Article
Mathematics
Miho Murata
Summary: In this paper, the unique existence of global strong solutions and decay estimates for the simplified Ericksen-Leslie system describing compressible nematic liquid crystal flows in R-N, 3 <= N <= 7 are proven. Firstly, the system is rewritten in Lagrange coordinates, and secondly, the global well-posedness for the transformed system is proved, which is the main task in this paper. The proof is based on the maximal Lp-Lq regularity and the L-p-L-q decay estimates to the linearized problem.
Article
Mathematics, Applied
Qiang Li, Mianlu Zou
Summary: This paper presents a regularity criterion for 3D nematic liquid crystal flows based on the horizontal components of velocity and molecular orientations. The criterion states that the smooth solution (u, d) can be extended beyond T if the integral(T)(0)(parallel to u(h)parallel to(2)(B infinity,infinity 0) + parallel to del d parallel to(2)(B infinity,infinity 0))dt < infinity.
Article
Mechanics
Chuong V. Tran, Xinwei Yu, David G. Dritschel
Summary: Incompressible fluid flows are characterized by high correlations between velocity and pressure, as well as between vorticity and pressure. This correlation plays a significant role in maintaining regularity in Navier-Stokes flows. The study suggests that as long as global pressure minimum (or minima) and velocity maximum (or maxima) are mutually exclusive, regularity is likely to persist.
JOURNAL OF FLUID MECHANICS
(2021)
Article
Mathematics, Applied
Giovanni P. Galdi
Summary: This paper discusses the motion of a sufficiently smooth rigid body B of arbitrary shape in an unbounded Navier-Stokes liquid under the action of prescribed external force F and torque M. It is proven that if the data is suitably regular and small, and F and M vanish for large times in the L-2 sense, then there exists at least one global strong solution to the corresponding initial-boundary value problem. Additionally, if B is a ball, this result has been previously known.
JOURNAL OF MATHEMATICAL FLUID MECHANICS
(2023)
Article
Mathematics, Applied
Nacer Aarach, Ning Zhu
Summary: Motivated by previous work, this paper investigates the global well-posedness of the 3D incompressible homogeneous and inhomogeneous magnetohydrodynamic (MHD) system with suitable small one-directional derivative of the initial data in some scaling invariant spaces. The authors prove that under certain conditions on the initial density, velocity, and magnetic field, the system has a unique global solution.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Minghao Li, Liuchao Xiao, Zhenzhen Li
Summary: In this paper, the supercloseness properties and global superconvergence results for the implicit Euler scheme of the transient Navier-Stokes equations are derived. The supercloseness properties of the Stokes projection for velocity and pressure are deduced based on a prior estimate of finite element solutions, properties of the Stokes projection and operator, derivative transforming skill, and H-1-norm estimate. The supercloseness properties of the interpolation operators for two pairs of rectangular elements are obtained, and global superconvergence results are achieved through interpolation postprocessing technique.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Applied
Yanghai Yu, Jinlu Li, Zhaoyang Yin
Summary: In this paper, a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations is derived and used to prove the existence of a unique global solution. Additionally, two examples of initial data satisfying the smallness condition are constructed, demonstrating that the norm of the initial data can be arbitrarily large.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Miho Murata, Yoshihiro Shibata
Summary: This paper proves the global well posedness and decay estimates for a Q-tensor model of nematic liquid crystals in R-N, N ≥ 3.
JOURNAL OF MATHEMATICAL FLUID MECHANICS
(2022)
Article
Mathematics, Applied
Hui Fang, Yihan Fan, Yanping Zhou
Summary: In this paper, the problem of energy equality in the two and three dimensional compressible Navier-Stokes-Korteweg equations with general pressure law is investigated. By using commutator estimation to handle the nonlinear terms, the sufficient conditions for the regularity of weak solutions to conserve energy are obtained.
