Article
Mathematics
Muhammad Zainul Abidin, Jiecheng Chen
Summary: This paper examines the generalized porous medium equation, obtaining global well-posedness results for small initial data in certain conditions, and demonstrating Gevrey class regularity of the solution.
Article
Mathematics, Applied
Caochuan Ma
Summary: In this paper, global well-posedness and optimal decay estimate of the solution to the 2D inviscid incompressible porous medium equation near a nontrivial equilibrium x2 are proven. The obtained decay rate is optimal, coinciding with that of the linear system. In particular, the L∞ estimate of u2 improves the associated work in (T.M. Elgindi, Arch. Ration. Mech. Anal. 2017).
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Youyi Zhao
Summary: In this paper, the global well-posedness of the system of incompressible viscous nonresistive magnetohydrodynamics (MHD) fluids in a three-dimensional horizontally infinite slab with finite height is investigated. The analysis is reformulated into Lagrangian coordinates and a new mathematical approach is developed to establish the global well-posedness of the MHD system without requiring nonlinear compatibility conditions on the initial data.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Multidisciplinary Sciences
Muhammad Zainul Abidin, Naeem Ullah, Omer Abdalrhman Omer
Summary: This paper considers the Cauchy problem of the three-dimensional primitive equations of geophysics and proves the global well-posedness in variable exponent Fourier-Besov spaces. It also demonstrates the Gevrey class regularity of the solution.
Article
Mathematics, Applied
Moncef Aouadi
Summary: This paper derives a nonlocal theory for porous elastic materials within the framework of Mindlin's strain gradient model. By considering the effect of nonlocal length scale parameters, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables. The existence and uniqueness of weak and strong solutions to the one-dimensional nonlinear problem are established under general assumptions on nonlinear source terms. The stability of the derived model is also investigated.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics, Applied
Youyi Zhao
Summary: This paper addresses the global well-posedness and time-decay of the system of full compressible viscous non-resistive MHD fluids in a three-dimensional horizontally infinite slab with finite height. The analysis is reformulated into Lagrangian coordinates and a new mathematical approach is developed to establish the global well-posedness of the MHD system, without requiring compatibility conditions on the initial data.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2022)
Article
Mathematics, Applied
Chenyin Qian, Beibei He, Ting Zhang
Summary: In this paper, the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids are investigated, using time-weighted global estimates and the Lagrangian approach. The results establish the global unique solvability of the initial data in the critical Besov space for the 2D incompressible inhomogeneous asymmetric fluids. The uniqueness of the solution is obtained without additional regularity assumptions on the initial density, which improves upon the recent result of Abidi and Gui (2021) for the 2D inhomogeneous incompressible Navier-Stokes system.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Mathematics
Yuhui Chen, Qinghe Yao, Minling Li, Zheng-an Yao
Summary: This paper investigates the initial value problem for the incompressible generalized Phan-Thien-Tanner (GPTT) model, which is related to the dynamic properties of polymeric fluids. A particular solution to the GPTT model is found under appropriate assumptions on a smooth function f. In three dimensions, the global existence and optimal time decay rates of strong solutions are established, provided that the initial data is close to the particular solution. These results generalize the network viscoelastic models.
ACTA MATHEMATICA SCIENTIA
(2023)
Article
Mathematics, Applied
Fei Jiang, Song Jiang, Youyi Zhao
Summary: In this paper, we study the global existence for an initial-boundary value problem in a horizontally periodic domain in three dimensions. By developing a new type of two-layer energy structure, we prove that the initial-boundary value problem admits a unique and global smooth solution. The solution decays exponentially in time to some rest state.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Physics, Mathematical
Raphael Danchin, Shan Wang
Summary: This paper aims to prove the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system under the condition of discontinuous initial density and critically regular initial velocity. Assuming the initial density is close to a positive constant, we obtain global existence and uniqueness in two dimensions when the initial velocity belongs to the critical homogeneous Besov space (1 < p < 2), and in three dimensions when the initial velocity is small in the same space (1 < p < 3). Furthermore, we establish a uniqueness statement in a critical functional framework for cases with large variations of density including vacuum. Interestingly, our result implies the uniqueness of Fujita-Kato type solutions constructed by Zhang (2020). Our work relies on interpolation results, time weighted estimates, and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Muhammad Zainul Abidin, Jie Cheng Chen
Summary: This paper considers a generalized incompressible magnetohydrodynamics system, and establishes the global well-posedness of the system with small initial condition in variable exponent Fourier-Besov-Morrey spaces using the Littlewood-Paley decomposition and Fourier localization method. Furthermore, it also achieves the Gevrey class regularity of the solution.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2022)
Article
Mathematics
Savin Treanta
Summary: In this paper, the well-posedness of a new class of variational problems with variational inequality constraints is studied by considering new forms of lower semicontinuity, pseudomonotonicity, hemicontinuity, and monotonicity for the considered scalar multiple integral functional. Specifically, by defining the set of approximating solutions for the class of variational problems under study, several results on well-posedness are established.
Article
Mathematics, Applied
Chenmin Sun, Nikolay Tzvetkov
Summary: The study focuses on cubic fractional NLS with weak dispersion and data distributed according to the Gibbs measure. By constructing root natural strong solutions, the research improves upon previous results and relies on new ideas to overcome difficulties caused by weakly dispersive effect.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics, Applied
Khaled Zennir, Aissa Boukarou, Rehab Nasser Alkhudhayr
Summary: The paper proves the existence of a unique local solution to the system of initial value problem described by integrable equations of modified Korteweg-de Vries (mKdV) in Bourgain type spaces using linear and trilinear estimates, as well as the contraction mapping principle. Additionally, the existence of a global solution is established due to the approximate conservation law.
JOURNAL OF FUNCTION SPACES
(2021)
Article
Mathematics
Jaewook Ahn, Kyungkeun Kang, Jihoon Lee
Summary: The passage discusses a class of logarithmic Keller-Segel type systems modeling the spatio-temporal behavior of chemotactic cells or criminal activities in spatial dimensions two and higher. It establishes the existence of classical solutions globally in time under certain assumptions on parameter values and given functions. The text also introduces a new type of small initial data to obtain global classical solutions and discusses the long-time asymptotic behaviors of solutions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
