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Improved uniform error bounds of an exponential wave integrator method for the Klein–Gordon–Schrödinger equation with the small coupling constant Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Jiyong Li
Recently, the long-time numerical simulation and error analysis of PDEs with weak nonlinearity (or small potentials) become an interesting topic. However, the existing results of long-time error analysis mostly focus on the single equations. In this paper, for the Klein–Gordon–Schrödinger equation (KGSE) with a small coupling constant $\varepsilon \in (0,1]$, we propose an exponential wave integrator
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Spatial manifestations of order reduction in Runge–Kutta methods for initial boundary value problems Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Rodolfo Ruben Rosales, Benjamin Seibold, David Shirokoff, Dong Zhou
This paper studies the spatial manifestations of order reduction that occur when timestepping initial-boundary-value problems (IBVPs) with high-order Runge–Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers
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Approximate primal-dual fixed-point based langevin algorithms for non-smooth convex potentials Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Ziruo Cai, Jinglai Li, Xiaoqun Zhang
The Langevin algorithms are frequently used to sample the posterior distributions in Bayesian inference. In many practical problems, however, the posterior distributions often consist of non-differentiable components, posing challenges for the standard Langevin algorithms, as they require to evaluate the gradient of the energy function in each iteration. To this end, a popular remedy is to utilize
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Global strong solutions to the compressible magnetohydrodynamic equations with slip boundary conditions in a 3D exterior domain Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Yazhou Chen, Bin Huang, Xiaoding Shi
In this paper we study the initial-boundary-value problem for the barotropic compressible magnetohydrodynamic system with slip boundary conditions in three-dimensional exterior domain. We establish the global existence and uniqueness of classical solutions to the exterior domain problem with the regular initial data that are of small energy but possibly large oscillations with constant state as far
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Analysis and computation for the scattering problem of electromagnetic waves in chiral media Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Gang Bao, Lei Zhang
This paper considers an obstacle scattering problem in a chiral medium under circularly polarized oblique plane wave incidence, which can be represented as a combination of a left-circularly polarized plane wave and a right-circularly polarized one. We apply a reduced model problem with coupled oblique derivative boundary conditions, describing the cross-coupling effect of electric and magnetic fields
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Stability of contact lines in 2D stationary Benard convection Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Yunrui Zheng
We consider the evolution of contact lines for thermal convection of viscous fluids in a two-dimensional open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of
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Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Ben Duan, Haoran Zheng
The Euler–Poisson equations with varying background charges in finitely long flat nozzles are investigated, for which two and only two transonic shock solutions are constructed. In href{https://dx.doi.org/10.4310/CMS.2012.v10.n2.a1}{[\textrm{T. Luo and Z.P. Xin, Commun. Math. Sci., 10:419–462, 2012}]}, Luo and Xin established the wellposedness of steady Euler–Poisson equations for the constant background
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Global existence and asymptotic behavior of the full Euler system with damping and radiative effects in $\mathbb{R}^3$ Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Shijin Deng, Wenjun Wang, Feng Xie, Xiongfeng Yang
In this paper, we study the global existence and the large-time behavior of solutions to the Cauchy problem of the full Euler system with damping and radiative effects around some constant equilibrium states. It is well-known that the solutions may blow up in finite time without the additional damping and radiative effects, and the global existence of the solutions obtained in this paper shows that
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Autonomous vehicles driving traffic: The Cauchy problem Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Mauro Garavello, Francesca Marcellini
This paper deals with the Cauchy problem for a PDE‑ODE model, where a system of two conservation laws, namely the Two-Phase macroscopic model proposed in [Rinaldo M. Colombo, Francesca Marcellini, and Michel Rascle, $\href{https://doi.org/10.1137/090752468}{\textrm{SIAM J. Appl. Math., 70(7):2652–2666, 2010}}$], is coupled with an ordinary differential equation describing the trajectory of an autonomous
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Hydrodynamic traffic flow models including random accidents: A kinetic derivation Commun. Math. Sci. (IF 1.0) Pub Date : 2024-03-04 Felisia Angela Chiarello, Simone Göttlich, Thomas Schillinger, Andrea Tosin
We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared
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Reproducing activation function for deep learning Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Senwei Liang, Liyao Lyu, Chunmei Wang, Haizhao Yang
We propose reproducing activation functions (RAFs) motivated by applied and computational harmonic analysis to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural
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Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Xinxiang Bian, Lingling Xie
This paper studies the large-time asymptotic stability and optimal time-decay rate of viscous contact wave to one-dimensional compressible Navier–Stokes equations. We prove that one-dimensional compressible Navier–Stokes equations are asymptotically stable for viscous contact wave with arbitrarily large strength, under large initial perturbations. The time optimal decay rate of viscous contact wave
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Linearized stability of planar rarefaction wave for 3D gas dynamics in thermal nonequilibrium Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Hua Zhong
For the three-dimensional gas flow in vibrational nonequilibrium, the linearized stability of the planar rarefaction waves is obtained in this paper in terms of the rarefaction wave strength is small enough. The main feature of the problem is that the $L^2$-norm of the perturbations may grow in time.
