Article
Mathematics
Simone Ciani, Sunra Mosconi, Vincenzo Vespri
Summary: We establish a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. By identifying the natural scalings and reducing the problem to a Fokker-Planck equation, we construct a self-similar Barenblatt solution. Utilizing translation invariance and a self-iteration method, we obtain positivity near the origin and derive a sharp anisotropic expansion. This ultimately leads to a scale-invariant Harnack inequality in an anisotropic geometry determined by the diffusion coefficients' speed. As a consequence, Holder continuity, an elliptic Harnack inequality, and a Liouville theorem are inferred.
JOURNAL D ANALYSE MATHEMATIQUE
(2023)
Article
Mathematics
Hongjie Dong, Tuoc Phan
Summary: The article proves mixed-norm Sobolev estimates for time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations in small cylinders, which also implies Caccioppoli type inequalities for Stokes systems with variable coefficients and a new ε-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations. This new criterion further implies some borderline cases of the well-known Serrin's regularity criterion.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Wojciech S. Ozanski
Summary: This paper proves that if u is a Leray-Hopf weak solution to the incompressible Navier-Stokes equations with hyperdissipation α ∈ (1, 5), then there exists a set S ⊂ R3 such that u remains bounded outside of S at each blow-up time, the Hausdorff dimension of S is bounded above by 5 - 4α, and its box-counting dimension is bounded by 31(-16α² + 16α + 5). The approach in this paper is inspired by the ideas presented in Katz and Pavlovic (Geom. Funct. Anal. 12:2 (2002), 355-379).
Article
Mathematics
Francesca Anceschi, Annalaura Rebucci
Summary: The aim of this work is to prove a Harnack inequality and Holder continuity for weak solutions to the Kolmogorov equation with measurable coefficients, integrable lower order terms, and nonzero source term. A functional space W is introduced for the study of weak solutions, which enables the proof of a weak Poincare inequality. The analysis is based on a weak Harnack inequality, a weak Poincare inequality combined with an L2 - L infinity estimate, and a classical covering argument.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Salvatore Leonardi, Nikolaos S. S. Papageorgiou
Summary: This study examines three types of anisotropic double phase problems with Dirichlet boundary condition. Two of these problems exhibit strong singularity and an unbounded coefficient. Utilizing variational method, truncation, comparison, and approximation techniques, we demonstrate the existence and multiplicity theorems for our problem.
RESULTS IN MATHEMATICS
(2023)
Article
Mathematics, Applied
Fiorella Rendon, Boyan Sirakov, Mayra Soares
Summary: In this paper, a global extension of the classical weak Harnack inequality is obtained, which extends and quantifies the Hopf-Oleinik boundary-point lemma for uniformly elliptic equations in divergence form. Among the consequences are a boundary gradient estimate, proposed by Krylov and well studied for non-divergence form equations, but completely novel in the divergence framework, and a new more general version of the Hopf-Oleinik lemma.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Multidisciplinary Sciences
Maria Alessandra Ragusa, Fan Wu
Summary: This paper investigates the regularity of weak solutions to the 3D incompressible MHD equations, providing a regularity criterion for weak solutions involving any two groups of functions in anisotropic Lorentz space.
Article
Mathematics
Nikolaos S. Papageorgiou, Andrea Scapellato
Summary: The above study proves a bifurcation-type result on the Dirichlet problem, describing the changes in the set of positive solutions as the parameter varies.
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
(2022)
Article
Mathematics, Applied
Giulio Ciraolo, Xiaoliang Li
Summary: In this paper, the authors consider a partially overdetermined problem for anisotropic N-Laplace equations in a convex cone. They prove the existence of a solution and a rigidity result using a prescribed logarithmic condition and the characterization of minimizers of anisotropic isoperimetric inequality inside convex cones.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Jamil Chaker, Minhyun Kim
Summary: We study the robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local Holder estimate.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics, Applied
Fan Wu
Summary: This paper discusses the global energy conservation for distributional solutions to incompressible Hall-MHD equations without resistivity. Building upon the works of Tan and Wu in [arXiv:2111.13547v2] and Wu in [J. Math. Fluid Mech. 24,111 (2022)], the energy balance is established for a distributional solution in whole spaces Rd(d ≥ 2), given that u, b ∈ L4L4 and Vb ∈ LaLa. Furthermore, as a corollary, the energy conservation criterion for a Leray-Hopf weak solution is also obtained.
Article
Mathematics, Applied
Hamid El Bahja
Summary: This paper studies the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. The existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces is proved, and the global boundedness of the weak solution is derived.
APPLICABLE ANALYSIS
(2023)
Article
Physics, Fluids & Plasmas
Luigi C. Berselli, Stefano Spirito
Summary: This paragraph presents a short and self-contained introduction to the global existence of Leray-Hopf weak solutions to the three-dimensional incompressible Navier-Stokes equations with constant density. A unified treatment is given for different domains, boundary conditions, and approximation methods, focusing on the compactness argument needed to show convergence of approximations to weak solutions.
Article
Mathematics, Applied
Harald Garcke, Patrik Knopf, Julia Wittmann
Summary: This article investigates the Cahn-Hilliard equation with anisotropic energy contributions, specifically the case with logarithmic free energy. Analytical results are presented for the existence, uniqueness, regularity, and separation properties of weak solutions. The analysis becomes intricate due to the highly non-linear nature of the equation and the non-smoothness of the relevant anisotropies, requiring new regularity results for quasilinear elliptic equations of second order.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2023)
Article
Mathematics
Ravi P. Agarwal, Ahmad M. Alghamdi, Sadek Gala, Maria Alessandra Ragusa
Summary: In this article, the regularity criteria for weak solutions of the Boussinesq equations are studied, focusing on the horizontal component of velocity or the horizontal derivatives of the two components of velocity in anisotropic Lorentz spaces. The results highlight the dominant role of the velocity field in the regularity theory of the Boussinesq equations.
DEMONSTRATIO MATHEMATICA
(2023)
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)