Global Regularity Research Articles

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

We study quasiconformal mappings in planar domains Ω and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary ∂Ω and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below 1, the same conditions give global regularity in Ω for the principal solutions with Beltrami coefficient supported in Ω.

Global Well-Posedness and Regularity of Weak Solutions to the Prandtl’s System

Global Well-Posedness and Regularity of Weak Solutions to the Prandtl’s System

Multispectral and hyperspectral images fusion based on subspace representation and nonlocal low-rank regularization

ABSTRACT Multispectral image (MSI) and hyperspectral image (HSI) fusion is a popular method for HSI super-resolution reconstruction (HSI-SR). MSI-HSI fusion problem is ill-posed and demands several image priors or regularization terms to solve accurately, which is a challenging issue. In this paper, we propose an MSI-HSI fusion model via subspace representation and nonlocal low-rank regularization (SRNLRR). SRNLRR model incorporates the global spectral correlations and spatial nonlocal similarities of HSI to improve the fusion results, where the priors complement each other. First, we use the mode-n tensor-matrix product to project latent high spatial resolution HSI (HR-HSI) into spectral subspace, which can capture spectral low-rankness and reduce computational complexity. Then, based on low-rank representation (LRR) and nonlocal processing strategy, we design a spatial nonlocal LRR regularization (spa-NLRR) and a spectral global LRR regularization (spe-GLRR). These two regularizations analyze the spatial nonlocal similarities and spectral global correlations from intermediate-level vision. Finally, we use the residual regularization program to obtain more image information and input it into the fusion model. We use alternate minimization (AM) methods to optimize the SRNLRR model and employ the alternating direction method (ADM) on spatial/spectral LRR learning. Comparison experiments between the SRNLRR method and six advanced methods on three HSI datasets indicate the superiority of our method.

Dissecting Mismatch Negativity: Early and Late Subcomponents for Detecting Deviants in Local and Global Sequence Regularities.

Mismatch negativity (MMN) is commonly recognized as a neural signal of prediction error evoked by deviants from the expected patterns of sensory input. Studies show that MMN diminishes when sequence patterns become more predictable over a longer timescale. This implies that MMN is composed of multiple subcomponents, each responding to different levels of temporal regularities. To probe the hypothesized subcomponents in MMN, we record human electroencephalography during an auditory local-global oddball paradigm where the tone-to-tone transition probability (local regularity) and the overall sequence probability (global regularity) are manipulated to control temporal predictabilities at two hierarchical levels. We find that the size of MMN is correlated with both probabilities and the spatiotemporal structure of MMN can be decomposed into two distinct subcomponents. Both subcomponents appear as negative waveforms, with one peaking early in the central-frontal area and the other late in a more frontal area. With a quantitative predictive coding model, we map the early and late subcomponents to the prediction errors that are tied to local and global regularities, respectively. Our study highlights the hierarchical complexity of MMN and offers an experimental and analytical platform for developing a multitiered neural marker applicable in clinical settings.

Weak Solutions to an Initial-Boundary Value Problem of a Consistent Three-phase-field Model

Global warming has caused the melting of glaciers, especially the Greenland sea ice which is a great threat to human beings. The phase-field method is used to describe the evolution of multi-phase microstructure of Greenland sea ice. This paper shall investigate the global existence, uniqueness and regularity of weak solutions to an initial-boundary value problem for a three-phase-field model based on criteria that lead to both physical and mathematical consistency. For global solutions, it is calculated by applying the method of continuation of local solutions. This model allows for the formulation of spurious phases, which is completely consistent with physical reality.

The two-and-a-half-dimensional incompressible Hall-MHD system with partial dissipation

In this paper, we investigate the Cauchy problem for the two-and-a-half-dimensional incompressible Hall-magnetohydrodynamic equations with partial dissipation. We prove the global regularity of a certain system of equations with horizontal or vertical dissipation and magnetic hyper-diffusion for large initial data.

Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L p L^{p} -type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures.

Leveraging deep neural networks to uncover unprecedented levels of precision in the diagnosis of hair and scalp disorders.

Hair and scalp disorders present a significant challenge in dermatology due to their clinical diversity and overlapping symptoms, often leading to misdiagnoses. Traditional diagnostic methods rely heavily on clinical expertise and are limited by subjectivity and accessibility, necessitating more advanced and accessible diagnostic tools. Artificial intelligence (AI) and deep learning offer a promising solution for more accurate and efficient diagnosis. The research employs a modified Xception model incorporating ReLU activation, dense layers, global average pooling, regularization and dropout layers. This deep learning approach is evaluated against existing models like VGG19, Inception, ResNet, and DenseNet for its efficacy in accurately diagnosing various hair and scalp disorders. The model achieved a 92% accuracy rate, significantly outperforming the comparative models, with accuracies ranging from 50% to 80%. Explainable AI techniques like Gradient-weighted Class Activation Mapping (Grad-CAM) and Saliency Map provided deeper insights into the model's decision-making process. This study emphasizes the potential of AI in dermatology, particularly in accurately diagnosing hair and scalp disorders. The superior accuracy and interpretability of the model represents a significant advancement in dermatological diagnostics, promising more reliable and accessible diagnostic methods.