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Lifespan estimates of solutions to the weakly coupled system of semilinear wave equations with space dependent dampings Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Sen Ming, Han Yang, Xiongmei Fan
$\def \lv{\lvert}\def\rv{\rvert}$ This paper is devoted to investigating the weakly coupled system of semilinear wave equations with space dependent dampings and power nonlinearities ${\lv v \rv}^p, {\lv u \rv}^q$, derivative nonlinearities ${\lv v_t \rv}^p, {\lv u_t \rv}^q$, mixed nonlinearities ${\lv v \rv}^q, {\lv u_t \rv}^p$, combined nonlinearities ${\lv v_t \rv}^{p_1} + {\lv v \rv}^{q_1}, {\lv
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Unified asymptotic analysis and numerical simulations of singularly perturbed linear differential equations under various nonlocal boundary effects Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Xianjin Chen, Chiun-Chang Lee, Masashi Mizuno
While being concerned with a singularly perturbed linear differential equation subject to integral boundary conditions, the exact solutions, in general, cannot be specified, and the validity of the maximum principle is unassurable. Hence, a problem arises: how to identify the boundary asymptotics more precisely? We develop a rigorous asymptotic method involving recovered boundary data to tackle the
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On global solutions to the inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent coefficients Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Bijun Zuo
In this paper, we study the initial-boundary value problem for the full inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent viscosity and heat conductivity coefficients. The viscosity coefficient may be degenerate in the sense that it may vanish in the region of absolutely zero temperature. Our main result is to prove the global existence of large weak solutions to such
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Uniqueness of global weak solutions to the frame hydrodynamics for biaxial nematic phases in $\mathbb{R}^2$ Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Sirui Li, Chenchen Wang, Jie Xu
We consider the hydrodynamics for biaxial nematic phases described by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. We prove the uniqueness of global weak solutions to the Cauchy problem of the frame hydrodynamics in dimension two. The proof is mainly based on the suitable weaker energy estimates within the Littlewood–Paley analysis. We take full advantage
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Weak solutions for a modified degenerate Cahn–Hilliard model for surface diffusion Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Xiaohua Niu, Yang Xiang, Xiaodong Yan
In this paper, we study the weak solutions of a modified degenerate Cahn–Hilliard type model for surface diffusion. With degenerate phase-dependent diffusion mobility and additional stabilizing function, this model is able to give the correct sharp interface limit. We introduce a notion of weak solutions for the nonlinear model. The existence of such solutions is obtained by approximations of the proposed
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A quadratic spline projection method for computing stationary densities of random maps Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Azzah Alshekhi, Jiu Ding, Noah Rhee
We propose a quadratic spline projection method that computes stationary densities of random maps with position-dependent probabilities. Using a key variation inequality for the corresponding Markov operator, we prove the norm convergence of the numerical scheme for a family of random maps consisting of the Lasota–Yorke class of interval maps. The numerical experimental results show that the new method
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Global existence of perturbed Navier–Stokes system around Landau solutions with slowly decaying oscillation Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Jiayan Wu, Cuili Zhai, Jingjing Zhang, Ting Zhang
In this paper, we consider the perturbed Navier–Stokes system around the Landau solutions. Using the energy method and the continuation method, we show the global existence of the $L^2$ local energy solution for the perturbed Navier–Stokes system with the oscillation decay initial data $v_0 \in E^2_{\sigma} + L^3_{\operatorname{uloc}} \,$.
