minimize the total weight of inter-cluster positively-weighted edges, while maximizing the total weight of the inter-cluster negatively-weighted edges. Our method scales to large sparse networks, and can be easily adjusted to incorporate labelled data information, as is often the case in the context of semisupervised learning. We tested our method on a number of both synthetic stochastic block models and real-world data sets (including financial correlation matrices), and obtained promising results that compare favourably against a number of state-of-the-art approaches from the recent literature.

• Lidan Wang, Guoli Zhou, Boling Guo
Asymptotic behavior of 3-D stochastic primitive equations of large-scale moist atmosphere with additive noise

The primitive equations (PEs) are a basic model in the study of large scale oceanic and atmospheric dynamics. Its highly non-linearity and anisotropic structure attracts much attention from mathematicians. \par In the present article, we consider the corresponding stochastic model. As studies from climate show that the complex multi-scale nature of the earth’s climate system results in many uncertainties that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. It is further pointed out by \cite{P, MTV} that stochastic modeling for climate is important for understanding the intrinsic variability of dominant low-frequency teleconnection patterns in climate, to provide cheap low-dimensional computational models for the coupled atmosphere-ocean system and to reduce model error in standard deterministic computer models for extended-range prediction through appropriate stochastic noise. This is the the first attempt to consider stochastic moist PEs defined on manifold. Using a new and general way, we prove the existence of random attractor (strong attractor) for the three dimensional stochastic moist primitive equations defined on a manifold $\D\subset\R^3$ improving the existence of weak attractor for the corresponding deterministic model $\cite{GH11}$. As an application of the result, we show the existence of the invariant measure. The technique presented in this work can be applied to common classes of dissipative stochastic partial differential equations. And it has some advantages over the common method of using compact Sobolev imbedding theorem, i.e., if the absorbing set in some Sobolev space do exist in view of the common method, our method would then further implies the existence of random attractor in this space.

• Oussama Amine, Emmanuel Coffie, Fabian Harang, Frank Norbert Proske
A Bismut-Elworthy-Li Formula for Singular SDEs Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling

In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDEs) with additive noise given by a multi- dimensional fractional Brownian motion with Hurst parameter H < 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 representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDEs. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed "local time variational calculus".

• Jian-Guo Liu, Jinhuan Wang
Global existence for Nernst-Planck-Navier-Stokes system in $\mathbb{R}^n$

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 the three energy-dissipation equalities. As their direct consequence, we obtain global existence for two components case in $\mathbb{R}^n$, $n\geq 1$, and multi components case in $\mathbb{R}^2$.

• Zefu Feng, Mei Zhang, Changjiang Zhu
Nonlinear stability of composite waves for one-dimensional compressible Navier-Stokes equations for a reacting mixture

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 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, and the rarefaction wave for the inflow problem of Navier-Stokes equations for a reacting mixture under some smallness conditions. The proofs are based on basic energy method.

• Manh Hong Duong, Bangti Jin
Wasserstein gradient flow formulation of the time-fractional Fokker-Planck equation

In this work, we investigate a variational formulation for a time-fractional Fokker- 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. Gorenflo, Y. Luchko, and M. Yamamoto, Fract. Calc. Appl. Anal.,2015]. We propose a JKO type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.

• Pengcheng Mu, Ming Mei, Kaijun Zhang
Subsonic and supersonic steady-states of bipolar hydrodynamic model of semiconductors with sonic boundary

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 case. With the developed technique based on the topological degree method, the following four kinds of stationary solutions under some conditions are proved to exist: the interiorly-subsonic-vs-fully-subsonic solution, the interiorly-supersonic-vs-fully-subsonic solution, the interiorly-subsonic-vs-fully-supersonic solution, and the interiorly-supersonic-vs-fully-supersonic solution. The non-existence of the above four kinds of solutions under some conditions is also technically proved. For the existence of these physical solutions, different from the previous studies by using traditional fixed-point argument via energy estimates, we recognize that such an approach is failed to our cases, due to that the effect of boundary degeneracy for the electrons causes hardly to estimate the upper and lower bounds for the holes. Instead of it, we develop a new idea to prove the existence of physical solutions by the topological degree method.