Optimal regularity and fine asymptotics for the porous medium equation in bounded domains

Abstract We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time T * {T^{*}} . More precisely, we show that solutions are C 2 , α ⁢ ( Ω ¯ ) {C^{2,\alpha}(\overline{\Omega})} in space, with α = 1 m {\alpha=\frac{1}{m}} , and C ∞ {C^{\infty}} in time (uniformly in x ∈ Ω ¯ {x\in\overline{\Omega}} ), for t > T * {t>T^{*}} . Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence O ⁢ ( t - 1 - γ ) {O(t^{-1-\gamma})} , and we prove that the convergence holds in the C 1 , α ⁢ ( Ω ¯ ) {C^{1,\alpha}(\overline{\Omega})} topology.

Contour Dynamics and Global Regularity for Periodic Vortex Patches and Layers

Contour Dynamics and Global Regularity for Periodic Vortex Patches and Layers

Global endpoint regularity estimates for the fractional Dirichlet problem

Global endpoint regularity estimates for the fractional Dirichlet problem

Global regularity for the 2D micropolar Rayleigh–Bénard convection system with velocity zero dissipation and temperature critical diffusion

This paper studies the global regularity problem for the 2D micropolar Rayleigh–Bénard convection system with velocity zero dissipation, micro‐rotation velocity Laplace dissipation, and temperature critical diffusion. By introducing a combined quantity and using the technique of Littlewood–Paley decomposition, we establish the global regularity result of solutions to this system.

Global regularity for nonlinear systems with symmetric gradients

We study global regularity of nonlinear systems of partial differential equations depending on the symmetric part of the gradient with Dirichlet boundary conditions. These systems arise from variational problems in plasticity with power growth. We cover the full range of exponents p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in (1,\\infty )$$\\end{document}. As a novelty the degenerate case for p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>2$$\\end{document} is included. We present a unified approach for all exponents by showing the regularity for general systems of Orlicz growth.

Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces

In this paper we study the stochastic Navier–Stokes equations on the d-dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case Bq,pd/q-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^{d/q-1}_{q,p}$$\\end{document} for q∈[2,2d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in [2,2d)$$\\end{document} and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions d⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\geqslant 2$$\\end{document}. Moreover, for d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.

Global regularity in parabolic weighted Orlicz-Morrey spaces of solutions to parabolic equations with VMO coefficients

Global regularity in parabolic weighted Orlicz-Morrey spaces of solutions to parabolic equations with VMO coefficients

Geometric Constraints for Global Regularity of 3D Shear Thickening Fluids

Geometric Constraints for Global Regularity of 3D Shear Thickening Fluids

Local and global regularity for the Stokes and Navier-Stokes equations with boundary data in the half-space

Local and global regularity for the Stokes and Navier-Stokes equations with boundary data in the half-space

Global regularity of the 3D generalized Navier-Stokes equations with damping term

Global regularity of the 3D generalized Navier-Stokes equations with damping term

THE REGULARITY OF SOLUTIONS PROBLEMS TO NONLINEAR ELLIPTIC EQUATIONS

Abstract. We obtained estimates in generalized Morrey spaces are used to study global regularity of the solution of the boundary problems of nonlinear elliptic equations in divergence form over a bounded non-smooth domain. For these is we used of Calderon-Zygmund theory. The main goal of the article is to obtain Hölder estimates for solutions to the boundary problems for nonlinear elliptic equations.

Global regularity criteria for the 2D Magneto-micropolar Equations with Partial Dissipation

The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field.These equations have been the focus of numerous analytical, experimental, and numerical investigations.One fundamental problem concerning these equations is whether their classical solutions are globally regular for all time or if they develop finite time singularities.The global regularity problem can be particularly challenging when there is only partial dissipation. In this paper, we study the 2D incompressible magneto-micropolar equations with partial dissipation prove two new regularity results. The first result addresses a weak solution, and the second result establishes global regularity criteria. As a consequence, we can single out one special partial dissipation case and establish the global regularity if (∂y u1, ∂y u2) ∈ L∞ [0, T ], R2. The proofs of our main results rely on anisotropic Sobolev-type inequalities and the appropriate combination and cancellation of terms.