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A stochastic Galerkin method for the direct and inverse random source problems of the Helmholtz equation Commun. Math. Sci. (IF 1.0) Pub Date : 2024-02-01 Ning Guan, Dingyu Chen, Peijun Li, Xinghui Zhong
This paper investigates a novel approach for solving both the direct and inverse random source problems of the one-dimensional Helmholtz equation with additive white noise, based on the generalized polynomial chaos (gPC) approximation. The direct problem is to determine the wave field that is emitted from a random source, while the inverse problem is to use the boundary measurements of the wave field
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Improved uniform error bound on the time-splitting method for the long-time dynamics of the fractional nonlinear Schrödinger equation Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Yue Feng, Ying Ma
We establish the improved uniform error bound on the time-splitting Fourier pseudospectral (TSFP) method for the long-time dynamics of the generalized fractional nonlinear Schrödinger equation (FNLSE) with $O(\varepsilon^2)$-nonlinearity, where $\varepsilon \in (0,1]$ is a dimensionless parameter. Numerically, we discretize the FNLSE by the second-order Strang splitting method in time and Fourier pseudospectral
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A general framework for nonlocal Neumann problems Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Guy Foghem, Moritz Kassmann
Within the framework of Hilbert spaces, we solve nonlocal problems in bounded domains with prescribed conditions on the complement of the domain. Our main focus is on the inhomogeneous Neumann problem in a rather general setting. We also study the transition from exterior value problems to local boundary value problems. Several results are new even for the fractional Laplace operator. The setting also
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Epidemic dynamics and wealth inequality under two feedback control strategies Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Lingling Wang, Chong Lai
A multi-agent wealth exchange model, which considers a varying trading propensity and a control of wealth inequality, is adopted to investigate the wealth distribution under infectious disease. Using the feedback control method, two saturated nonlinear incidence rates are obtained to explore the impact of the government contact control measures on epidemic dynamics and wealth distribution. We find
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Description of random level sets by polynomial chaos expansions Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Markus Bambach, Stephan Gerster, Michael Herty, Aleksey Sikstel
We present a novel approach to determine the evolution of level sets under uncertainties in their velocity fields. This leads to a stochastic description of level sets. To compute the quantiles of random level sets, we use the stochastic Galerkin method for a hyperbolic reformulation of the equations for the propagation of level sets. A novel intrusive Galerkin formulation is presented and proven to
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Global mild solutions of the non-cutoff Vlasov–Poisson–Boltzmann system Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Hao Wang, Guangqing Wang
This paper is concerned with the Cauchy problem on the Vlasov–Poisson–Boltzmann system in the torus domain. The Boltzmann collision kernel is assumed to be angular non-cutoff with $0 \leq \gamma \lt 1$ and $1/2 \leq s \lt 1$, where $\gamma, s$ are two parameters describing the kinetic and angular singularities, respectively. We obtain the global-in-time unique mild solutions, and prove that the solutions
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Energy method for the Boltzmann equation of monatomic gaseous mixtures Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Laurent Boudin, Bérénice Grec, Milana Pavić-Čolić, Srboljub Simić
In this paper, we present an energy method for the system of Boltzmann equations in the multicomponent mixture case, based on a micro-macro decomposition. More precisely, the perturbation of a solution to the Boltzmann equation around a global equilibrium is decomposed into the sum of a macroscopic and a microscopic part, for which we obtain a priori estimates at both lower and higher orders. These
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A new priori error estimation of nonconforming element for two-dimensional linearly elastic shallow shell equations Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Rongfang Wu, Xiaoqin Shen, Qian Yang, Shengfeng Zhu
In this paper, we mainly propose a new priori error estimation for the two-dimensional linearly elastic shallow shell equations, which rely on a family of Kirchhoff–Love theories. As the displacement components of the points on the middle surface have different regularities, the nonconforming element for the discretization shallow shell equations is analysed. Then, relying on the enriching operator
