Interior Regularity Research Articles

On the interior regularity conditions of a suitable weak solution to 3D MHD equations

On the interior regularity conditions of a suitable weak solution to 3D MHD equations

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We investigate the interior regularity to nonlinear subelliptic systems in divergence form with drift term for the case of superquadratic controllable structure conditions in the Heisenberg group. On the basis of a generalization of the A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{A}$\\end{document}-harmonic approximation technique, C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C^{1}$\\end{document}-regularity is established for horizontal gradients of vector-valued solutions to the subelliptic systems with drift term. Specially, our result is optimal in the sense that in the case of Hölder continuous coefficients we directly attain the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.

Interior Regularity for Two-Dimensional Stationary Q-Valued Maps

We prove that 2-dimensional Q-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions.

Weighted estimates of commutators of singular operators in generalized Morrey spaces beyond Muckenhoupt range and applications

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

Local near-field scattering data enables unique reconstruction of rough electric potentials

The focus of this paper is the study of the inverse point-source scattering problem, specifically in relation to a certain class of electric potentials. Our research provides a novel uniqueness result for the inverse problem with local data, obtained from the near field pattern. Our work improves the work of Caro and Garcia, who investigated both the direct problem and the inverse problem with global near field data for critically singular and δ -shell potentials. The primary contribution of our research is the introduction of a Runge approximation result for the near field data on the scattering problem which, in combination with an interior regularity argument, enables us to establish a uniqueness result for the inverse problem with local data. Additionaly, we manage to consider a slightly wider class of potentials.

On the interior regularity criteria for the viscoelastic fluid system with damping

Abstract We consider a system of PDEs that model a viscoelastic fluid with damping mechanism. In ℝ 3 {\mathbb{R}^{3}} , we construct some new local energy bounds that enable us to improve several ϵ-regularity criteria for the Caffarelli–Kohn–Nirenberg theorem for weak solutions of this system.

Remarks on interior regularity criteria without pressure for the Navier-Stokes equations

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the LtpLxq(Q1) norm of the velocity is sufficiently small, where 1≤2p+3q<2 and 2≤p≤∞. It improves the recent result of p,q>2 by Kwon [15], and also generalizes Chae-Wolf's Lt∞Lx32+ criterion [3].

Global well-posedness and interior regularity of 2D Navier–Stokes equations with stochastic boundary conditions

The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.

Regularity for a geometrically nonlinear flat Cosserat micropolar membrane shell with curvature

We consider the rigorously derived thin shell membrane \Gamma -limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation m\colon \omega\subset \mathbb{R}^{2}\to \mathbb{R}^{3} and the orthogonal microrotation tensor field R\colon \omega\subset \mathbb{R}^{2}\to \operatorname{SO}(3) . The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet-type energy term |\mathrm{D}R|^{2} . We use Rivière’s regularity techniques of harmonic-map-type systems for our system which couples harmonic maps to \operatorname{SO}(3) with a linear equation for m . The additional coupling term in the harmonic map equation is of critical integrability and can only be handled because of its special structure.

Interior regularity to the signed solution of the singular doubly nonlinear parabolic equations

We study doubly nonlinear parabolic equation with sign changing solutions. We establish the Hölder regularity of the singular parabolic equations within a parabolic domain. J. Bangladesh Acad. Sci. 47(2); 241-248: December 2023

Full Double Hölder Regularity of the Pressure in Bounded Domains

Abstract We consider Hölder continuous weak solutions $u\in C^{\gamma }(\Omega )$, $u\cdot n|_{\partial \Omega }=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega \subset{\mathbb{R}}^{d}$. If $\Omega $ is of class $C^{2,1}$ then the corresponding pressure satisfies $p\in C^{2\gamma }_{*}(\Omega )$ in the case $\gamma \in (0,\frac{1}{2}]$, where $C^{2\gamma }_{*}$ is the Hölder–Zygmund space, which coincides with the usual Hölder space for $\gamma &amp;lt;\frac 12$. This result, together with our previous one in [ 11] covering the case $\gamma \in (\frac 12,1)$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C^{2\gamma }_{*}$ estimate for the pressure on the whole space ${\mathbb{R}}^{d}$, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case $\gamma =\frac{1}{2}$. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood–Paley analysis of the modified equation in the new coordinate system. We also discuss the relation between different notions of weak solutions, a step that plays a major role in our approach.

