Eventual Regularity Research Articles

Global boundedness and eventual regularity of chemotaxis-fluid model driven by porous medium diffusion

Global boundedness and eventual regularity of chemotaxis-fluid model driven by porous medium diffusion

Upper and lower [formula omitted] estimates for solutions to parabolic equations

In this article we prove results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and Zingano, we use eventual regularity of solutions to directly work with smooth solutions in physical space, bootstrapping decay estimates from the L2 norm to higher order derivatives. Besides obtaining upper and lower bounds through this method, we also obtain reverse results: from higher order derivatives decay estimates, we deduce bounds for the L2 norm. We use these general results to prove new decay estimates for some equations and to recover some well known results.

Global well-posedness, regularity and blow-up for the β-CCF model

The paper is concerned with the β-CCF model which is a 1D nonlocal nonlinear transport model. The velocity is coupled via a composition of the Hilbert transform and Riesz potentials, giving rise to three basic cases that reflect the balance between the nonlinearity and dissipation, namely the subcritical, critical and supercritical ranges. In our analysis we consider those three cases and obtain a set of results containing local well-posedness, global smoothness, eventual regularity and blow-up of solutions.

Experimental Capture Width Ratio on Unit Module System of Hybrid Wave Energy Converter for Nearshore

This study proposes a new hybrid wave energy converter composed of a horizontal cylinder and a swing plate to improve the capture width ratio. The horizontal cylinder generates electrical energy by using the potential energy of the incident wave, whereas the swing plate produces electrical energy by using the kinetic energy of the water particles. The converter can improve the capture width ratio of the wave energy by efficiently combining the energies generated by these two different sources. The power-generating performance of the proposed hybrid wave energy converter is evaluated experimentally through a hydraulic model test at a scale ratio of 0.3 in a two-dimensional wave tank using direct conversion by a dynamo PTO (Power Take-Off) system. The dynamic power-generation characteristics of the hybrid wave energy converter are analyzed with respect to the eventual regularity of the incident wave (regular and irregular wave conditions), and the data necessary for the design of the generator and control system are acquired.

Global Weak Solutions of the Navier–Stokes Equations for Intermittent Initial Data in Half-Space

We prove the existence of global-in-time weak solutions of the incompressible Navier–Stokes equations in the half-space \(\mathbb {R}^3_+\) with initial data in a weighted space that allows non-uniformly locally square integrable functions that grow at large scales in an intermittent sense. The space for initial data is built on cubes whose sides R are proportional to the distance to the origin and the square integral of the data is allowed to grow as a power of R. The existence is obtained via a new a priori estimate and a stability result in the weighted space, as well as new pressure estimates. Also, we prove eventual regularity of such weak solutions, up to the boundary, for (x, t) satisfying \(t\ge \epsilon _0 |x|^2 + M \), where \(\epsilon _0 >0\) is arbitrarily small and \(M>0\). By adding conditions on the data within a weighted \(L^2\) framework, we improve the algebraic bounds on the size of this region and we refine the pointwise decay rate of the solution within this region. As an application of the existence theorem, we construct global discretely self-similar solutions, thus extending the theory on the half-space to the same generality as the whole space.

Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation

We consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form for parameter choices kappa ge 0 and mu >0. Herein, we moreover impose a nonnegative and time-constant prescribed concentration c_star in C^2({overline{Omega }}) for the signal chemical on the boundary of the domain Omega subset {mathbb {R}}^{mathcal {N}} with {mathcal {N}}in {2,3}. After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of c_star being also constant in space. We show that sufficiently strong logistic influence, in the sense that for omega >0 and mu _0>0 there is some eta =eta (omega ,mu _0,c_star )>0 with the property that whenever μ0≤μandκmin{μ,μN+66+ω}<η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mu _0\\le \\mu \\quad \ ext {and}\\quad \\frac{\\kappa }{\\min \\{\\mu ,\\mu ^{\\frac{{\\mathcal {N}}+6}{6}+\\omega }\\}}<\\eta \\end{aligned}$$\\end{document}are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on omega ,mu _0,eta ,c_star and the initial data.