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Unipolar Euler–Poisson equations with time-dependent damping: blow-up and global existence Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Jianing Xu, Shaohua Chen, Ming Mei, Yuming Qin
This paper is concerned with the Cauchy problem for one-dimensional unipolar Euler–Poisson equations with time-dependent damping, where the time-asymptotically degenerate damping in the form of $-\dfrac{\mu}{(1+t)^\lambda} \rho \mu$ for $\lambda \gt 0$ with $\mu \gt 0$ plays a crucial role for the structure of solutions. The main issue of the paper is to investigate the critical case with $\lambda=1$
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On the properties of affine solutions of cold plasma equations Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Olga S. Rozanova, Marko K. Turzynsky
We study the affine solutions of the equations of plane oscillations of cold plasma, which, under the assumption of electrostaticity, correspond to the Euler–Poisson equations in the repulsive case. It is proved that the zero equilibrium state of the cold plasma equations, both with and without the assumption of electrostaticity, is unstable in the class of all affine solutions. It is also shown that
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A variational approach for price formation models in one dimension Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Yuri Ashrafyan, Tigran Bakaryan, Diogo Gomes, Julian Gutierrez
In this paper, we study a class of first-order mean-field games (MFGs) that model price formation. Using Poincaré lemma, we eliminate one of the equations of the MFGs system and obtain a variational problem for a single function. We prove the uniqueness of the solutions to the variational problem and address the existence of solutions by applying relaxation arguments. Moreover, we establish a correspondence
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Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity Commun. Math. Sci. (IF 1.0) Pub Date : 2023-12-07 Xia Ye, Mingxuan Zhu
This paper is concerned with the initial value problem of the three-dimensional nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity. We prove the existence of global strong solution on $\mathbb{R}^3 \times (0,\infty)$ under the initial norm ${\lVert u_0 \rVert}_{\dot{H}^\alpha} + {\lVert \nabla d_0 \rVert}_{\dot{H}^\alpha} (1/2 \lt \alpha \leq 1)$ being suitably
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The Neumann boundary condition for the two-dimensional Lax–Wendroff scheme Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Antoine Benoit, Jean-François Coulombel
We study the stability of the two-dimensional Lax–Wendroff scheme with a stabilizer that approximates solutions to the transport equation. The problem is first analyzed in the whole space in order to show that the so-called energy method yields an optimal stability criterion for this finite difference scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the
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A Cahn-Hilliard model coupled to viscoelasticity with large deformations Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Abramo Agosti, Pierluigi Colli, Harald Garcke, Elisabetta Rocca
We propose a new class of phase field models coupled to viscoelasticity with large deformations, obtained from a diffuse interface mixture model composed by a phase with elastic properties and a liquid phase. The model is formulated in the Eulerian configuration and it is derived by imposing the mass balance for the mixture components and the momentum balance that comes from a generalized form of the
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Gamma convergence for the de Gennes–Cahn–Hilliard energy Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Shibin Dai, Joseph Renzi, Steven M. Wise
The degenerate de Gennes–Cahn–Hilliard (dGCH) equation is a model for phase separation which may more closely approximate surface diffusion than others in the limit when the thickness of the transition layer approaches zero. As a first step to understand the limiting behavior, in this paper we study the $\Gamma$-limit of the dGCH energy. We find that its $\Gamma$-limit is a constant multiple of the
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Optimal large-time behavior of the compressible Phan–Thein–Tanner model Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Yin Li, Ruiying Wei, Guochun Wu, Zheng-An Yao
In this paper, we investigate global existence and optimal decay rates of strong solutions to the three dimensional compressible Phan–Thein–Tanner model. We prove the global existence of the solutions by the standard energy method under the small initial data assumptions. Furthermore, if the initial data belong to $L^1 (\mathbb{R}^3)$, we establish the optimal time decay rates of the solution as well
