Regularity Results Research Articles

Poisson problems involving fractional Hardy operators and measures

In this paper, we study the Poisson problem involving a fractional Hardy operator and a measure source. The complex interplay between the nonlocal nature of the operator, the peculiar effect of the singular potential and the measure source induces several new fundamental difficulties in comparison with the local case. To overcome these difficulties, we perform a careful analysis of the dual operator in the weighted distributional sense and establish fine properties of the associated function spaces, which in turn allow us to formulate the Poisson problem in an appropriate framework. In light of the close connection between the Poisson problem and its dual problem, we are able to establish various aspects of the theory for the Poisson problem including the solvability, a priori estimates, variants of Kato’s inequality and regularity results.

A bifurcation result for a Keller-Segel-type problem

We consider a parametric elliptic problem governed by the spectral Neumann fractional Laplacian on a bounded domain of RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^N$$\\end{document}, N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}, with a general nonlinearity. This problem is related to the existence of steady states for Keller-Segel systems in which the diffusion of the chemical is nonlocal. By variational arguments we prove the existence of a weak solution as a local minimum of the corresponding energy functional and we derive some qualitative properties of this solution. Finally, we prove a regularity result for weak solutions of the problem under consideration, which is of independent interest.

The final value problem for anomalous diffusion equations involving weak-valued nonlinearities

We are concerned with the final value problem governed by a class of semilinear anomalous diffusion equations, where the nonlinearity may take values in Hilbert scales with negative orders. The solvability and Hölder regularity results are proved by establishing estimates in Hilbert scales for resolvent operators in connection with nonlinearity function. The obtained results are applicable for some concrete problems modeling subdiffusion phenomena.

Erratum to “An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs”

Erratum to “An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs”

An optimal control problem for the continuity equation arising in smart charging

This paper is focused on the mathematical modeling and solution of the optimal charging of a large population of identical plug-in electric vehicles (PEVs) with mixed state variables (continuous and discrete). A mean field assumption is formulated to describe the evolution interaction of the PEVs population. The optimal control of the resulting continuity equation of the mixed system under state constraints is investigated. We prove the existence of a minimizer. We then characterize the solution as the weak solution of a system of two coupled PDEs: a continuity equation and of a Hamilton-Jacobi equation. We provide regularity results of the optimal feedback control.

Hölder regularity of the Boltzmann equation past an obstacle

AbstractRegularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].

Unconditional Regularity and Trace Results for the Isentropic Euler Equations with \({\boldsymbol{\gamma = 3}}\)

Unconditional Regularity and Trace Results for the Isentropic Euler Equations with \({\boldsymbol{\gamma = 3}}\)

A Numerical Study on the Performance of the 66k DWT Bulk Carrier in Regular and Irregular Waves

Accurate resistance performance evaluation is essential to predict the minimum propulsion horsepower required for ships in real sea. Therefore, in this paper, the added resistance performance of K-Supramax in various wave conditions was calculated using CFD (Computational Fluid Dynamics). First, to compare the resistance performance according to the DOF (Degree of Freedom), the added resistance performance in the head regular wave was investigated and compared with the experiment. The 2DOF and 6DOF results in the head regular wave had no significant differences. Second, to compare the resistance performance among various wave conditions, the effect of the heading angle was analyzed by comparing resistance and motion RAOs (Response Amplitude Operators) in bow quartering sea conditions using 6DOF. The added resistance showed the highest value near the resonance frequency of 170°. As the heading angle increased, the added resistance tended to decrease, regardless of the wavelength ratio. Also, the added resistance performance in irregular waves was investigated with reference to the adverse conditions. The spectral method, which linearly superimposes regular wave results, and the added resistance generated during irregular waves directly through CFD were compared. The resistance in irregular waves increased by approximately 92% for the spectral method and by approximately 72% for the direct irregular wave calculation compared to the resistance in calm water. In addition, the PSD (Power Spectral Density) of resistance and the distribution of motion RAOs obtained from direct irregular wave calculations were investigated.

Generic ill-posedness of the energy–momentum equations and differential inclusions

We show that the energy–momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion into the orthogonal group and in particular these solutions can be nowhere $C^1$ . We prove that these solutions are not stationary points if the Lagrangian $W$ is $C^1$ and strictly rank-one convex. In view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for solution of the energy–momentum equation in dimension 2, we give a sufficient condition for the non-existence of a partial $C^1$ -regularity result even under the condition that the mappings satisfy a positive Jacobian determinant condition. Finally, we consider a number of well-known functionals studied in non-linear elasticity and geometric function theory and show that these do not satisfy this obstruction to partial regularity.

Free boundary partial regularity in the thin obstacle problem

AbstractFor the thin obstacle problem in , , we prove that at all free boundary points, with the exception of a ‐dimensional set, the solution differs from its blow‐up by higher order corrections. This expansion entails a C1, 1‐type free boundary regularity result, up to a codimension 3 set.

Radiological and Functional Pulmonary Evolution in Post-COVID-19 Patients: An Observational Study.

