Regularity Criterion Research Articles

On the critical regularity of nonlinearities for semilinear classical wave equations

In this paper, we consider the Cauchy problem for semilinear classical wave equations utt-Δu=|u|pS(n)μ(|u|)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_{tt}-\\Delta u=|u|^{p_S(n)}\\mu (|u|) \\end{aligned}$$\\end{document}with the Strauss exponent pS(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_S(n)$$\\end{document} and a modulus of continuity μ=μ(τ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu =\\mu (\ au ),$$\\end{document} which provides an additional regularity of nonlinearities in u=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u=0$$\\end{document} comparing with the power nonlinearity |u|pS(n).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|u|^{p_S(n)}.$$\\end{document} We obtain a sharp condition on μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} as a threshold between global (in time) existence of small data radial solutions by deriving polynomial-logarithmic type weighted Lt∞Lr∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }_tL^{\\infty }_r$$\\end{document} estimates, and blow-up of solutions in finite time even for small data by applying iteration methods with slicing procedure. These results imply a conjecture for the critical regularity of source nonlinearities for semilinear classical wave equations. We verify this conjecture in the 3d case.

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].

Eigenvalue Regularity Criteria of the Three-Dimensional Micropolar Fluid Equations

In this paper, we consider several improved regularity criteria to the 3D micropolar fluid equations. In particular, we prove regularity criteria that only require control of the middle eigenvalue of strain tensor in critical Besov spaces, which can be regarded as improvement and extension of results very recently obtained.

THERMODYNAMIC FORMALISM FOR AMENABLE GROUPS AND COUNTABLE STATE SPACES

Abstract Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties, such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that nontrivially satisfy the conditions for having this equivalence and a nonempty range of inverse temperatures where uniqueness holds.

Phase transitions in the fractional three-dimensional Navier–Stokes equations

The fractional Navier–Stokes equations on a periodic domain [0,L]3 differ from their conventional counterpart by the replacement of the −νΔu Laplacian term by νsAsu , where A=−Δ is the Stokes operator and νs=νL2(s−1) is the viscosity parameter. Four critical values of the exponent s⩾0 have been identified where functional properties of solutions of the fractional Navier–Stokes equations change. These values are: s=13 ; s=34 ; s=56 and s=54 . In particular: (i) for s>13 we prove an analogue of one of the Prodi–Serrin regularity criteria; (ii) for s⩾34 we find an equation of local energy balance and; (iii) for s>56 we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi–Serrin criterion for s>13 suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range 0<s<13 .

Design optimization of irregularity RC structure based on ANN-PSO

Seismic design principles advocate for simple and regular structures to minimize earthquake damage. However, this frequently does not lead to unique and aesthetically pleasing designs, leading some engineers to select irregular structures despite the potential risks. The primary aim of this investigation is to achieve the optimal design of torsional irregularity coefficients for planar irregular reinforced concrete (RC) frames under static and dynamic loads, utilizing a 3D 6-layer model. Structural ground vibration analysis was conducted using the ETABS software. By imposing limits on the torsional irregularity coefficients for each layer of the frame layout, we subsequently applied the combination of artificial neural networks (ANN) with the particle swarm optimization (PSO) algorithm, namely ANN-PSO, to address the size distribution issue across the structure. The design variables included the dimensions of the columns located in each layer of the layout. The results demonstrate that the ANN-PSO algorithm optimizes the cross-sectional area of columns with significant variations. The coefficients of the torsion inequality rule in the optimized solution closely approach the minimum values. The dimensions and orientations of the optimized columns slightly differ from the pre-optimized scheme. In the optimized scheme, the coefficients of the torsional irregularity in the Y-direction meet the requirements, preventing any torsional irregularities from occurring. The research presented an effective method, including an innovative combination of ANN-PSO and the finite element method (FEM), for designing RC structures. The findings of the research provided a practical solution to fulfill torsional regularity criteria, indicating the proposed approach is an effective method for the economical and safe design of RC structures in earthquake-prone areas. The outcomes of the present study highlighted the innovative framework to achieve optimal and safe designs for irregular RC structures while minimizing torsional damage during earthquakes.

Conditional regularity for the 3D magnetic Bénard system in Vishik spaces

In this paper, we consider the Cauchy problem associated to a magnetic Bénard system in R3. By using Littlewood–Paley decomposition technique, we provide regularity criteria involving the gradient of the velocity in Vishik spaces, which generalize some well-known results.

A regularity criterion to a mathematical model in superfluidity in $\mathbb{R}^n$

A regularity criterion to a mathematical model in superfluidity in $\mathbb{R}^n$

On the regularity criteria for the 3D axisymmetric Hall-MHD system in Lorentz spaces

The current paper establishes the regularity criteria for the 3D Hall-MHD system in the axisymmetric setting. When the horizontal angular component of the velocity belongs to some Lorentz spaces, then strong solutions to this system can be smoothly extended beyond the possible blow-up time T. It should be pointed out that our result does not need any additional assumption on the magnetic field.

The Book of Samuel and the Three-Actor Rule in Classical Greek Tragedy

ABSTRACT Following a two-part 2022 publication which argues that portions of biblical literature may be written in the style of Classical Greek theatre plays, this paper seeks to demonstrate that 1 and 2 Samuel consistently adhere to the distinctive three-actor rule of Greek tragedy. The number of countable speaking actors present in any given scene from 1 or 2 Samuel never appears to exceed three actors at a time, provided that (1) only speaking actors are included in the tally, (2) group speech is treated as that of a single actor, and (3) scenes are parsed into episodes following regular criteria.

Regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term

&lt;abstract&gt;&lt;p&gt;This paper was devoted to establishing some regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term. We focused on considering the role of the damping term in regularity criteria and the global existence brought by these criteria.&lt;/p&gt;&lt;/abstract&gt;

Some regularity criteria via the signed curl operators for the 3D Navier-Stokes equations

Some regularity criteria via the signed curl operators for the 3D Navier-Stokes equations

Regularity criterion of three dimensional magneto-micropolar fluid equations with fractional dissipation

&lt;p&gt;In this paper, we investigate the regularity criterion of weak solutions to three-dimensional magneto-micropolar fluid equations with fractional dissipation. A regularity criterion is established via the third component of the velocity fields, the micro-rotational velocity fields, and the magnetic fields.&lt;/p&gt;

Two regularity criteria of the 3D magneto-micropolar equations in Vishik spaces

Two regularity criteria of the 3D magneto-micropolar equations in Vishik spaces

Global weighted regularity for the 3D axisymmetric non-resistive MHD system

We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in $ \mathbb{R}^{3} $. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time $ T $ if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.

The Double‐Logarithmic Regularity Criterion of Pressure for the 3D Navier–Stokes Equations in the Besov Space

In the paper, we consider the regularity of the weak solutions for the incompressible 3D Navier–Stokes (N–S) equations. Our main result is the double‐logarithmic regularity criterion of pressure for the 3D Navier–Stokes equations in the Besov space. Our results extend and generalize previous works.

One new regularity criterion to two-dimensional full compressible magnetohydrodynamic system

One new regularity criterion to two-dimensional full compressible magnetohydrodynamic system

Several refined regularity criteria for the Navier–Stokes equations

Several refined regularity criteria for the Navier–Stokes 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.

Geometric regularity criteria for the Navier–Stokes equations in terms of velocity direction

In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier–Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and C rdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier–Stokes equations.