• Zilai Li, Yulin Ye
On the free boundary problem of 1D compressible Navier-Stokes equations with heat conductivity dependent of temperature

The free boundary problem of one-dimensional heat conducting compressible Navier-Stokes equations with large initial date is investigated. We obtain the global existence of strong solutions 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 density$(\mu=\overline{\mu}(1+\rho^{a}), \ b\in[0,\infty))$. Moreover, the large time behavior of the free boundary for the full compressible Navier-Stokes equations is considered when the viscosity is constant and it is first shown that the interfaces which separate the gas from vacuum will expand outwards at an algebraic rate in time for all $\gamma >1$.

Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropic systems

The derivation of different models of non linear acoustic in thermo-ellastic 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 \textit{ansatz}. The use of small, to compare to the constant state perturbations, correctors allows to obtain the approximation results for the solutions of these models and to estimate the time during which they keep closed in the $L^2$ norm. In the aim to compare the solutions of the exact and approximated systems in found approximation domains a global well-posedness result for the Navier-Stokes system in a half-space with time periodic initial and boundary data was obtained.

• Zhilei Liang
On the existence of weak solutions to non-local Cahn-Hilliard/Navier-Stokes equations and its local asymptotics

Cahn-Hilliard/Navier-Stokes system is the combination of the Cahn-Hilliard equation with the Navier-Stokes equations. It describes the motion of un-steady mixing fluids and has a wide range of applications ranging from turbulent two-phase flows to microfluidics. Here we consider the non-local Cahn-Hilliard equation (the gradient term of the order parameter in the free energy is replaced with its spatial convolution) coupled with the Navier-Stokes equations. Assume that the densities of the incompressible fluids are constant and the double well potential is singular, we establish the existence of global weak solutions to the non-local system in three dimensional torus. In addition, we show that, under suitable initial assumptions, the solutions are asymptotic to those of the local Cahn-Hilliard/Navier-Stokes equations

• Alina Chertock, Alexander Kurganov, Tong Wu
Operator Splitting Based Central-Upwind Schemes for Shallow Water Equations with Moving Bottom Topography

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 are typically much faster than the speed at which the changes in the bottom topography occur. This imposes a severe stability restriction on the size of time steps, which, in turn, leads to excessive numerical diffusion that affects the computed bottom structure. In order to overcome this difficulty, we develop an operator splitting approach for the underlying coupled system, which allows one to treat slow and fast waves in a different manner and using different time steps. Our method is based on the application of a finite-volume central-upwind scheme introduced in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), pp. 133-160], and incorporates a staggered grid strategy needed for a proper approximation of the bottom topography function. A number of one- and two-dimensional numerical examples are presented to demonstrate the performance of the proposed method.

• Guiqiong Gong, Lin He
Stability of a composite wave of viscous contact wave and rarefaction waves for radiative and reactive gas without viscosity

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 highly nonlinearity, together with the lack of viscosity, which result in much more difficulties. When the solution to the corresponding Riemann problem of the Euler equation consists of a contactdiscontinuity and rarefaction waves, we proved that there exists a unique global-in-time solution and which tends to the combination of a viscous contact wave and rarefaction waves asymptotically with small initial data. The proof is given by the elementary energy method.

• Leif Arkeryd, Anne Nouri
On stationary solutions to normal, coplanar discrete Boltzmann equation models

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 L1 compactness for the integrated collision frequency and gain term. L1 compactness of a sequence of approximations is obtained using the Kolmogorov-Riesz theorem and replaces the L1 compactness of velocity averages in the continuous case, not available when the velocities are discrete.

• Guoliang He, Yanqing Wang, Daoguo Zhou
Lower bounds of blow up solutions in $\dot{H}^{1}_{p}(\mathbb{R}^{3})$ of the Navier-Stokes equations and the quasi-geostrophic equation

In this paper, we derive some new lower bounds of possible blow up solutions in $\dot{H}^{1}_{p}(\mathbb{R}^{3})$ with $3/2< p< \infty$ 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 $1\leq s< 5/2$ in \cite{[CZ],[CM],[MORRVZ],[RSS]}. We apply this to discuss the upper box dimension of the set of singular times of weak solutions. In addition, blow up rates of solutions to the 2D supercritical surface quasi-geostrophic equation in $\dot{H}^{1}_{p}(\mathbb{R}^{2})$ are established.