Well-posedness of a hydrodynamic phase-field system for functionalized membrane-fluid interaction

&lt;p style='text-indent:20px;'&gt;We study a hydrodynamic phase-field system modeling the deformation of functionalized membranes in incompressible viscous fluids. The governing PDE system consists of the Navier–Stokes equations coupled with a convective sixth-order Cahn–Hilliard type equation driven by the functionalized Cahn–Hilliard free energy, which describes the phase separation process in mixtures with an amphiphilic structure. In the three dimensional case, we prove existence of global weak solutions provided that the initial total energy is finite. Then we establish uniqueness of weak solutions under suitable regularity assumptions that are only imposed on the velocity field or its gradient. Next, we prove existence and uniqueness of local strong solutions for arbitrary regular initial data and derive some blow-up criteria. Finally, we show the eventual regularity of global weak solutions for large time. The results are obtained in a general setting with variable fluid viscosity and diffusion mobility.&lt;/p&gt;

On eventual regularity properties of operator-valued functions

Let $\mathcal {L}(X;Y)$ be the space of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Given an operator-valued function $u:\mathbb {R}_{\geq 0}\rightarrow \mathcal {L}(X;Y)$, suppose that every orbit $t\mapsto u(t)x$ has a re

Reaction-Driven Relaxation in Three-Dimensional Keller\u2013Segel\u2013Navier\u2013Stokes Interaction

The Keller–Segel–Navier–Stokes system nt+u·∇n=Δn-χ∇·(n∇c)+ρn-μn2,ct+u·∇c=Δc-c+n,ut+(u·∇)u=Δu+∇P+n∇ϕ+f(x,t),∇·u=0,(⋆)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{rcll} n_t + u\\cdot \\nabla n &{}=&{} \\Delta n - \\chi \\nabla \\cdot (n\\nabla c) + \\rho n-\\mu n^2,\\\\ c_t + u\\cdot \\nabla c &{}=&{} \\Delta c-c+n, \\\\ u_t + (u\\cdot \\nabla )u &{}=&{} \\Delta u + \\nabla P + n \\nabla \\phi + f(x,t), \\qquad \\nabla \\cdot u=0, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}is considered in a smoothly bounded convex domain Omega subset mathbb {R}^3, with phi in W^{2,infty }(Omega ) and fin C^1(bar{Omega }times [0,infty );mathbb {R}^3), and with chi >0, rho in mathbb {R} and mu >0. As recent literature has shown, for all reasonably mild initial data a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem possesses a global generalized solution, but the knowledge on its regularity properties has not yet exceeded some information on fairly basic integrability features. The present study reveals that whenever omega >0, requiring that ρmin{μ,μ32+ω}<η\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{\\rho }{\\min \\{\\mu ,\\mu ^{\\frac{3}{2}+\\omega }\\}} < \\eta \\end{aligned}$$\\end{document}with some eta =eta (omega )>0, and that f satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical. Furthermore, under these hypotheses (star ) is seen to admit an absorbing set with respect to the topology in L^infty (Omega ). By trivially applying to the case when mu >0 is arbitrary and rho le 0, these results especially assert essentially unconditional statements on eventual regularity in taxis-reaction systems interacting with liquid environments, such as arising in contexts of models for broadcast spawning discussed in recent literature.

On a repulsion Keller–Segel system with a logarithmic sensitivity

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T &gt; 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties

In this paper, we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the forager group. Specifically, we will investigate an initial boundary-value problem for a prototypical forager–exploiter model of the form [Formula: see text] in a smoothly bounded domain [Formula: see text], where [Formula: see text], [Formula: see text] is nonnegative and the functions [Formula: see text] are assumed to satisfy [Formula: see text], [Formula: see text] as well as [Formula: see text] respectively, with constants [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text]. Assuming that [Formula: see text], [Formula: see text] and that [Formula: see text] satisfies certain structural conditions, we establish the global solvability of this system with respect to a suitable generalized solution concept and then, for the more restrictive case of [Formula: see text] and [Formula: see text], investigate an eventual regularity effect driven by the decay of the nutrient density [Formula: see text].

Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L2-based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical L2-based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.

Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data

We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space $\mathring M^{2,2}_{\mathcal C}$ introduced in [Z. Bradshaw and I. Kukavica, Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data, J. Math. Fluid Mech. to appear]. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely self-similar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.

Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation

We study the qualitative properties of the unique global viscosity solution of the superquadratic diffusive Hamilton-Jacobi equation with (generalized) homogeneous Dirichlet conditions. We are interested in the phenomena of gradient blow-up (GBU), loss of boundary conditions (LBC), recovery of boundary conditions and eventual regularization, and in their mutual connections.In any space dimension, we establish the sharp minimal rate of GBU. Only partial results were previously known except in one space dimension. We also obtain the corresponding minimal regularization rate.In one space dimension, under suitable conditions on the initial data, we give a quite detailed description of the behavior of solutions for all t>0. In particular, we show that nonminimal GBU solutions immediately lose the boundary conditions after the blow-up time and are immediately regularized after recovering the boundary data. Moreover, both GBU and regularization occur with the minimal rates, while loss and recovery of boundary data occur with linear rate. We describe further the intermediate singular life of those solutions in the time interval between GBU and regularization.We also study minimal GBU solutions, for which GBU occurs without LBC: those solutions are immediately regularized, but their blow-up and regularization rates are more singular.Most of our one-dimensional results crucially depend on zero-number arguments, which do not seem to have been used so far in the context of viscosity solutions of Hamilton-Jacobi equations.

Regularity results for a class of generalized surface quasi-geostrophic equations

We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high frequencies and inner/outer spatial scales to pass to the limit; while the proof of both the global regularity result and the eventual regularity for the supercritical diffusion are essentially based on the use of the so-called modulus of continuity method.

On the Regularity Issues of a Class of Drift-Diffusion Equations with Nonlocal Diffusion

In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type L\'evy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly.

Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D

We study the chemotaxis–fluid system{nt+u⋅∇n=Δn−∇⋅(nc∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, where Ω⊂R2 is a bounded domain with smooth boundary and ϕ∈C2(Ω¯). From recent results it is known that for suitable regular initial data, the corresponding initial–boundary value problem possesses a global generalized solution. We will show that for small initial mass ∫Ωn0 these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties.Moreover, from the analysis of certain energy-type inequalities arising during the investigation of the eventual regularity, we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of n0 in L1(Ω) and in Llog⁡L(Ω), u0 in L4(Ω), and of ∇c0 in L2(Ω).

Flatness implies smoothness for solutions of the porous medium equation

One of the major problems in the theory of the porous medium equation $$\partial _t\rho =\Delta _x\rho ^m,\,m > 1$$ , is the regularity of the solutions $$\rho (t,x)\ge 0$$ and the free boundaries $$\Gamma =\partial \{(t,x): \rho >0\}$$ . Here we assume flatness of the solution and derive $$C^\infty $$ regularity of the interface after a small time, as well as $$C^\infty $$ regularity of the solution in the positivity set and up to the free boundary for some time interval. The proof starts from Caffarelli’s blueprint of an improvement of flatness by rescaling, and combines it with the Carleson measure approach applied to the degenerate subelliptic equation satisfied by the pressure of the porous medium equation in transformed coordinates. The improvement of flatness finally hinges on Gaussian estimates for the subelliptic problem. We use these facts to prove the following eventual regularity result: solutions defined in the whole space with compactly supported initial data are smooth after a finite time $$T_r$$ that depends on $$\rho _0$$ . More precisely, we prove that for $$t \ge T_r$$ the pressure $$\rho ^{m-1}$$ is $$C^\infty $$ in the positivity set and up to the free boundary, which is a $$C^\infty $$ hypersurface. Moreover, $$T_r$$ can be estimated in terms of only the initial mass and the initial support radius. This regularity result eliminates the assumption of non-degeneracy on the initial data that has been carried on for decades in the literature. Let us recall that regularization for small times is false, and that as $$t\rightarrow \infty $$ the solution increasingly resembles a Barenblatt function and the support looks like a ball.

Global regularity for the supercritical active scalars

This paper focuses on the global regularity problem for a family of active scalar equations with fractional dissipation. We obtain an explicit lower bound on the local existence of solutions, the eventual regularity of solutions and the global regularity of solutions for large initial data for the supercritical active scalar equations. In addition, a new regularity criterion is presented for the supercritical case which is used to study the eventual regularity.

On the global regularity for the supercritical SQG equation

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation @t +R ? r + = 0; ( ; 0) = 0 on T 2 = (0; 1) 2 , with 2 (0; 1). The coefficient in front of the dissipative term = ( ) = 2 is normalized to 1. We show that given a smooth initial datum withk 0k = 2 L2 k 0k 1 = 2 _ H2 R, where R is arbitrarily large, there exists 1 = 1(R)2 (0; 1) such that for 1, the solution of the supercritical SQG equation with dissipation does not blow up in finite time. The main ingredient in the proof is a new concise proof of eventual regularity for the supercritical SQG equation, that relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.