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Operator shifting for model-based policy evaluation Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Xun Tang, Lexing Ying, Yuhua Zhu
In model-based reinforcement learning, the transition matrix and reward vector are often estimated from random samples subject to noise. Even if the estimated model is an unbiased estimate of the true underlying model, the value function computed from the estimated model is biased. We introduce an operator shifting method for reducing the error introduced by the estimated model. When the error is in
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A non-equilibrium multi-component model with miscible conditions Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Jean Bussac
This paper concerns the study of a full non-equilibrium model for a compressible mixture of any number of phases. Miscible conditions are considered in one phase, which lead to non-symmetric constraints on the statistical fractions. These models are subject to the choice of interfacial and source terms. We show that under a standard assumption on the interfacial velocity, the interfacial pressures
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A low Mach two-speed relaxation scheme for the compressible Euler equations with gravity Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Claudius Birke, Christophe Chalons, Christian Klingenberg
We present a numerical approximation of the solutions of the Euler equations with a gravitational source term. On the basis of a Suliciu type relaxation model with two relaxation speeds, we construct an approximate Riemann solver, which is used in a first order Godunov-type finite volume scheme. This scheme can preserve both stationary solutions and the low Mach limit to the corresponding incompressible
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Global strong solutions to the compressible Navier–Stokes system with potential temperature transport Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Xiaoping Zhai, Yongsheng Li, Fujun Zhou
We study the global strong solutions to the compressible Navier–Stokes system with potential temperature transport in $\mathbb{R}^n$. Different from the Navier–Stokes–Fourier system, the pressure being a nonlinear function of the density and the potential temperature, we can not exploit the special quasi-diagonalization structure of this system to capture any dissipation of the density. Some new ideas
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Stability of planar rarefaction wave for viscous vasculogenesis model Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Qingqing Liu, Yuxiu Tian
In this paper, we are concerned with a two-dimensional quasi-linear hyperbolicparabolic-elliptic system modelling vasculogenesis. We first derive a two-dimensional inviscid system as the asymptotic equations in large time by ignoring all the viscous terms. Then we show that this inviscid system admits a planar rarefaction wave when the pressure function satisfies some suitable structure conditions
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Determination for the 2D incompressible Navier–Stokes equations in Lipschitz domain Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Xin-Guang Yang, Meng Hu, To Fu Ma, Jinyun Yuan
The number of determining modes is estimated for the 2D Navier–Stokes equations subject to an inhomogeneous boundary condition in Lipschitz domains by using an appropriate set of points in the configuration space to represent the flow by virtue of the Grashof number and the measure of Lipschitz boundary based on a stream function and some delicate estimates. The asymptotic determination via finite
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Flocking behavior of the Cucker–Smale model under a general digraph on the infinite cylinder Commun. Math. Sci. (IF 1.0) Pub Date : 2023-11-15 Xiaoyu Li, Lining Ru
In this paper, we generalize the Cucker–Smale model under a general digraph on the infinite cylinder with the help of the Lie group structure of the infinite cylinder and study the flocking behavior of this model. We show that for $0 \leq \beta \lt 1 / (2h)$ unconditional flocking occurs, where h is the shortest height of the spanning trees of the digraph, and conditional flocking occurs for $\beta
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A kind of time-inconsistent corporate international investment problem with discontinuous cash flow Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Haiyang Wang, Zhen Wu
In this paper, we study a kind of time-inconsistent corporate international investment problem with discontinuous cash flow in consideration of the exchange risk, information costs and taxes. The time-inconsistency arises from the presence of investment risk in the cost functional. We first define the time-consistent equilibrium strategy for this problem and establish a sufficient condition for it