• Petr Plechac, Gideon Simpson
Sampling from Rough Energy Landscapes

Rough energy landscapes appear in a variety of applications including disordered media and soft matter. In this work, we examine challenges to sampling from Boltzmann distributions associated with rough energy landscapes. Here, the roughness will correspond to highly oscillatory, but bounded, perturbations of a smooth landscape. Through a combination of numerical experiments and asymptotic analysis we demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can be severely attenuated as the roughness increases. In contrast, we prove, rigorously, that Random Walk Metropolis is insensitive to such roughness. We also formulate two alternative sampling strategies that incorporate large scale features of the energy landscape, while resisting the impact of roughness; these also outperform Random Walk Metropolis. Numerical experiments on these landscapes are presented that confirm our predictions. Open analysis questions and numerical challenges are also highlighted.

• Giacomo Dimarco, Nicolas Crouseilles, Anais Crestetto, Mohammed Lemou
A new deviational Asymptotic Preserving Monte Carlo method for the homogeneous Boltzmann equation

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 in solving a suitable equation for the per- turbation from the local thermodynamic equilibrium. This equation is then discretized by using unconditionally stable exponential schemes in time which project the solution over the corresponding equilibrium state when the time step is sent to infinity. The Monte Carlo method is designed on this time integration method and it only describes the perturbation from the final state. In this way, the number of samples diminishes during the time evolution of the solution and when the final equilibrium state is reached, the number of statistical sam- ples becomes automatically zero. The resulting method is computationally less expensive as the solution approaches the equilibrium state as opposite to standard methods for kinetic equations which computational cost increases with the number of interactions. At the same time, the statistical error decreases as the system approaches the equilibrium state. In a last part, we show the behaviors of this new approach in comparison with standard Monte Carlo techniques and in comparison with spectral methods on different prototype problems.

• Yaohua Zang, Gang Bao, Xiaojing Ye, Hongyuan Zha, Haomin Zhou
A Jump Stochastic Differential Equation Approach for Influence Prediction on Heterogenous Networks

We propose a novel problem formulation of continuous-time information propagation on heterogeneous networks based on jump stochastic differential equations (JSDEs). The structure of the network and activation rates between nodes are naturally taken into account in the jump SDE system. This new formulation allows for an efficient and stable algorithm for many challenging information propagation problems, including estimations of individual activation probability and influence level, by solving the JSDE numerically. To this end, we develop an efficient numerical algorithm incorporating variance reduction; furthermore, we provide theoretical bounds for its sample complexity. Numerical experiments on a variety of propagation networks show that the proposed method is more accurate and efficient compared with the state-of-the-art methods, and more importantly, it can be applied to solve other critical information propagation problems where existing methods cannot.

• Roberta Bianchini
Relative entropy in diffusive relaxation for a class of discrete velocities BGK models

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 has been successfully used to show the strong convergence of a vector-BGK model to the 2D incompressible Navier-Stokes equations.

• Xin Zhong
$L^\infty$ continuation principle for two-dimensional compressible nematic liquid crystal flows

We consider the 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.

• Sanae Rujivan
Analytically pricing variance swaps in commodity derivative markets under stochastic convenience yields

In this paper we present an analytical formula for pricing discretely-sampled variance swaps with the realized variance is 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 (PDE) formulated for the pricing $n^{\text{th}}$ moment swaps is analytically solved in real space to obtain our solution as a special case for variance swaps when $n=2$. Interestingly, we successfully manage to establish an interrelationship equation for variance swap and futures prices that would be beneficial for practitioners in commodity markets who prefer to hedge price volatility risk using futures contracts. We further discuss the validity of our solution as well as propose a methodology to characterize a feasible parameter subspace, ensuring the model parameters extracted from market data produce finiteness and strictly positiveness of variance swap prices when they are applied to our solution. Numerical tests are provided to confirm the correctness as well as efficiency of our solution. Finally, we investigate how sensitive our solution to the change of the model parameters.

• Feng Cheng, Ning Jiang, Yi-Long Luo
On dissipative solutions to a simplified hyperbolic Ericksen-Leslie system of liquid crystals

We study dissipative solutions to a 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 of global-in-time weak solutions. Finally, we prove that there is at least one dissipative solution for this simplified hyperbolic Ericksen-Leslie system.