Article
Mathematics, Applied
Pranava Chaitanya Jayanti, Konstantina Trivisa
Summary: The study examines the behavior of Hall-Vinen-Bekharevich-Khalatnikov equations for superfluidity at nonzero temperatures. It proves the global well-posedness of strong solutions for smooth, compactly supported data, and discusses the sufficient conditions on a 2D vorticity field for finite kinetic energy.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Mathematics
Shuichi Kawashima, Ryosuke Nakasato, Takayoshi Ogawa
Summary: The study focuses on the global existence of solution for the initial value problem of the compressible Hall-magnetohydrodynamic system in the whole space R-3, showing both the manner of existence and time decay of the solution. Results also demonstrate the pointwise estimate of the solution in the Fourier space, utilizing systematic use of product estimates in Chemin-Lerner spaces and applying the energy method by Matsumura-Nishida.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mechanics
A. Viudez
Summary: Exact solutions of the time-dependent three-dimensional nonlinear vorticity equation for Euler flows with spherical geometry are provided. The velocity solution is the sum of a multipolar oscillatory function and a rigid cylindrical motion with swirl. These solutions are important for understanding inertial oscillations and nonlinear effects in multipolar flows.
JOURNAL OF FLUID MECHANICS
(2022)
Article
Mathematics
Cholmin Sin, Evgenii S. Baranovskii
Summary: This article demonstrates that a weak solution for unsteady 3D shear thickening flows becomes a strong solution if the velocity field u belongs to the critical space L-beta (0, T; L-alpha (Omega)), and 2 <= p < 11/5. Here, Omega is R-3 or the periodic domain [0, 1](3). The key is to prove and utilize a variant of Gagliardo-Nirenberg's inequality including the terms ||u||(alpha) and || | Du| (p-2/2) del(2)u||(2), where Du is the deformation rate tensor.
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics, Applied
Yanqing Wang, Yulin Ye
Summary: In this paper, an energy conservation criterion is derived for weak solutions of both the incompressible and compressible Navier-Stokes equations. The criterion is based on a combination of velocity and its gradient. For the incompressible case, it extends known results on periodic domain, including the famous Lions' energy conservation criterion. For the compressible case, it improves recent results and extends criteria for energy conservation from incompressible to compressible flow.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Physics, Multidisciplinary
Mojtaba Rajabi, Hend Baza, Taras Turiv, Oleg D. Lavrentovich
Summary: This study demonstrates that active droplets containing swimming bacteria can achieve unidirectional motion when placed in an inactive nematic liquid-crystal medium, by converting the random motion of the bacteria inside the droplet into directional self-locomotion through interactions with the surrounding nematic. The trajectory of the active droplet can be predesigned by patterning the molecular orientation of the nematic, showing that the broken spatial symmetry of the medium can control directional microscale transport.
Article
Mathematics
Francesco De Anna, Stefano Scrobogna
REVISTA MATEMATICA IBEROAMERICANA
(2020)
Article
Mathematics, Applied
Radu Ignat, Luc Nguyen, Valeriy Slastikov, Arghir Zarnescu
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2020)
Article
Multidisciplinary Sciences
Xian Chen, Irene Fonseca, Miha Ravnik, Valeriy Slastikov, Claudio Zannoni, Arghir Zarnescu
Summary: The article discusses various research directions in the mathematical design of new materials, including phase-transforming materials, semiconductor materials, soft matter, magnetic materials, liquid crystals, and liquid crystal colloids. It emphasizes the potential for exciting progress that mathematical approaches could bring to these design themes in both soft and hard condensed matter.
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2021)
Article
Multidisciplinary Sciences
Arghir Zarnescu
Summary: Mathematical studies of nematic liquid crystals are approached from two different perspectives: fluid mechanics and calculus of variations, focusing on dynamical and stationary problems respectively. This review aims to introduce practitioners to results and issues from the other perspective, as well as presenting research topics that bridge the gap between the two communities.
