In this paper the regularity of weak solutions and the blow-up criteria of smooth solutions to the micropolar fluid equations on three dimension space are studied in the Lorentz space $L^{p,\infty}(\mathbb{R}^3)$. We obtain that if $u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 1$ with $3<p\le \infty$; or $\nabla u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 2$ with $\frac32<p\le \infty$; or the pressure $P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 2$ with $\frac32<p\le \infty$; or $\nabla P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 3$ with $1<p\le \infty$, then the weak solution $(u,ω)$ satisfying the energy inequality is a smooth solution on $[0,T)$.
In this work, we prove a regularity criterion for micropolar fluid flows in terms of the one partial derivative of the velocity in Morrey-Campanato space.
The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Δ)^αu$ and $(-Δ)^βw$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $α\geq \frac{5}{4}$, $β\geq 0$ and $α+β\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addit…
In the analysis of PDEs, regularity of often measured in terms of Sobolev, H{ö}lder, Besov or Lipschitz spaces, etc. However, sometimes a gain of regularity can also be expressed just in terms of Lebesgue spaces, by passing from a singular setting to a less singular one. In this article we will obtain a gain of integrability for weak solutions of the micropolar fluid equations using as general framework Morrey spaces, which is a very useful language to study regularity in PDEs. An interesting point is that the two variables of the micropolar fluid equations can be studied separately.
In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if \begin{equation*} \int^{T}_{0}\|\partial_{3}u\|^{\frac{2}{1-r}}_{\dot{B}^{-r}_{\infty,\infty}} dt<\infty \quad \text{with} \quad 0< r<1, \end{equation*} then, the solutions of the micropolar fluid equations actually are smooth on $(0, T)$. This improves and extends many previous results.
The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field u while the second equation gives the evolution of the microrotation field $ω$. In this article we will consider regularity problems for weak solutions of this system. For this we will introduce the new notion of partial suitable solutions, which imposes a specific behavior for the velocity field u only, and under some classical hypotheses over the pressure, we will obtain a h{ö}lderian gain for both variables u and $ω$.
We develop here a particular version of the partial regularity theory for the Magneto-Micropolar equations (MMP) where a perturbation term is added. These equations are used in some special cases, such as in the study of the evolution of liquid cristals or polymers, where the classical Navier-Stokes equations are not an accurate enough model. The incompressible Magneto-Micropolar system is composed of three coupled equations: the first one is based in the Navier-Stokes system, the second one considers mainly the magnetic field while the last equation introduces the microrotation field represen…
We consider the rigorously derived thin shell membrane $Γ$-limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation $m:ω\subset{\mathbb R}^2\to{\mathbb R}^3$ and the orthogonal microrotation tensor field $R:ω\subset{\mathbb R}^2\to SO(3)$. The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet type energy term $|DR|^2$. We use Rivière's regularity techniques of harmonic map type systems for our system which couples harmonic m…
In this paper, we consider weak solutions of the Euler-Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solutions is stationary, then its singular set is discrete for $2<p<3$ and has zero $1$-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.
This paper focuses on the initial- and boundary-value problem for the two-dimensional micropolar equations with only angular velocity dissipation in a smooth bounded domain. The aim here is to establish the global existence and uniqueness of solutions by imposing natural boundary conditions and minimal regularity assumptions on the initial data. Besides, the global solution is shown to possess higher regularity when the initial datum is more regular. To obtain these results, we overcome two main difficulties, one due to the lack of full dissipation and one due to the boundary conditions. In ad…
In this paper, we are concerned with the initial-boundary value problem to the 2D magneto-micropolar system with zero angular viscosity in a smooth bounded domain. We prove that there exists a unique global strong solution of such a system by imposing natural boundary conditions and regularity assumptions on the initial data, without any compatibility condition.