• Wei Liu, Ziqing Xie, Wenfan Yi
Normalized Goldstein-type Local Minimax Method for Finding Multiple Unstable Solutions of Semilinear Elliptic PDEs

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 to be suitable for finding multiple unstable solutions of semilinear elliptic PDEs both in numerical implementation and theoretical analysis. Compared with the normalized Armijo-type local minimax method (NA-LMM), which was first introduced in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3):865--885, 2002] and then modified in [Z.Q. Xie, Y.J. Yuan, and J. Zhou, SIAM J. Sci. Comput., 34(1):A395--A420, 2012], our approach can prevent the step-size from being too small automatically and then ensure that the iterations make reasonable progress by taking full advantage of two inequalities. The feasibility of the NG-LMM is verified strictly. Further, the global convergence of the NG-LMM is proven rigorously under a weak assumption that the peak selection is only continuous. Finally, it is implemented to solve several typical semilinear elliptic boundary value problems on square or dumbbell domains for multiple unstable solutions and the numerical results indicate that this approach performs well.

• Jingrun Chen, Zhiwei Sun, Yun Wang, Lei Yang
A spin-wave solution to the Landau-Lifshitz-Gilbert equation

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, a spin wave for this equation is obtained based on the assumption that the spin wave maintains its periodicity in space when propagating at a varying velocity. In the absence of magnetic field, an explicit form of spin wave is provided. When a magnetic field is applied, the spin wave does not have such an explicit form but its stability is justified rigorously. Moreover, an approximate explicit solution is constructed with approximation error depending quadratically on the strength of magnetic field and being uniform in time.

• Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim, Jinwook Jung
Emergence of stochastic flocking for the discrete Cucker-Smale model with randomly switching topologies

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 for the law of switching times, we use the Poisson process and the geometric process and show that these two processes satisfy the required conditions in a given framework so that we have a stochastic flocking with probability one. As a corollary of our flocking analysis, we improve the earlier result [15] on the continuous C-S model.

• Feng Cheng, Ning Jiang, Yi-Long Luo

We study dissipative solutions to a 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 of global-in-time weak solutions. Finally, we prove that there is at least one dissipative solution for this simplified hyperbolic Ericksen-Leslie system.

• Weinan E, Chao Ma, Qingcan Wang
Rademacher Complexity and the Generalization Error of Residual Networks

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, which suggest that they are nearly optimal in the high dimensional setting. They are achieved by constraining the hypothesis space with an appropriately defined path norm such that the constrained space is large enough for the approximation error rates to be optimal and small enough for the estimation error rates to be optimal at the same time. Comparisons are made with other norm-based bounds.

• Ling-Bing He, Zheng-an Yao, Yu-Long Zhou
Asymptotic analysis of the Boltzmann equation with very soft potentials from angular cutoff to non-cutoff

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, we rigorously give the asymptotic formula between the solutions to cutoff and non-cutoff equations with an explicit convergence rate.

• Wei Liu, Ziqing Xie, Wenfan Yi
Normalized Goldstein-type Local Minimax Method for Finding Multiple Unstable Solutions of Semilinear Elliptic PDEs

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 to be suitable for finding multiple unstable solutions of semilinear elliptic PDEs both in numerical implementation and theoretical analysis. Compared with the normalized Armijo-type local minimax method (NA-LMM), which was first introduced in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3):865--885, 2002] and then modified in [Z.Q. Xie, Y.J. Yuan, and J. Zhou, SIAM J. Sci. Comput., 34(1):A395--A420, 2012], our approach can prevent the step-size from being too small automatically and then ensure that the iterations make reasonable progress by taking full advantage of two inequalities. The feasibility of the NG-LMM is verified strictly. Further, the global convergence of the NG-LMM is proven rigorously under a weak assumption that the peak selection is only continuous. Finally, it is implemented to solve several typical semilinear elliptic boundary value problems on square or dumbbell domains for multiple unstable solutions and the numerical results indicate that this approach performs well.

• Ruimeng Hu
Deep Fictitious Play for Stochastic Differential Games

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 \emph{deep fictitious play}, a multi-stage learning process. Specifically at each stage, we propose the strategy of letting individual player optimize her own payoff subject to the other players' previous actions, equivalent to solve N decoupled stochastic control optimization problems, which are approximated by DNNs. Therefore, the fictitious play strategy leads to a structure consisting of N DNNs, which only communicate at the end of each stage. The resulted deep learning algorithm based on fictitious play is scalable, parallel and model-free, {\it i.e.}, using GPU parallelization, it can be applied to any N-player stochastic differential game with different symmetries and heterogeneities ({\it e.g.}, existence of major players). We illustrate the performance of the deep learning algorithm by comparing to the closed-form solution of the linear-quadratic game. Moreover, we prove the convergence of fictitious play under appropriate assumptions, and verify that the convergent limit forms an open-loop Nash equilibrium. We also discuss the extensions to other strategies designed upon fictitious play and closed-loop Nash equilibrium in the end.