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2021)
Article
Mathematics
Francesco De Anna, Joshua Kortum, Anja Schloemerkemper
Summary: This work examines the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The study showed that the weak solutions are smooth everywhere except for at a discrete set of time values, and uniqueness is based on suitable energy estimates within a less regular functional framework. Techniques of harmonic analysis and paradifferential calculus were employed in obtaining these estimates.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Nicholas D. Alikakos, Dimitrios Gazoulis, Arghir Zarnescu
Summary: We study entire minimizers of the Allen-Cahn systems with potentials having a finite number of global minima and sub-quadratic behavior locally near their minima. We show the existence of entire solutions in an equivariant setting, connecting the minima of the potential at infinity and modeling many coexisting phases with free boundaries. The existence of a free boundary can be related to the presence of a specific sub-quadratic feature, a dead core, whose size is quantified.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Razvan-Dumitru Ceuca, Jamie M. Taylor, Arghir Zarnescu
Summary: In this study, we investigate the impact of boundary rugosity in nematic liquid crystalline systems. A highly general formulation is employed to handle multiple liquid crystal theories simultaneously. Utilizing Gamma convergence techniques, we demonstrate that the fine-scale surface oscillations can be substituted by an effective homogenized surface energy in a simpler domain. Convergence rates are then quantitatively examined in a simplified setting.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2023)
Article
Physics, Condensed Matter
A. Earls, M-C Calderer, M. Desroches, A. Zarnescu, S. Rodrigues
Summary: We propose a minimalist phenomenological model to describe the 'interfacial water' phenomenon near hydrophilic polymeric surfaces. By combining a Ginzburg-Landau approach with Maxwell's equations, we derive a well-posed model that offers a macroscopic interpretation of experimental observations. The unknown parameters in the derived governing equations are estimated using experimental measurements, and the resulting profiles are found to be in agreement with experimental results. This proposed model serves as the first step towards a more complete and parsimonious macroscopic model, which can help elucidate the effects of interfacial water on cells, infrared neural stimulation, and drug interactions.
JOURNAL OF PHYSICS-CONDENSED MATTER
(2022)
Article
Mathematics
Nicholas D. Alikakos, Zhiyuan Geng, Arghir Zarnescu
Summary: We study globally bounded entire minimizers u : R-n -> R-m of Allen-Cahn systems for potentials W >= 0. We establish estimates and bounds for the diffuse interface I-0 and the free boundary partial derivative I-0, and provide results for the case when alpha = 1.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Zhiyuan Geng, Arghir Zarnescu
Summary: This paper investigates the properties of the Landau-de Gennes functional on 3D domains, focusing on the structure and behavior of its minimizers near a point defect. The results obtained provide convergence and approximation properties for the minimizers in the neighborhood of the defect.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Simone Rusconi, Christina Schenk, Arghir Zarnescu, Elena Akhmatskaya
Summary: Rational computer-aided design of multiphase polymer materials is crucial for various applications, and while property predictive models have been developed, they lack computational efficiency and accurate prediction of material properties. This study explores the feasibility of enhancing the performance of the LPMF PBM model by reducing its complexity through disregarding the aggregation terms. The resulting models demonstrate a significant improvement in computational efficiency compared to the original LPMF PBM.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Eduard Feireisl, Arnab Roy, Arghir Zarnescu
Summary: In this paper, we study the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. By assuming the object is a ball of small radius e, we demonstrate that the behavior of the fluid is independent of the object in the limit of e approaching 0. This result holds for the isentropic pressure law p(Q)=aQ(?) for any ?>3/2, with mild assumptions regarding the rigid body density. Notably, the density can be bounded as soon as ?>3. The proof utilizes a novel method for constructing test functions in the weak formulation of the problem, including a new form of the Bogovskii operator.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu
Summary: This article studies a system of nonlinear PDEs modeling the electrokinetics of a nematic electrolyte material consisting of various ion species in a nematic liquid crystal. It focuses on the two-species case and proves apriori estimates providing weak sequential stability, the main step towards proving the existence of weak solutions.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2021)
Article
Mathematics
Radu Ignat, Luc Nguyen, Valeriy Slastikov, Arghir Zarnescu
ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE
(2020)
Article
Mathematics, Interdisciplinary Applications
Giacomo Canevari, Arghir Zarnescu
MATHEMATICS IN ENGINEERING
(2020)
Article
Mathematics
Daniele Cassani, Zhisu Liu, Giulio Romani
Summary: This article investigates the strongly coupled nonlinear Schrodinger equation and Poisson equation in two dimensions. The existence of solutions is proved using a variational approximating procedure, and qualitative properties of the solutions are established through the moving planes technique.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Giovanni Alessandrini, Romina Gaburro, Eva Sincich
