Article
Mathematics, Applied
Yuhui Chen, Minling Li, Qinghe Yao, Zheng-an Yao
Summary: In this paper, the large-time behavior of strong solutions to one-dimensional compressible isentropic magnetohydrodynamic equations near a stable equilibrium is investigated. A new energy estimate and upper and lower bounds are established for the solutions. The results are also applicable to compressible Navier-Stokes equations.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Mathematics, Applied
Weixuan Shi, Jianzhong Zhang
Summary: This paper is dedicated to optimal time-decay estimates of global strong solutions near constant equilibrium to the compressible magnetohydrodynamic (MHD) equations in critical Besov spaces. A new low-frequency assumption is claimed to play a key role in the large-time behavior of solutions. The proof relies mainly on sharp time-weighted energy estimates for solutions with both low and high frequencies.
APPLICABLE ANALYSIS
(2021)
Article
Mathematics, Applied
Weixuan Shi
Summary: This paper investigates the large-time asymptotic behavior of global strong solutions to the compressible Navier-Stokes-Poisson equations near constant equilibrium in the critical L-p framework. The proof relies on the pure energy argument without spectral analysis, improving upon previous results by removing the usual smallness assumption on low frequencies of initial data.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Limei Zhu, Ruizhao Zi
Summary: The time decay rates of the smooth solution to the compressible MHD system on T-3 were considered, with the conclusion that all derivatives of the smooth solution decay exponentially fast to equilibrium when the density and magnetic field are uniformly bounded. A global smooth solution obeying these decay rates was also constructed, with small energy but potentially large oscillations.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Fuyi Xu, Meiling Chi, Jiqiang Jiang, Yujun Cui
Summary: This paper revisits the optimal time decay rates of classical solutions to the 3D compressible Navier-Stokes equations, obtaining conditions for the optimal time decay rates of the solution and its first order derivative in L-2-norm. Compared with previous works, this paper simplifies the conditions but still requires smallness of initial perturbations in specific norms.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2021)
Article
Mathematics, Applied
Haifeng Shang
Summary: This paper investigates the large time behavior of solutions to the n-dimensional generalized magnetohydrodynamic equations. By considering initial data in negative Sobolev or Besov spaces, we establish upper and lower bounds for optimal decay estimates of the solutions and their higher order derivatives. The lower bound for the decay estimate provides a positive answer to the problem proposed in Jiang et al. (J Math Fluid Mech 22:9, 2020). Furthermore, non-uniform decay and the lower and upper bounds for optimal decay rates of the weak solutions are obtained.
JOURNAL OF MATHEMATICAL FLUID MECHANICS
(2023)
Article
Mathematics, Applied
Hai-Liang Li, Ling-Yun Shou
Summary: In this paper, the Cauchy problem of the multidimensional compressible Navier-Stokes-Euler system for two-phase flow motion is considered. The system consists of the isentropic compressible Navier-Stokes equations and the isothermal compressible Euler equations coupled with each other through a relaxation drag force. The local existence and uniqueness of the strong solution for general initial data in a critical homogeneous Besov space is established, and the global existence of the solution is proven if the initial data are a small perturbation of the equilibrium state. Moreover, the optimal time-decay rates of the global solution toward the equilibrium state are obtained under an additional condition on the regularity of the initial perturbation.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Juan Wang, Yinghui Zhang
Summary: This work investigates the optimal decay rates for higher-order spatial derivatives of the compressible viscous quantum magnetohydrodynamic model. The findings suggest that the decay rates in this model are faster than those in existing models.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics
Lintao Ma, Juan Wang, Yinghui Zhang
Summary: This work investigates the optimal decay rates of higher-order derivatives of solutions to the 3D compressible Navier-Stokes equations with large initial data. The main findings are twofold: firstly, it is shown that the second-order spatial derivative of the solution converges to zero at a certain rate for specific initial data, improving upon previous results. Secondly, if additional low-frequency assumptions are satisfied by the initial data, the optimal lower decay rates of the first- and second-order spatial derivatives of the solution are obtained, which are new compared to prior works.
