Y spaces and global smooth solution of fractional Navier–Stokes equations with initial value in the critical oscillation spaces

For fractional Navier–Stokes equations and critical initial spaces X, one used to establish the well-posedness in the solution space which is contained in $C({\mathbb{R}}_{+},X)$. In this paper, for heat flow, we apply parameter Meyer wavelets to introduce Y spaces $Y^{m,\beta }$ where $Y^{m,\beta }$ is not contained in $C({\mathbb{R}}_{+},{\dot{B}}_{\infty }^{1?2\beta ,\infty })$. Consequently, for $\frac{1}{2}<\beta <1$, we establish the global well-posedness of fractional Navier–Stokes equations with small initial data in all the critical oscillation spaces. The critical oscillation spaces may be any Besov–Morrey spaces ${({\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ or any Triebel–Lizorkin–Morrey spaces ${({\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ where $1\le p,q\le \infty ,0\le {\gamma }_2\le \frac{n}{p},{\gamma }_1?{\gamma }_2=1?2\beta$. These critical spaces include many known spaces. For example, Besov spaces, Sobolev spaces, Bloch spaces, Q-spaces, Morrey spaces and Triebel–Lizorkin spaces etc.     

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.     

Formality theorem for g-manifolds

With any $\mathfrak{g}$-manifold M are associated two dglas $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))$ and $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$, whose cohomologies $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ and $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_{\infty }$ quasi-isomorphism $\mathrm{\Phi }:\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))\to \mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$ whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold M is an isomorphism of Gerstenhaber algebras from $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ to $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$.     

Superconcentration,and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in ${\mathbb{R}}^n$in the ?-position, and such that the space $({\mathbb{R}}^n,{6?6}_B)$ admits a 1-unconditional basis. Then for any $\epsilon \in (0,1/2]$, and for random $c\epsilon \log ?n/\log ?\frac{1}{\epsilon }$-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\epsilon )$-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ?-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and ${\mathbb{E}}_{{\gamma }_n}{6?6}_B\le n^c{\mathbb{E}}_{{\gamma }_n}{6{\mathrm{grad}}_B(?)6}_2$ for a small universal constant $c>0$, where ${\mathrm{grad}}_B(?)$ is the gradient of ${6?6}_B$ and ${\gamma }_n$ is the standard Gaussian measure in ${\mathbb{R}}^n$. Then for any $p\in [1,c\log ?n]$ the p-th power of the norm ${6?6}_B^p$ is $\frac{C}{\log ?n}$-superconcentrated in the Gauss space.     

Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption

The chemotaxis system

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\mathrm{?}?\left(\frac{u}{v}\mathrm{?}v\right),\hfill \\ v_t=\mathrm{\Delta }v?uv,\hfill \end{array}(?)$

is considered under homogeneous Neumann boundary conditions in the ball $\mathrm{\Omega }=B_R(0)?{\mathbb{R}}^n$, where $R>0$ and $n\ge 2$.Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data $(u_0,v_0)$ satisfying $u_0\ge 0$ and $v_0>0$, the present paper establishes the existence of a globally defined pair $(u,v)$ of radially symmetric functions which are continuous in $(\overline{\mathrm{\Omega }}?\{0\})\times [0,\infty )$ and smooth in $(\overline{\mathrm{\Omega }}?\{0\})\times (0,\infty )$, and which solve the corresponding initial-boundary value problem for (?) with $(u(?,0),v(?,0))=(u_0,v_0)$ in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (?) involving arbitrarily large initial data.     

Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

A class of chemotaxis-Stokes systems generalizing the prototype

$\{\begin{array}{ccc}\hfill n_t+u?\mathrm{?}n& \hfill =\hfill & \mathrm{?}?\left(n^{m?1}\mathrm{?}n\right)?\mathrm{?}?\left(n\mathrm{?}c\right),\hfill \\ \hfill c_t+u?\mathrm{?}c& \hfill =\hfill & \mathrm{\Delta }c?nc,\hfill \\ \hfill u_t+\mathrm{?}P& \hfill =\hfill & \mathrm{\Delta }u+n\mathrm{?}?,\mathrm{?}?u=0,\hfill \end{array}$

is considered in bounded convex three-dimensional domains, where $?\in W^{2,\infty }(\mathrm{\Omega })$ is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that

(0.1)$m>\frac{9}{8}.$

Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state $(\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{n}_0,0,0)$ in the large time limit.This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1).     