Stability of vertically charged steady magnetic field in 3D incompressible magneto-micropolar fluids without magnetic and angular viscosity in a strip domain
Chongqing Normal University（重庆师范大学）；South China University of Technology（华南理工大学）；University of Notre Dame（圣母大学）；Harbin Engineering University（哈尔滨工程大学）
Local and global strong solutions to a reduced model for inviscid micropolar fluids
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Ikerbasque – Basque Foundation for Science（伊卡巴斯基奎科学基金会）；Université Claude Bernard Lyon 1 , Institut Camille Jordan – UMR 5208（里昂第一大学克莱尔·让·约旦研究所）；Universidad de Chile , Center for Mathematical Modeling（智利大学数学建模中心）
Global Yudovich-type solutions to a reduced model for micropolar fluids with zero viscosity
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Ikerbasque – Basque Foundation for Science（伊卡巴斯克科学基金会）；Université Claude Bernard Lyon 1 , Institut Camille Jordan – UMR 5208（里昂第一大学克洛德·贝尔纳大学、坎耶·若尔丹研究所）
This article examines the existence and uniqueness of weak solutions to the d-dimensional micropolar equations ($d=2$ or $d=3$) with general fractional dissipation $(-Δ)^αu$ and $(-Δ)^βw$. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $α\ge \frac12$ and $β\ge \frac12$, any initial data $(u_0, w_0)$ in the critical Besov space $u_0\in B^{1+\frac{d}{2}-2α}_{2,1}(\mathbb R^d)…
For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model.
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Leray's blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,\infty}(-1,0;L^{q,l}(\mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<\infty$ and $q\leq l <\infty$ is finite.
In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain $Ω\subset\R^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,...,m-1$, we will show $(u,v)$ is smooth.
In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on $(0,T]$ provided that either the norm $\|Π\|_{L^{p,\infty}(0,T; L ^{q,\infty}(\mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=2$ $({3}/{2}<q<\infty)$ or $\|\nablaΠ\|_{L^{p,\infty}(0,T; L ^{q,\infty}(\mathbb{R}^{3}))} $ with $ {2}/{p}+{3}/{q}=3$ $(1<q<\infty)$ is small. This gives an affirmative answer to a question proposed by Suzuki in [26, Remark 2.4, p.3850]. Moreover, regular conditions in terms of $\nab…
In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% \begin{equation} u_{3}\in L^{\frac{30α}{7α-45}}\left( 0,T;L^{α,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{45}{7}% \leq α\leq \infty , \label{eq01} \end{equation}% and \begin{equation} j_{3}\in L^{\frac{2β}{2β-3}}\left( 0,T;L^{β,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{3}{2}\leq β\le…
In this paper we derive regular criteria in Lorentz spaces for Leray-Hopf weak solutions $v$ of the three-dimensional Navier-Stokes equations based on the formal equivalence relation $π\cong|v|^2$, where $π$ denotes the fluid pressure and $v$ the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem). It is shown that if $\fπ{(e^{-|x|^2}+|v|)^θ}\in L^p(0,T;L^{q,\infty})\,,$ where $0\leqθ\leq1$ and $\f2p+\f3q=2-θ$, then $v$ is regular on $(0,T]$. Note that, if $\Om$ is periodic, we may replace $\,e^{-|x|^2} \,$ by a positive constant. This result improves a 2018 stat…
Let $E$ be a bounded domain in $\mathbb R^d$. We study regularity property of $χ_E$ and integrability of $\widehat {χ_E }$ when its boundary $\partial E$ satisfies some conditions. At the critical case these properties are generally known to fail. By making use of Lorentz and Lorentz-Sobolev spaces we obtain the endpoint cases of the previous known results. Our results are based on a refined version of Littlewood-Paley inequality, which makes it possible to exploit cancellation effectively.
We are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $L_{q,r}(0,T;L_p(Ω))$ spaces.
In this paper, we investigate the 3D inhomogeneous Navier-Stokes flows with vacuum, and obtain regularity criteria and Liouville type theorems in the Lorentz space if a smooth solution $(ρ, \mathbf{u})$ satisfies suitable conditions.
The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type \begin{align*} \begin{cases} \mathrm{div}(|\nabla u|^{p-2}\nabla u) &= f+ \ \mathrm{div}(|\mathbf{F}|^{p-2}\mathbf{F}) \qquad \text{in} \ \ Ω, \\ \hspace{1.2cm} u &=\ g \hspace{3.1cm} \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz space, with given data $\mathbf{F} \in L^p(Ω;\mathbb{R}^n)$, $f \in L^{\frac{p}{p-1}}(Ω)$, $g \in W^{1,p}(Ω)$ for $p>1$ and $Ω\subset \ma…
In this paper, we study regularity of weak solutions to the incompressible Boussinesq equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces.