Summary: This paper considers the inverse problem of determining the conductivity of a possibly anisotropic body Ω, subset of R-n, by means of the local Neumann-to-Dirichlet map on a curved portion Σ of its boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities, the paper provides a Hölder stability estimate on Σ when the conductivity is a priori known to be a constant matrix near Σ.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nuno Costa Dias, Cristina Jorge, Joao Nuno Prata
Summary: This article studies the time dependent Euler-Bernoulli beam equation with discontinuous and singular coefficients, and obtains an explicit formulation of the differential problem using an extension of the Hormander product of distributions. The dynamics of the Euler-Bernoulli beam model with discontinuous flexural stiffness and structural cracks are further explored, and the relationship between the characteristic frequencies of the beam and the singularities in the flexural stiffness is investigated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Baoquan Zhou, Hao Wang, Tianxu Wang, Daqing Jiang
Summary: This paper is Part I of a two-part series that presents a mathematical framework for approximating the invariant probability measures and density functions of stochastic generalized Kolmogorov systems with small diffusion. It introduces two new approximation methods and demonstrates their utility in various applications.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Yun Li, Danhua Jiang, Zhi-Cheng Wang
Summary: In this study, a nonlocal reaction-diffusion equation is used to model the growth of phytoplankton species in a vertical water column with changing-sign advection. The species relies solely on light for metabolism. The paper primarily focuses on the concentration phenomenon of phytoplankton under conditions of large advection amplitude and small diffusion rate. The findings show that the phytoplankton tends to concentrate at certain critical points or the surface of the water column under these conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Monica Conti, Stefania Gatti, Alain Miranville
Summary: The aim of this paper is to study a perturbation of the Cahn-Hilliard equation with nonlinear terms of logarithmic type. By proving the existence, regularity and uniqueness of solutions, as well as the (strong) separation properties of the solutions from the pure states, we finally demonstrate the convergence to the Cahn-Hilliard equation on finite time intervals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Xiaolong He
Summary: This article employs the CWB method to construct quasi-periodic solutions for nonlinear delayed perturbation equations, and combines the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, thus extending the applicability of the CWB method. As an application, it studies the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nicolas Camps, Louise Gassot, Slim Ibrahim
Summary: In this paper, we consider the probabilistic local well-posedness problem for the Schrodinger half-wave equation with a cubic nonlinearity in quasilinear regimes. Due to the lack of probabilistic smoothing in the Picard's iterations caused by high-low-low nonlinear interactions, we need to use a refined ansatz. The proof is an adaptation of Bringmann's method on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, ill-posedness results for this equation are discussed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Elie Abdo, Mihaela Ignatova
Summary: In this study, we investigate the Nernst-Planck-Navier-Stokes system with periodic boundary conditions and prove the exponential nonlinear stability of constant steady states without constraints on the spatial dimension. We also demonstrate the exponential stability from arbitrary large data in the case of two spatial dimensions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Peter De Maesschalck, Joan Torregrosa
Summary: This paper provides the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. The key idea is the perturbation of a vector field with many cusp equilibria, which is constructed using elements of catastrophe theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Leyi Jiang, Taishan Yi, Xiao-Qiang Zhao
Summary: This paper studies the propagation dynamics of a class of integro-difference equations with a shifting habitat. By transforming the equation using moving coordinates and establishing the spreading properties of solutions and the existence of nontrivial forced waves, the paper contributes to the understanding of the propagation properties of the original equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Mckenzie Black, Changhui Tan
Summary: This article investigates a family of nonlinear velocity alignments in the compressible Euler system and shows the asymptotic emergent phenomena of alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are studied, resulting in a variety of different asymptotic behaviors.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Lorenzo Cavallina
Summary: In this paper, the concept of variational free boundary problem is introduced, and a unified functional-analytical framework is provided for constructing families of solutions. The notion of nondegeneracy of a critical point is extended to this setting.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Ying-Chieh Lin, Kuan-Hsiang Wang, Tsung-Fang Wu
Summary: In this study, we investigate a linearly coupled Schrodinger system and establish the existence of positive ground states under suitable assumptions and by using variational methods. We also relax some of the conditions and provide some results on the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