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
(2022)
Article
Mathematics
Yazhou Chen, Hai-Liang Li, Houzhi Tang
Summary: In this paper, the global existence and long-time behavior of the solution to the non-isentropic compressible Navier-Stokes-Allen-Cahn (NSAC) system describing the motion of a two-phase immiscible viscous compressible heat-conducting flow are studied. The global well-posedness of the Cauchy problem is established and the decay rates of the phase field, density, velocity, and temperature are analyzed. The results provide a foundation for further research.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Shengbin Fu, Wenting Huang, Weiwei Wang
Summary: In this paper, we establish the optimal temporal decay rates of all-order spatial derivatives for the Cauchy problem of three-dimensional compressible viscoelastic flows under weaker initial conditions. The main contribution of this paper is the use of spectral analysis and energy methods to obtain optimal decay estimates for the highest-order derivatives of the solution, which extends the previous work [X. Hu and G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal. 45 (2013) 2815-2833] that only considered lower-order derivative estimates.
ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Suhua Lai, Jiahong Wu, Xiaojing Xu, Jianwen Zhang, Yueyuan Zhong
Summary: This paper investigates the impact of Boussinesq systems on buoyancy-driven fluids, finding that temperature can stabilize fluids and cause velocity to decay over time, with optimal decay rates obtained.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Mathematics
Guochun Wu, Han Wang, Yinghui Zhang
Summary: The paper focuses on the Cauchy problem of the 3D compressible Navier-Stokes-Poisson system. It aims to prove the optimal decay rates of higher spatial derivatives of the solution and investigate the influences of the electric field on the qualitative behaviors of the solution. The results show that the density and high frequency part of the momentum have the same L-2 decay rates as the compressible Navier-Stokes equation and heat equation, but the L-2 decay rate of the momentum is slower due to the effect of the electric field.
ELECTRONIC RESEARCH ARCHIVE
(2021)
Article
Mathematics
Zhouping Xin, Jiang Xu
Summary: This study focuses on the large time behavior of solutions to the barotropic compressible Navier-Stokes equations in R-d (d >= 2). A pure energy argument has been developed in a more general L-p framework, leading to optimal time-decay rates and removing the smallness of low frequencies of initial data required in previous studies.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Leilei Tong, Ronghua Pan, Zhong Tan
Summary: This study focuses on compressible micropolar fluids system in three-dimensional space, examining the asymptotic behavior of the solution to the Cauchy problem near the constant equilibrium state with sufficiently small initial perturbation. Under certain assumptions of the initial data, it is shown that the solution converges to its constant equilibrium state at the exact same L-2-decay rates as the linearized equations, demonstrating optimal convergence rates. The proof is based on spectral analysis of the semigroup generated by the linearized equations and nonlinear energy estimates.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Torsten Lindstrom
Summary: This paper aims to analyze the mechanism for the interplay of deterministic and stochastic models in contagious diseases. Deterministic models usually predict global stability, while stochastic models exhibit oscillatory patterns. The study found that evolution maximizes the infectiousness of diseases and discussed the relationship between herd immunity concept and vaccination programs.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Dong Deng, Hongxun Wei
Summary: This paper investigates the existence and nonexistence of time-periodic traveling waves for a diffusive influenza model with treatment and seasonality. By utilizing the next generation operator theory and Schauder's fixed point theorem, the conditions for the existence of time-periodic traveling wave solutions are obtained, along with the proof of nonexistence in certain cases and exponential decay for waves with critical speed.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Xuan Ma, Yating Wang
Summary: In this paper, the dynamics of a rarefied gas in a finite channel is studied, specifically focusing on the phenomenon of Couette flow. The authors demonstrate that the unsteady Couette flow for the Boltzmann equation converges to a 1D steady state and derive the exponential time decay rate. The analysis holds for all hard potentials.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Meng Zhao
Summary: In this paper, a reaction-diffusion waterborne pathogen model with free boundary is studied. The existence of a unique global solution is proved, and the longtime behavior is analyzed through a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are obtained, which differs from the previous results by Zhou et al. (2018) stating that the epidemic will spread when the basic reproduction number is larger than 1.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Gulsemay Yigit, Wakil Sarfaraz, Raquel Barreira, Anotida Madzvamuse
Summary: This study presents theoretical considerations and analysis of the effects of circular geometry on the stability of reaction-diffusion systems with linear cross-diffusion on circular domains. The highlights include deriving necessary and sufficient conditions for cross-diffusion driven instability and computing parameter spaces for pattern formation. Finite element simulations are also conducted to support the theoretical findings. The study suggests that linear cross-diffusion coupled with reaction-diffusion theory is a promising mechanism for pattern formation.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Miaoqing Tian, Lili Han, Xiao He, Sining Zheng
