We consider the operator $-Δ+b \cdot \nabla$ with $b:\mathbb R^d \rightarrow \mathbb R^d$ ($d \geq 3$) in the class of form-bounded vector fields (containing vector fields having critical-order singularities), and characterize quantitative dependence of the $\mathcal W^{1+\frac{2}{q},p}$ ($2 \leq p < q$) and the $C^{0,γ}$ regularity of solutions to the corresponding elliptic equation in $L^p$ on the value of the form-bound of $b$.
We obtain the basic results concerning the problem of constructing operator realizations of the formal differential expression $\nabla \cdot a \cdot \nabla - b \cdot \nabla$ with measurable matrix $a$ and vector field $b$ having critical-order singularities as the generators of Markov semigroups in $L^p$ and $C_\infty$.
In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} - \nabla \cdot a \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - a \cdot \nabla^2 + b \cdot \nabla, \end{equation} where $a=I+c \mathsf{f} \otimes \mathsf{f}$, the vector fields $\nabla_i \mathsf{f}$ ($i=1,2,\dots,d$) and $b$ are form-bounded (this includes the sub-critical class $[L^d + L^\infty]^d$ as well as vector fields having critical-order singularities). We characterize quantitative dependence on $c$ and the values of the form-bounds of the …
Remarks on the paper: Ikehata, M., Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(σ-iωε)\nabla u=0$, Inverse Problems, 18(2002), 1281-1290
In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} -Δ- \nabla \cdot (a-I) \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - Δ- (a-I) \cdot \nabla^2 + b \cdot \nabla, \end{equation} where the second order perturbations are given by the matrix $$a-I=c|x|^{-2}x \otimes x, \quad c>-1.$$ The vector field $b:\mathbb R^d \rightarrow \mathbb R^d$ is form-bounded with the form-bound $δ>0$ (this includes a sub-critical class $[L^d + L^\infty]^d$, as well as vector fields having critical-order singularities). We…
This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion \begin{equation}\nonumber \left\{ \begin{aligned} &n_t+u\cdot\nabla n=\nabla\cdot\left(|\nabla n|^{p-2}\nabla n\right)-\nabla\cdot(n\nabla c), &x\inΩ,\ t>0,\ \ &c_t+u\cdot\nabla c=Δc-nc,&x\inΩ,\ t>0,\ \ &u_t=Δu+\nabla P+n\nablaϕ,&x\inΩ,\ t>0,\ \ &\nabla\cdot u=0, &x\inΩ,\ t>0,\ \ \end{aligned} \right. \end{equation} where $ϕ\in W^{2,\infty}(Ω)$ is the gravitational potential. It is proved that global bounded weak solutions …
We establish a result which provides the criteria for a divergence operator dependent both on the gradient and the function itself, to satisfy the S+ property. The result is complemented by several examples and an application.
In bounded smooth domains $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, we consider the Keller-Segel-Stokes system \begin{align*} n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c),\\ c_t + u\cdot \nabla c &= Δc - c + n,\\ u_t &= Δu + \nabla P + n\nabla ϕ, \qquad \nabla \cdot u=0, \end{align*} and prove global existence of generalized solutions if \[ χ<\begin{cases} \infty,&N=2,\\ \frac{5}{3},&N=3. \end{cases} \] These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.
This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion \begin{eqnarray} \left\{\begin{array}{lll} n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(nχ(c)\nabla c),& x\inΩ,\ t>0, c_t+u\cdot\nabla c=Δc-nf(c),& x\inΩ,\ t>0, u_t+(u\cdot\nabla) u=Δu+\nabla P+n\nablaΦ,& x\inΩ,\ t>0, \nabla\cdot u=0,& x\inΩ,\ t>0 \end{array}\right. \end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $Ω\subset \mathbb{R}^3$ with smooth boundary. Here, $Φ\in …
We consider an inverse boundary value problem for the equation $\nabla\cdot(σ-iωε)\nabla u=0$ in a given bounded domain $Ω$ at a fixed $ω>0$. $σ$ and $ε$ denote the conductivity and permittivity of the material forming $Ω$, respectively. We give some formulae for extracting information about the location of the discontinuity surface of $(σ,ε)$ from the Dirichlet-to-Neumann map. In order to obtain results we make use of two methods. The first is the enclosure method which is based on a new role of the exponentially growing solutions of the equation for the background material. The second is a g…