COVID-19 has generated a scenario for global health with multiple systemic impairments. This retrospective study evaluated the clinical, radiological, and pulmonary functional evolution in 302 post-COVID-19 patients. Regarding post-COVID-19 pulmonary symptoms, dry cough, dyspnea, and chest pain were the most frequent. Of the associated comorbidities, asthma was more frequent (23.5%). Chest tomography (CT) initially showed a mean pulmonary involvement of 69.7%, and evaluation in the subsequent months showed improvement in the evolutionary image. With less than six months post-pathology, there was a commitment of 37.7% from six to twelve months it was 20%, and after 12 months it was 9.9%. As for most of the sample, 50.3% of the patients presented CT normalization less than six months after infection, 23% were normalized between six and twelve months, and 5.2% presented with normalized images after twelve months, with one remaining. A percentage of 17.3% maintained post-COVID-19 pulmonary residual sequelae. Regarding spirometry, less than six months after pathology, 59.3% of the patients presented regular exam results, 12.3% had their function normalized within six to twelve months, and 6.3% had normal exam results twelve months after their post-pathology evaluation. Only 3.6% of the patients still showed some alteration during this period.

Integral inequalities with an extended Poisson kernel and the existence of the extremals

Abstract In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.

Fracture Characteristics of EN AW 1200 Tensile Test Specimens Joined with FSW and SFSW Processes

The aim of the current study is to perform a preliminary analysis on specimens extracted from 4 mm sheets of EN AW 1200 alloys that were butt-joined using FSW and two SFSW processes. The purpose of this scientific investigation is to promote the suitability and advantages of new ecological joining processes by implementing SFSW in the automotive, naval, railway and aerospace industry. The papers authors performed destructive testing on specimens extracted from sheets joined with the above mentions processes, namely tensile strength test according to SR EN ISO 6892-1:2020. Researchers also performed topography analysis on samples extracted from the AW 1200 but-joined similar sheets using scanning electron microscopy – SEM, combined with energy dispersive X-ray analysis EDAX. Results from the tensile tests highlighted an improvement in the ultimate tensile strength of the welded butt-joint, if compared to regular FSW results. The highest tensile strength value resulted from the samples extracted from joints performed using the second version of SFSW process. Researchers also noticed that samples broke in the weld or transition area. There was also a change in the chemical composition of the sample performed using the first SFSW method, namely, a high concentration of oxide appeared (≥18%), leading to the conclusion that the module used induces rapid oxidation of joints. Experimental data doesn’t comply with automotive, railway, naval and aerospace-imposed compliances, thus further investigations should be done to improve the application of the technology.

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 $\gamma <\frac 12$. 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.

Optimal regularity of isoperimetric sets with Hölder densities

We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild (alpha -)Hölder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach in any dimension the regularity class C^{1,frac{alpha }{2-alpha }}. This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends alpha -Hölder continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint).

Multi–component Cahn–Hilliard Systems with Singular Potentials: Theoretical Results

We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of Nge 2 components occupying a bounded region Omega subset mathbb {R}^{d},dle 3. Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if dle 2, we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if d=3, we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If d=3, this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application.

Optimal partial regularity of discontinuous subelliptic systems with VMO coefficients related to Hörmander’s vector fields

In this paper, we study discontinuous subelliptic systems with VMO coefficients related to Hörmander’s vector fields. In the case of growth exponential pgeq 2 , the regularity results of the partial Hölder continuity of weak solutions are established based on the mathcal{A}-harmonic approximation method under the controllable growth condition.

Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients

In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order n≥2. We show that every matrix function of this class is associated with some differential expression. Moreover, we construct the family of associated matrices for a fixed differential expression. Furthermore, our regularization results are applied to inverse spectral theory. We study a new type of inverse spectral problems, which consist of the recovery of distribution coefficients from the spectral data independently of the associated matrix. The uniqueness theorems are proved for the inverse problems by the Weyl–Yurko matrix and by the discrete spectral data. As examples, we consider the cases n=2 and n=4 in more detail.

Singularity extraction for elliptic equations with coefficients with jumps and Dirac sources

We explore the mathematical structure of the solution to an elliptic diffusion problem with point-wise Dirac sources. The conductivity parameter is space-varying, may have jumps and the Dirac sources may be located along the discontinuity curves of that parameter. The variational problem, issued by duality, is proven to be well posed using a sharp elliptic regularity result by Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The paper is aimed at a key expansion into a split singular/regular contributions. The singular part is calculated by an explicit formula, while the regular correction can be computed as the solution to a standard variational Poisson problem. The latter can be successfully approximated by most of the numerical methods practiced nowadays. Some analytical examples are discussed at last to assess the minimality of the assumptions we use to establish our theoretical results.

Regularity Results on the Flow Maps of Periodic Dispersive Burgers Type Equations and the Gravity–Capillary Equations

Regularity Results on the Flow Maps of Periodic Dispersive Burgers Type Equations and the Gravity–Capillary Equations