Summary: This paper studies the attraction-repulsion chemotaxis system of two-species with two chemical substances. The behavior of solutions is determined by the interactions among diffusion, attraction, repulsion, logistic sources, and nonlinear productions in the system. The paper provides conditions for the global boundedness of solutions.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Michal Borowski, Iwona Chlebicka, Blazej Miasojedow
Summary: This article provides a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff-Havin-Maz'ya type. It characterizes the potentials that are bounded between rearrangement invariant spaces via a one-dimensional inequality of Hardy-type. By controlling very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, the special case of the mentioned potential infers the local regularity properties of solutions in rearrangement invariant spaces for prescribed classes of data.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Young-Pil Choi, Jinwook Jung
Summary: This study investigates the global-in-time well-posedness of the pressureless Euler-alignment system with singular communication weights. A global-in-time bounded solution is constructed using the method of characteristics, and uniqueness is obtained via optimal transport techniques.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Chuangxia Huang, Xiaodan Ding
Summary: In this paper, a diffusive Mackey-Glass model with distinct diapause and developmental delays is proposed based on the diapause effect. Some sufficient conditions for the existence of traveling wave fronts are obtained by constructing appropriate upper and lower solutions and employing inequality techniques. Two numerical examples are provided to demonstrate the reliability and feasibility of the proposed model.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Hongxing Zhao
Summary: This paper investigates the flow of fluid through a thin corrugated domain saturated with porous medium, governed by the Navier-Stokes model. Asymptotic models are derived by comparing the relation between a and the size of the periodic cylinders. The homogenization technique based on the generalized Poincare inequality is used to prove the main results.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Evgenii S. Baranovskii, Roman V. Brizitskii, Zhanna Yu. Saritskaia
Summary: This paper proves the solvability of optimal control problems for both weak and strong solutions of a boundary value problem associated with the nonlinear reaction-diffusion-convection equation with variable coefficients. In the case of strong solutions, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of solvability of the corresponding boundary value problems and the qualitative analysis of their solution properties. The paper establishes existence results for weak solutions with large data, the maximum principle, and local existence and uniqueness of a strong solution. Furthermore, an optimal feedback control problem is considered, and sufficient conditions for its solvability in the class of weak solutions are obtained using methods of the theory of topological degree for set-valued perturbations.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Antonia Chinni, Beatrice Di Bella, Petru Jebelean, Calin Serban
Summary: This article focuses on the multiplicity of solutions for differential inclusions involving the p-biharmonic operator, applying a variational approach and relying on non-smooth critical point theory.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhong Tan, Saiguo Xu
Summary: This paper investigates the Rayleigh-Taylor instability of three-dimensional inhomogeneous incompressible Euler equations with damping in a horizontal slab. It is shown that the Euler system with damping is nonlinearly unstable around the given steady state if the steady density profile is non-monotonous along the height. A new variational structure is developed to construct the growing mode solution, and the difficulty in proving the sharp exponential growth rate is overcome by exploiting the structures in linearized Euler equations. Combined with error estimates and a standard bootstrapping argument, the nonlinear instability is established.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Samuele Ricco, Andrea Torricelli
Summary: This paper presents a solution method for the autonomous obstacle problem, finding a necessary condition for the extremality of the unique solution using a primal-dual formulation. The proof is based on classical arguments of Convex Analysis and Calculus of Variations' techniques.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Shuxin Ge, Rong Yuan, Xiaofeng Zhang
Summary: This paper studies an initial boundary value problem for a nonlocal parabolic equation with a diffusion term and convex-concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, the existence of global solutions is proven and some blow-up conditions are obtained. Using the variational structure of the problem, the Mountain-pass theorem is utilized to demonstrate the existence of nontrivial steady-state solutions. The dynamical behavior of global solutions with relatively compact trajectories in H01 (Ω) is also established, showing uniform convergence to a non-zero steady state after a long time due to the energy functional satisfying the P.S. condition. Finally, an unstable steady states sequence is derived using another minimax theorem.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)