In this paper, we consider the following system $$\left\{\begin{array}{ll} n_t+u\cdot\nabla n&=Δn-\nabla\cdot(n\mathcal{S}(|\nabla c|^2)\nabla c)-nm,\\ c_t+u\cdot\nabla c&=Δc-c+m,\\ m_t+u\cdot\nabla m&=Δm-mn,\\ u_t&=Δu+\nabla P+(n+m)\nablaΦ,\qquad \nabla\cdot u=0 \end{array}\right.$$ which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where $\mathcal{S}$ is a given function fulfilling $$|\mathcal{S}(σ)|\leq K_{\mathcal{S}}(1+σ)^{-\fracθ{2}},\qquad σ\geq 0$$ with some $K_{\mathcal{S}}>0.$ Based on conditional estimates of the quantity $c$ and the gra…
We consider the system \begin{align*} \begin{cases} n_t + u \cdot \nabla n = Δn - χ\nabla \cdot (\frac{n}{c} \nabla c), \\ c_t + u \cdot \nabla c = Δc - nf(c), \\ u_t + (u \cdot \nabla) u = Δu + \nabla P + n \nabla ϕ, \quad \nabla \cdot u = 0, \end{cases} \end{align*} in smooth bounded domains $Ω\subset \mathbb R^N$, $N \in \mathbb N$, for given $f \ge 0$, $ϕ$ and complemented with initial and homogeneous Neumann--Neumann--Dirichlet boundary conditions, which models aerobic bacteria in a fluid drop. We assume $f(0) = 0$ and $f'(0) = 0$, that is, that $f$ decays slower than linearly near $0$, a…
A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations
本文研究了二维广义磁流体动力学方程组\begin{eqnarray*} \left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - ν(-\triangle)^α u b_t + u \cdot \nabla b & = & b \cdot \nabla u - κ(-\triangle)^β b \end{array}\right. \end{eqnarray*}的整体正则性，得到了当$ 0\leqslantα< 1 / 2,\,\, β\geqslant 1, \,\,3α+ 2β>3 $时的整体正则解，改进了\cite{TYZ2013}的结果.特别地，我们得到了当$α=0$和$β>\frac 32$满足时二维广义MHD的整体正则性。
In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$ time differentiability in an expontential Besov-Orlicz space for the solution process $u$. Furthermore, we prove $1/2$ time differentiability of the nonlinear gradient $|\nabla u|^{\frac{p-2}{2}} \nabla u$ in a Nikolskii space.
This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =Δn^m - χ\nabla\cdot(n\nabla c) + κn -μn^2$, $x \in Ω,\ t>0$, $c_t + u\cdot\nabla c = Δc - nc$, $x \in Ω,\ t>0$, $u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ, \quad \nabla\cdot u = 0$, $x \in Ω,\ t>0$, where $Ω\subset \mathbb{R}^3$ is a bounded domain and $χ,κ\ge 0$ and $m, μ>0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1…
In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{δE}{δh}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution prese…
In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid: \begin{equation*} \left\{ \begin{split} \partial_t n + \mathbf{u}\cdot\nabla n & = Δn - \nabla\cdot\left(χ(c)n\nabla c\right) + \nabla\cdot(n\nablaϕ), \\ \partial_t c + \mathbf{u}\cdot\nabla c &= Δc - nf(c), \\ \partial_t\mathbf{u} + κ(\mathbf{u}\cdot\nabla)\mathbf{u} + \nabla P & = Δ\mathbf{u} - n\nablaϕ+ χ(c)n\nabla c, \\ \nabla\cdot\mathbf{u} &= 0 \end{split} \right. \end{equation*} in $\mathbb{R}^d\times(0,T)\, (d=2,3)$. One of the nove…
We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=Δn-\nabla\!\cdot(n\nabla c)+κn-μn^{2},\ &x\inΩ,& t>0,\\ c_{t}+u\cdot\!\nabla c&=Δc-nc,\ &x\inΩ,& t>0,\\ u_{t}&=Δu+\nabla P+n\nablaϕ,\ &x\inΩ,& t>0,\\ \nabla\cdot u&=0,\ &x\inΩ,& t>0,\\ \big(\nabla n-n\nabla c\big)\cdotν&=0,\quad c=c_{\star}(x),\quad u=0, &x\in\partialΩ,& t>0, \end{array}\right. \end{align*} where $κ\geq0$, $μ>0$ and, in contrast to the commonly investigated variants of chemotaxi…
This paper considers the chemotaxis-Navier--Stokes system with nonlinear diffusion and logistic-type degradation term \begin{align*} \begin{cases} n_t + u\cdot\nabla n = \nabla \cdot(D(n)\nabla n) - \nabla\cdot(n χ(c) \nabla c) + κn - μn^α, & x\in Ω,\ t>0, \\ c_t + u\cdot\nabla c = Δc - nf(c), & x \in Ω,\ t>0, \\ u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ+ g, \ \nabla\cdot u = 0, & x \in Ω,\ t>0, \end{cases} \end{align*} where $Ω\subset \mathbb{R}^3$ is a bounded smooth domain; $D \ge 0$ is a given smooth function such that $D_1 s^{m-1} \le D(s) \le D_2 s^{m-1}$ for all $s\ge 0$ with som…
In this paper, we obtain $C^{1,α}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switch…
We shall establish the interior Hölder continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are \begin{equation} u_t= \nabla \cdot \bigg( |\nabla u|^{p-2} \nabla u \bigg), \quad \text{ for } \quad 1<p<2, \end{equation} and \begin{equation} u_{t}- \nabla \cdot ( u^{m-1} | \nabla u |^{p-2} \nabla u ) =0 , \quad \text{for} \quad m+p>3-\frac{p}{N}, \end{equation} via a new and simplified proof using recent techniques on expansion of positivity and $L^{1}$-Harnack estimates.