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Energy stability of variable-step L1-type schemes for time-fractional Cahn–Hilliard model Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Bingquan Ji, Xiaohan Zhu, Hong-Lin Liao
The positive definiteness of discrete time-fractional derivatives is fundamental to the numerical stability (in the energy sense) for time-fractional phase-field models. A novel technique is proposed to estimate the minimum eigenvalue of discrete convolution kernels generated by the non-uniform L1, half-grid based L1 and time-averaged L1 formulas of the fractional Caputo’s derivative. The main discrete
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Recovery of a distributed order fractional derivative in an unknown medium Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Bangti Jin, Yavar Kian
In this work, we study an inverse problem of recovering information about the weight in distributed-order time-fractional diffusion from the observation at one single point on the domain boundary. In the absence of an explicit knowledge of the medium, we prove that the one-point observation can uniquely determine the support bound of the weight. The proof is based on asymptotics of the data, analytic
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High friction limits of Euler–Navier–Stokes–Korteweg equations for multicomponent models Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Giada Cianfarani Carnevale, Corrado Lattanzio
In this paper we analyze the high friction regime for the Navier–Stokes–Korteweg equations for multicomponent systems. According to the shape of the mixing and friction terms, we shall perform two limits: the high friction limit toward an equilibrium system for the limit densities and the barycentric velocity, and, after an appropriate time scaling, the diffusive relaxation toward parabolic, gradient
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Dual quaternion matrices in multi-agent formation control Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Liqun Qi, Xiangke Wang, Ziyan Luo
Three kinds of dual quaternion matrices associated with the mutual visibility graph, namely the relative configuration adjacency matrix, the logarithm adjacency matrix and the relative twist adjacency matrix, play important roles in multi-agent formation control. In this paper, we study their properties and applications. We show that the relative configuration adjacency matrix and the logarithm adjacency
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Emergence of phase-locked states for a deterministic and stochastic Winfree model with inertia Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Myeongju Kang, Marco Rehmeier
We study the emergence of phase-locking for Winfree oscillators under the effect of inertia. It is known that in a large coupling regime, oscillators governed by the deterministic second-order Winfree model with inertia converge to a unique equilibrium. In contrast, in this paper we show the asymptotic emergence of non-trivial synchronization in a suitably small coupling regime. Moreover, we study
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Error estimates of the sav method for the coupled Cahn–Hilliard system in copolymer/homopolymer mixtures Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Jin Huang, Guanghua Ji
In this paper, we consider the fully discrete scheme based on the scalar auxiliary variable (SAV) approach in time and the Fourier spectral method in space, for solving the phase-field model of the blend consisting of AB diblock copolymers and C homopolymers. We establish the error estimates of the numerical scheme rigorously, and show that the fully discrete scheme converges with order $O(\tau^2 +
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Fully discrete low-regularity integrator for the Korteweg–de Vries equation Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Yongsheng Li, Fangyan Yao
In this paper we propose a fully discrete low-regularity integrator for the Korteweg-de Vries equation on the torus. This is an explicit scheme and can be computed with a complexity of $\mathcal{O}(N \log N)$ operations by fast Fourier transform, where $N$ is the degrees of freedom in the spatial discretization. We prove that the scheme is first-order convergent in both time and space variables in
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Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Piotr B. Mucha, Wojciech S. Ożański
We consider a pressureless hydrodynamic model of collective behaviour, which is concerned with a density function $\rho$ and a velocity field $v$ on the torus, and is described bythe continuity equation for $\rho, \rho_t + \mathrm{div}(v \rho) = 0$, and a compressible hydrodynamic equation for $v, \rho v_t + \rho v \cdot \nabla v - \Delta v = − \rho \nabla K\rho$ with a forcing modelling collective