Strong unique continuation and global regularity estimates for nanoplates

In this paper, we analyze some properties of a sixth-order elliptic operator arising in the framework of the strain gradient linear elasticity theory for nanoplates in flexural deformation. We first rigorously deduce the weak formulation of the underlying Neumann problem as well as its well posedness. Under some suitable smoothness assumptions on the coefficients and on the geometry, we derive interior and boundary regularity estimates for the solution of the Neumann problem. Finally, for the case of isotropic materials, we obtain new Strong Unique Continuation results in the interior, in the form of doubling inequality and three spheres inequality by a Carleman estimates approach.

Regularity for convex viscosity solutions of special Lagrangian equation

AbstractWe establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.

Interior regularity for strong solutions to a class of fully nonlinear elliptic equations

We obtain local pointwise second derivative estimates for W 2 , p W^{2,p} -strong solutions to a class of fully nonlinear elliptic equations on Euclidean domains, motivated by problems in conformal geometry.

The Neumann problem for the fractional Laplacian: regularity up to the boundary

We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-\Delta)^s u=f$ in $\Omega$, $\mathcal N_s u=0$ in $\Omega^c$, then $u$ is $C^\alpha$ up tp the boundary for some $\alpha>0$. Moreover, in case $s>\frac12$, we then show that $u\in C^{2s-1+\alpha}(\overline\Omega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.

Harnack inequality and interior regularity for Markov processes with degenerate jump kernels

Harnack inequality and interior regularity for Markov processes with degenerate jump kernels

Image regularity conditions for nonconvex multiobjective optimization problems with applications

The main purpose of this paper is to investigate Lagrange saddle points and calmness properties for a nonconvex multiobjective optimization problem by virtue of image regularity conditions. Following along with the image space analysis, the quasi-interior, quasi-relative interior and linear image regularity conditions are presented. Simultaneously, some equivalent characterizations to Fritz John and Karush/Kuhn-Tucker saddle points are established respectively by virtue of the proposed image regularity conditions. Moreover, some calmness properties are also obtained by means of local conic image regularity conditions in the image space of the multiobjective optimization problem.

Stable solutions to semilinear elliptic equations for operators with variable coefficients

In this paper we extend the interior regularity results for stable solutions in [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear elliptic equation $ a_{ij}(x)u_{ij} + b_i(x) u_i + f(u) = 0 $ are Hölder continuous in the optimal range of dimensions $ n \leq 9 $. Our bounds are independent of the nonlinearity $ f \in C^1 $, which we assume to be nonnegative.The main achievement of our work is to make the constants in our estimates depend on the $ C^1 $ norm of $ a_{ij} $ and the $ C^0 $ norm of $ b_i $, instead of their $ C^2 $ and $ C^1 $ norms, respectively, which arise in a first approach to the computations.

On the interior regularity criterion for suitable weak solutions to the 3d magneto-hydrodynamics equations in terms of the velocity gradient

On the interior regularity criterion for suitable weak solutions to the 3d magneto-hydrodynamics equations in terms of the velocity gradient

Local Regularity Criteria in Terms of One Velocity Component for the Navier–Stokes Equations

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier–Stokes equations in three dimensions. It is shown that the velocity is regular near a point z if its scaled $$L^p_tL^q_x$$ -norm of some quantities related to the velocity field is finite and the scaled $$L^p_tL^q_x$$ -norm of one velocity component is sufficiently small near z.