We prove two new functional inequalities of the forms\[ \int_G φ(ψ- \overlineψ) \leq \frac{1}{a}\int_G ψ\ln \left(\frac{\;ψ\;}{ \overlineψ}\right) + \frac{a}{4β_0} \left\{ \int_G ψ\right\}\int_G|\nabla φ|^2 \] and \[ \int_G ψ\ln \left(\frac{\;ψ\;}{ \overlineψ}\right) \leq \frac{1}{β_0}\left\{ \int_G ψ\right\}\int_G |\nabla \ln(ψ)|^2 \] for any finitely connected, bounded $C^2$-domain $G \subseteq \mathbb{R}^2$, a constant $β_0 > 0$, any $a > 0$ and sufficiently regular functions $φ$, $ψ$. We then illustrate their usefulness by proving long time stabilization and eventual smoothness properties …
We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux $$\left\{\begin{array}{l} n_t+u\cdot\nabla n=Δn-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in Ω, t>0,\\ c_t+u\cdot\nabla c=Δc-c+m,\quad x\in Ω, t>0,\\ m_t+u\cdot\nabla m=Δm-nm,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+(n+m)\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.$$ in a bounded domain $Ω\subset \mathbb{R}^3$ with smooth boundary, where $κ\in \mathbb{R}$ is given constant, $S$ is a matrix-valued sensitiv…
Systems of the type $$\begin{cases} u_t = \nabla \cdot (D_1(u) \nabla u - S_1(u) \nabla v) + f_1(u, v),\\ v_t = \nabla \cdot (D_2(v) \nabla v + S_2(v) \nabla u) + f_2(u, v) \end{cases} \qquad (\star)$$ can be used to model pursuit-evasion relationships between predators and prey. Apart from local kinetics given by $f_1$ and $f_2$, the key components in this system are the taxis terms $-\nabla \cdot (S_1(u) \nabla v)$ and $+\nabla \cdot (S_2(v) \nabla u)$; that is, the species are not only assumed to move around randomly in space but are also able to partially direct their movement depending on…
This paper deals with convergence of solutions to a class of parabolic Keller-Segel systems, possibly coupled to the (Navier-)Stokes equations in the framework of the full model \begin{eqnarray*} \left\{ \begin{array}{lcl} \, \, \partial_t n_ε + u_ε \cdot \nabla n_ε &=& Δn_ε - \nabla \cdot \Big( n_ε S(x, n_ε, c_ε)\cdot\nabla c_ε\Big) + f(x, n_ε, c_ε), \\[1mm] ε\partial_t c_ε + u_ε\cdot\nabla c_ε &=& Δc_ε - c_ε + n_ε , \\[1mm] \,\,\partial_t u_ε + κ(u_ε\cdot\nabla) u_ε &=& Δu_ε + \nabla P_ε + n_ε \nablaϕ, \qquad \nabla\cdot u_ε=0 \end{array} \right. \end{eqnarray*} to solutions of the parabolic…
The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-χ\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-μn), c_t+u\cdot \nabla c=\triangle c-nc, u_t+ (u\cdot \nabla) u=Δu+\nabla P+n\nablaϕ, \nabla\cdot u=0, \end{array}\right. \end{equation*} is considered in a bounded smooth domain $Ω\subset \mathbb{R}^2$, where $ϕ\in W^{1,\infty}(Ω)$, $χ>0$, $r\in \mathbb{R}$ and $μ> 0$ are given parameters. It is shown that there exists a value $μ_*(Ω,χ, r)\geq 0$ such that whenever $ μ>μ_*(Ω,χ, r)$, the global-in-time classi…
We consider the $\nabla ϕ$ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least $C^{2,β}$ for some $β>0$. We also prove a fluctuation-dissipation relation by identifying its Hessian with the covariance matrix characterizing the scaling limit of the model. Finally, we obtain a quantitative rate of convergence for the Hessian of the finite-volume surface tension to that …
In this note we comment on the homogenization of a random elliptic operator in divergence form $-\nabla \cdot a\nabla$, where the coefficient field $a$ is distributed according to a stationary, but not necessarily ergodic, probability measure $P$. We generalize the well-known case for $P$ stationary and ergodic by showing that the operator $-\nabla \cdot a(\frac{\cdot}{\varepsilon})\nabla$ almost surely homogenizes to a constant-coefficient, random operator $-\nabla \cdot A_h\nabla$. Furthermore, we use a disintegration formula for $P$ with respect to a family of ergodic and stationary probabi…