This paper is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this paper is to establish some global estimates for non-uniformly elliptic in divergence form as follows \begin{align*} -\mathrm{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u) = - \mathrm{div}(|\mathbf{F}|^{p-2}\mathbf{F} + a(x)|\mathbf{F}|^{q-2}\mathbf{F}), \end{align*} that arises from double phase functional problems. In particular, the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators. T…
We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation $-{\rm div}(A\nabla u+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$, $d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with "good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary sma…
In this paper we continue the investigation of the regularity of the so-called weak $\frac{n}{p}$-harmonic maps in the critical case. These are critical points of the following nonlocal energy \[ {\mathcal{L}}_s(u)=\int_{\mathbb{R}^n}| ( {-Δ})^{\frac{s}{2}} u(x)|^p dx\,, \] where $u\in \dot{H}^{s,p}(\mathbb{R}^n,\mathcal{N})$ and ${\mathcal{N}}\subset\mathbb{R}^N$ is a closed $k$ dimensional smooth manifold and $s=\frac{n}{p}$. We prove Hölder continuity for such critical points for $p \leq 2$. For $p > 2$ we obtain the same under an additional Lorentz-space assumption. The regularity theory i…
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 $L^2$-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 $L^2…
Consider the nonlinear parabolic equation in the form $$ u_t-{\rm div} \mathbf{a}(D u,x,t)={\rm div}\,(|F|^{p-2}F) \quad \text{in} \quad Ω\times(0,T), $$ where $T>0$ and $Ω$ is a Reifenberg domain. We suppose that the nonlinearity $\mathbf{a}(ξ,x,t)$ has a small BMO norm with respect to $x$ and is merely measurable and bounded with respect to the time variable $t$. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates ex…
We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density is discontinuous and the initial velocity has critical regularity. Assuming that the initial density is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity belongs to some critical homogeneous Besov space (and in small in the three-dimensional case). Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the…
It is shown both locally and globally that $L_t^{\infty}(L_x^{3,q})$ solutions to the three-dimensional Navier-Stokes equations are regular provided $q\not=\infty$. Here $L_x^{3,q}$, $0<q\leq\infty$, is an increasing scale of Lorentz spaces containing $L^3_x$. Thus the result provides an improvement of a result by Escauriaza, Seregin and {\v S}verák ((Russian) Uspekhi Mat. Nauk {\bf 58} (2003), 3--44; translation in Russian Math. Surveys {\bf 58} (2003), 211--250), which treated the case $q=3$. A new local energy bound and a new $ε$-regularity criterion are combined with the backward uniquenes…
We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of the previous works of the authors. This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a G-gamma space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of linear equation in a domain Omega with data in the space of the integrable fun…
We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Solutions can be physically realized because they have finite local $L^2$-mass. Moreover, we analyze the asymptotic stability of solutions and, although the equation has no scaling, show the existence of a class of solutions asymptotically self-similar w.r.t. the scali…
In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}(\mathcal{A}(x, \nabla u))&=& μ\quad \text{in} ~Ω, u&=&0 \quad \text{on}~ \partial Ω, \end{array}\right. \end{eqnarray*} where $μ$ is a finite signed Radon measure in $Ω$, $Ω\subset \mathbb{R}^n$ is a bounded domain such that its complement $\mathbb{R}^n\backslashΩ$ is uniformly $p$-thick and $\mathcal{A}$ is a Carathéodory vector valued function satisfying growth and monotonic…
This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain a few new results concerning the $n,1$ relative and global capacities. We obtain sharp estimates for the $n,1$ relative capacity of the concentric condensers $(\overline{B}(0,r), B(0,1))$ for all $r$ in $[0,1).$ As a consequence we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R…
This paper continues the development of regularity results for quasilinear measure data problems \begin{align*} \begin{cases} -\mathrm{div}(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ \quad \quad \qquad u &=0 \quad \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz and Lorentz-Morrey spaces, where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), $μ$ is a finite Radon measure on $Ω$, and $A$ is a monotone Carathéodory vector valued operator acting between $W^{1,p}_0(Ω)$ and its dual $W^{-1,p'}(Ω)$. It emphasizes that this paper studies the `very singular' case $1<p \le \frac{3n-2}{2n-1}$ and th…