• Liqun Qi, Xinzhen Zhang, Yannan Chen
A Tensor Rank Theory and Maximum Full Rank Subtensors

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 we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

• Xiaofeng Yang, Liquan Mei
Efficient numerical scheme for the anisotropic modified phase-field crystal model with a strong nonlinear vacancy potential

In this paper, we consider numerical approximations for the anisotropic modified phase-field 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 scheme with unconditional energy stability, we combine the multiple scalar auxiliary variable (MSAV) approach with stabilization technique for achieving an efficient and linear numerical scheme, in which two new scalar auxiliary variables are introduced to reformulate the model and a linear stabilization term is added to enhance the stability and keep the required accuracy while using the large time steps. The scheme leads to decoupled linear equations with constant coefficients at each time step, and its unique solvability and unconditional energy stability are proved. Various numerical experiments are performed to show the accuracy, stability, and efficiency of the proposed scheme.

• Chunyan Huang, Boling Guo, Yi Heng
Scattering for 3D quantum Zakarov system in L2

In this paper, we study the Cauchy problem for quantum Zakharov system in three space dimension. 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.

• Iulian Cimpean, Andreea Grecu
The nonlinear Schrodinger equation with white noise dispersion on quantum graphs

We show that the nonlinear Schrodinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in L^2 once the free deterministic Schrodinger group satises 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 white noise dispersion is the scaling limit of the solution to the NLSE with random dispersion.

• Yajun Zhou
Two inequalities for convex equipotential surfaces

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 inequalities generalize a geometric conservation law for equipotential curves in dimension two, and offer solutions to two free boundary problems in three-dimensional electrostatics.

• Jianli Liu, Jingjie Wang, Manwai Yuen
Blowup for C^1 solutions of compressible Euler equations with time-dependent damping in R^n

In this paper, we will show the blowup phenomenon of solutions of the compressible Euler equations with time-dependent damping. Firstly, under the assumptions on the radially symmetric initial data and initial density contains in vacuum states, the singularity of the classical solutions will formed in finite time in Rn(n \ge 2). Furthermore, we can also find a sufficient condition for the functional of initial data such that smooth solution of the irrotational compressible Euler equations with time-dependent damping breaks down in finite time for all kinds of fractional coefficients in Rn(n \ge 1).

• Yuan Gao, Kay Kirkpatrick, Jeremy Louis Marzuola, Jonathan C. Mattingly, Katherine Newhall
Limiting Behaviors of high dimensional stochastic spin ensembles

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 when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the convergence of the M-H dynamics to the Harmonic map heat flow equation in two steps: First, with fixed lattice size and proper choice of proposal size in one M-H step, the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations (SDE). Second, with proper scaling of the inverse temperature in the Gibbs distribution and taking the lattice size to infinity, it will be shown that this SDE system converges to the deterministic Harmonic map heat flow equation. Our results are not unexpected, but show remarkable connections between the M-H steps and the SDE Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.

• Hiroyoshi Mitake, Hung V. Tran, Truong-Son Van
Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model

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 results under certain natural conditions on the initial data, and a nonconvergence result when such conditions fail.

• Chia-Yu Hsieh, Ho-Man Tai, Yong Yu
Singular Solutions to Some Semilinear Elliptic Equations: An Approach of Born-Infeld Approximation

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. Supposing that $P_1, ..., P_N$ are $N$ locations of singularities, we classify all these singular solutions by the order of $u$ near each $P_j$. Here $u$ satisfies (1.1) with some natural assumptions on the nonlinearity $f(e^u)$. To show the existence of singular solutions to (1.1), we introduce an inverse problem (1.10)--(1.12) and utilize a Born-Infeld approximation scheme. Ruling out a possible occurrence of bubbling phenomenon, we show that as the Born-Infeld parameter $b \rightarrow \infty$, solutions of (1.10) on a subdomain of $\Omega$ with finitely many holes can be used to approximate singular solutions to (1.1), provided that the size of each hole is carefully given and satisfies the constraint condition (1.12). Our work rigorously justifies the Born-Infeld-Higgs approximation to the abelian Maxwell-Higgs theory. When $b = 1$, we also find finite-energy solutions to the equation of Born-Infeld gauged harmonic maps, which have finitely many magnetic singularities in $\Omega$.