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $\displaystyle -\operatorname{div}(A(|\nabla u|)\nabla u)+B\left( |\nabla u|\right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
We consider initial-boundary value problems for the $κ$-dependent family of chemotaxis-(Navier--)Stokes systems \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=Δn-\nabla\!\cdot(n\nabla c),\ &x\inΩ,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=Δc-cn,\ &x\inΩ,& t>0,\\ u_{t}&+&κ(u\cdot\nabla)u&=Δu+\nabla P+n\nablaϕ,\ &x\inΩ,& t>0,\\ &&\nabla\cdot u&=0,\ &x\inΩ,& t>0, \end{array}\right. \end{align*} in a bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary and given potential function $ϕ\in C^{1+β}(\overlineΩ)$ for some $β>0$. It is known that…
In this work, we study a chemotaxis-Navier-Stokes model in a two-dimensional setting as below, \begin{eqnarray} \left\{ \begin{array}{llll} \displaystyle n_{t}+\mathbf{u}\cdot\nabla n=Δn-\nabla \cdot(n\nabla c)+f(n), &&x\inΩ,\,t>0,\\ \displaystyle c_{t}+\mathbf{u}\cdot\nabla c=Δc - c+ n, &&x\inΩ,\,t>0,\\ \displaystyle \mathbf{u}_{t}+κ(\mathbf{u}\cdot\nabla)\mathbf{u}=Δ\mathbf{u} +\nabla P+ n\nablaϕ, &&x\inΩ,\,t>0,\\ \displaystyle \nabla\cdot\mathbf{u}=0,&&x\inΩ,\,t>0.\\ \end{array} \right. \end{eqnarray} Motivated by a recent work due to Winkler, we aim at investigating generalized solvability…
This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(n\nabla c),\quad x\in Ω, t>0, \\ c_t+u\cdot\nabla c=Δc-c+n,\quad x\in Ω, t>0,\\ u_t+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0, \quad x\in Ω, t>0 \end{array}\right.$$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $Ω\subseteq \mathbb{R}^3$ with smooth boundary, where $ϕ\in W^{1,\infty}(Ω),m>0$. It is proved that if $m>\frac{4}{3}$, then for a…
In this paper, the three-dimensional chemotaxis-stokes system \begin{eqnarray*} \left\{\begin{array}{lll} \medskip n_{t}+u\cdot\nabla n=Δn^m-\nabla\cdot(n S(x,n,c)\cdot\nabla c),&x\inΩ,\ \ t>0, \medskip c_t+u\cdot\nabla c=Δc-nf(c),&x\inΩ,\ \ t>0, \medskip u_t+\nabla P=Δu +n\nablaϕ,&x\inΩ,\ \ t>0, \nabla\cdot u=0, &x\inΩ,\ \ t>0,, \end{array}\right. \end{eqnarray*} posed in a bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary is considered under the no-flux boundary condition for $n$, $c$ and the Dirichlect boundary condition for $u$ under the assumption that the Frobenius norm of the t…
In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in $\mathbb{R}^n=\mathbb{R}^2$) for $p>2$ in the presence of Dirichlet as well as Neumann boundary conditions and with $\mathbf{h}\in C^{1-n/q}$, $f\in L^q$, $2=n<q\leq\infty$. The regularity assumptions on the principal part $A$ as well as that on the Dirichlet/Neumann conditions are exactly the same as in the linear case and therefore sharp (see Remark 2.5 below). O…
In this article we prove existence, uniqueness and regularity for the singular equation \begin{eqnarray*} \begin{cases} |\nabla u|^α(F(D^{2}u)+h(x)\cdot\nabla u)+c(x)|u|^αu+p(x)u^{-γ}=0 \ \mbox{ in } \ Ω\\ u>0 \ \mbox{ in } \ Ω, \ u=0 \ \mbox{ on } \ \partialΩ\end{cases} \end{eqnarray*} when $p$ is some continuous and positive function, $c$ and $h$ are continuous, $α> -1$ and $F$ is Fully non linear elliptic. Some conditions on the first eigenvalue for the operator $|\nabla u|^α(F(D^{2}u)+h(x)\cdot\nabla u)+c(x)|u|^αu$ are required. The results generalizes the well known results of Lazer and M…