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Dissipative solutions to the compressible isentropic Navier–Stokes equations Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Liang Guo, Fucai Li, Cheng Yu
The dissipative solutions to the compressible isentropic Navier–Stokes equations are introduced in this paper. This notion was inspired by the concept of dissipative solutions to the incompressible Euler equations of Lions ($\href{https://global.oup.com/academic/product/mathematical-topics-in-fluid-mechanics-9780199679218}{[\textrm{P.-L. Lions, Oxford Science Publication, Oxford, 1996}]}$, Section
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Existence of positive solutions for a class of quasilinear elliptic equations with parameters Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Yinbin Deng, Youjun Wang
This paper is devoted to investigating the existence of positive solutions for a class of parameter-dependent quasilinear elliptic equations\[-\Delta u+V(x)u-\frac{\gamma u}{2\sqrt{1+u^2}}\Delta\sqrt{1+u^2}= \lambda |u|^{p-2}u,\ \ u\in H^1(\mathbb{R}^N),\]where $\gamma,\lambda$ are positive parameters, $N\ge 3$. For a trapping potential $V(x)$ and $p\in (2,2^\ast)$, by controlling the range of $\gamma$
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On connection among quantum-inspired algorithms of the Ising model Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Bowen Liu, Kaizhi Wang, Dongmei Xiao, Zhan Yu
Various combinatorial optimization problems can be reduced to find the minimizer of an Ising model without external fields. This Ising problem is NP‑hard and discrete. It is an intellectual challenge to develop algorithms for solving the problem. Over the past decades, many quantum and classical computations have been proposed from physical, mathematical or computational views for finding the minimizer
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Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump Commun. Math. Sci. (IF 1.0) Pub Date : 2023-10-09 Renata Bunoiu, Karim Ramdani, Claudia Timofte
We study the homogenization of a scalar problem posed in a composite medium made up of two materials, a positive and a negative one. An important feature is the presence of a flux jump across their oscillating interface. The main difficulties of this study are due to the sign-changing coefficients and the appearance of an unsigned surface integral term in the variational formulation. A proof by contradiction
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Global small solutions to heat conductive compressible nematic liquid crystal system: smallness on a scaling invariant quantity Commun. Math. Sci. (IF 1.0) Pub Date : 2023-09-22 Jinkai Li, Qiang Tao
In this paper, we consider the Cauchy problem to the three dimensional heat conducting compressible nematic liquid crystal system in the presence of vacuum and with vacuum far fields. Global well-posedness of strong solutions is established under the condition that the scaling invariant quantity\begin{flalign*}(\|\rho_0\|_\infty+1)\big[\|\rho_0\|_3+(\|\rho_0\|_\infty+1)^2(\|\sqrt{\rho_0}u_0\|_2^2+
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Sharp interface limit for compressible Navier–Stokes/Allen–Cahn system with shock wave Commun. Math. Sci. (IF 1.0) Pub Date : 2023-09-22 Yunkun Chen, Bin Huang, Xiaoding Shi
In this paper, the sharp interface limit for the diffusion interface model system of immiscible two-phase flow called compressible Navier–Stokes/Allen–Cahn system is studied in one dimension. The results show that, for the initial perturbations with small energy but possibly large oscillations of shock wave solutions, and the strength of initial phase field is allowed to vary arbitrarily within its
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Discrete perturbed gradient flow and its application Commun. Math. Sci. (IF 1.0) Pub Date : 2023-09-22 Lingzhi Hao, Xiongtao Zhang
We study discrete dynamical system with perturbed gradient flow structure and its related applications. We prove that states with uniform bound will eventually converge to an equilibrium state, where Łojasiewicz inequality plays an important role. Moreover, the convergence rate is uniform with respect to the mesh size, which implies uniform transition from discrete time model to continuous time model
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Uniqueness of weak solutions to the Boussinesq equations with fractional dissipation Commun. Math. Sci. (IF 1.0) Pub Date : 2023-09-22 Ruihong Ji, Dan Li, Jiahong Wu
This paper examines the existence and uniqueness of weak solutions to the ddimensional Boussinesq equations with fractional dissipation $(-\Delta)^{\alpha}u$ and fractional thermal diffusion $(-\Delta)^{\beta}\theta$. The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting $u\in L^{\infty}(0