• Bojan Dimitrov Popov, Yuchen Hua
Invariant domain preserving central schemes for nonlinear hyperbolic systems

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. SIAM J. Anal., 54(4):2466-2489, 2016] adjusted to the finite volume framework. There are three novelties in this work. The first one is that any classical second-order central scheme can be modified to satisfy an invariant domain property of the first order scheme via a process which we call convex limiting. This is done by using convex flux limiting along the lines of [J.-L Guermond, B. Popov and I. Tomas. Comput. Methods Appl. Mech. Engrg., 347:143-175, 2019]. The second novelty is the design of a new second order method based on slope limiting only. The new local slope reconstruction technique is based on convex limiting so that the cell interface values are corrected to fit into a local invariant domain of the hyperbolic system. This new type of slope limiting depends on the hyperbolic system and to the best of our knowledge is the only one to guarantee local invariant domain preservation. Both schemes, flux and slope limiting based, are shown to be second order accurate for smooth solutions in the $L^\infty$-norm and robust in all test cases. The third novelty is a new second order method based on the MAPR limiter from [I. Christov and B. Popov. J. Comput. Phys., 227(11):5736-5757, 2008] and adaptive slope limiting in the spirit of [A. Kurganov, G. Petrova and B. Popov. SIAM Journal on Scientific Computing, 29(6):2381-2401, 2007] but based on an entropy commutator. This new method can be used as an underlying high-order method and combined with convex flux limiting to guarantee a local invariant domain property. The time stepping of all methods is done by using Strong Stability Preserving Runge-Kutta methods and the invariant domain property is proved under a standard CFL condition.

• Lihui Chai, James Hateley, Emmanuel Lorin, Xu Yang
On the Convergence of Frozen Gaussian Approximation for linear non-strictly hyperbolic systems

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 Numer. Anal., 57, 2383--2412, 2019]. However, the theory of convergence is still incomplete for non-strictly hyperbolic systems, where the latter can be interpreted as a diabatic (or more) coupling. In this paper, we establish the convergence theory for FGA for linear non-strictly hyperbolic systems, with an emphasis on the elastic wave equations and the Dirac system. Unlike the convergence theory of FGA for strictly linear hyperbolic systems, the key estimate lies in the boundness of intraband transitions in diabatic coupling.

• Boling Guo, Xinglong Wu
Global Existence of Solution in the Besov space to the Nonlinear Wave Equations in $\mathbb{R}^d$

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 $\mathbb{R}^d,\;d >3$. We first recall some useful lemmas in Besov spaces. Next, the local well-posedness of Eq.(1.1) is obtained in $\dot{B}_{2,1}^{\frac{d}{2}-\frac{1}{2}} \cap\dot{B}_{2,1}^{\frac{d}{2}}$, and a lower bound of blow-up rate of blow-up solution in the space is established. Finally, by construction of the space $\mathscr{X}_R(M)$, thanks to a priori estimates, we show the global solution in the Besov space for the Cauchy problem of Eq.(1.1) if the initial data is sufficiently small.

• Yong Chen, Jinqiao Duan, Hongjun Gao
Global well-posedness of the stochastic Camassa-Holm equation

We establish the existence of global martingale solutions of the stochastic Camassa-Holm equation in $H^1(\mr)$. The construction of the solution is based on the regularization method, the stopping times, the stochastic compactness method, the Jakubowski-Skorokhod theorem and the stochastic renormalized formulations. Under the regularization effect from a special noise, we also obtain the global existence and uniqueness of stochastic Camassa-Holm equation in $L^2(\mr)$.

• Kai Du, Zhen Wu, Detao Zhang
Dynkin Game For Callable-Puttable Convertible Bonds: The Valuation and Sensitivity Analysis

This paper studies the pricing problem for callable-puttable convertible bonds with call-put 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 bond. Based on that, we further obtain the valuation model of callable-puttable convertible bonds and investigate how the call and put clauses affect the value of convertible bonds. In addition, the sensitivity analysis of callable-puttable convertible bonds' price about some key parameters is considered in virtue of the comparison theorem of doubly reflected BSDEs and is verified by numerical simulations through an obstacle problem for a parabolic partial differential equation (PDE).