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$. These groups contain as special cases the odd dimensional general linear groups ${\mathrm{GL}}_{2n+1}(R)$ where R is any ring, the odd dimensional orthogonal and symplectic groups ${\mathrm{O}}_{2n+1}(R)$ and ${\mathrm{Sp}}_{2n+1}(R)$ where R is any commutative ring and further the first author's even dimensional unitary groups ${\mathrm{U}}_{2n}(R,\mathrm{\Lambda })$ where $(R,\mathrm{\Lambda })$ is any form ring. We classify the E-normal subgroups of the groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ (i.e. the subgroups which are normalized by the elementary subgroup ${\mathrm{EU}}_{2n+1}(R,\mathrm{\Delta })$), under the condition that R is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ by conjugation on the set of all E-normal subgroups.     

Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant

$\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u?\mathrm{?}?(u{\chi }_1(w)\mathrm{?}w)+{\mu }_{1}u(1?u?a_{1}v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v?\mathrm{?}?(v{\chi }_2(w)\mathrm{?}w)+{\mu }_{2}v(1?a_{2}u?v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w?(\alpha u+\beta v)w,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary, where the initial data $(u_0,v_0)\in {(C^0(\stackrel{￣}{\mathrm{\Omega }}))}^2$ and $w_0\in W^{1,\infty }(\mathrm{\Omega })$ are non-negative and the parameters $d_1,d_2,d_3>0$, ${\mu }_1,{\mu }_2>0$, $a_1,a_2>0$ and $\alpha ,\beta >0$. The chemotactic function ${\chi }_i(w)$ ($i=1,2$) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for $i=1,2$,(i) ${\chi }_i(w)={\chi }_{0,i}>0$ and

${6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}<\frac{\pi \sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}?\frac{2\sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}\arctan ?\frac{d_i?d_3}{2}\sqrt{\frac{n+1}{d_i{d}_3}};$

(ii) $0<{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}\le \frac{d_3}{3(n+1){6{\chi }_{i}6}_{L^{\infty }[0,{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}]}}\min ?\{\frac{2d_i}{d_i+d_3},1\}$.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If $a_1,a_2\in (0,1)$ and $u_0\ne 0\ne v_0$, then any global bounded solution exponentially converge to $(\frac{1?a_1}{1?a_1{a}_2},\frac{1?a_2}{1?a_1{a}_2},0)$ as $t\to \infty$;? If $a_1>1>a_2>0$ and $v_0\ne 0$, then any global bounded solution exponentially converge to $(0,1,0)$ as $t\to \infty$;? If $a_1=1>a_2>0$ and $v_0\ne 0$, then any global bounded solution algebraically converge to $(0,1,0)$ as $t\to \infty$.     

Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger–Newton equation)

$?\mathrm{\Delta }u+u=(I_{\alpha }?|u{|}^p)|u{|}^{p?2}u.$

Here $I_{\alpha }$ stands for the Riesz potential of order $\alpha \in (0,N)$, and $\frac{N?2}{N+\alpha }<\frac{1}{p}\le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when α is either close to 0 or close to N.     

A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities

This paper is devoted to study the planar polynomial system:

$\dot{x}=ax?y+P_n(x,y),\dot{y}=x+ay+Q_n(x,y),$

where $a\in \mathbb{R}$ and $P_n,Q_n$ are homogeneous polynomials of degree $n\ge 2$. Denote $\psi (\theta )=\cos ?(\theta )?Q_n(\cos ?(\theta ),\sin ?(\theta ))?\sin ?(\theta )?P_n(\cos ?(\theta ),\sin ?(\theta ))$. We prove that the system has at most 1 limit cycle surrounding the origin provided $(n?1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0$. Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin.     

Positive effects of repulsion on boundedness in a fully parabolic attraction–repulsion chemotaxis system with logistic source

In this paper we study the global boundedness of solutions to the fully parabolic attraction–repulsion chemotaxis system with logistic source: $u_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)+\xi \mathrm{?}?(u\mathrm{?}w)+f(u)$, $v_t=\mathrm{\Delta }v?\beta v+\alpha u$, $w_t=\mathrm{\Delta }w?\delta w+\gamma u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$), where χ, α, ξ, γ, β and δ are positive constants, and $f:\mathbb{R}\to \mathbb{R}$ is a smooth function generalizing the logistic source $f(s)=a?b{s}^{\theta }$ for all $s\ge 0$ with $a\ge 0$, $b>0$ and $\theta \ge 1$. It is shown that when the repulsion cancels the attraction (i.e. $\chi \alpha =\xi \gamma$), the solution is globally bounded if $n\le 3$, or $\theta >{\theta }_n:=\min ?\{\frac{n+2}{4},\frac{n\sqrt{n^2+6n+17}?n^2?3n+4}{4}\}$ with $n\ge 2$. Therefore, due to the inhibition of repulsion to the attraction, in any spatial dimension, the exponent θ is allowed to take values less than 2 such that the solution is uniformly bounded in time.     

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

We study the chemotaxis effect vs. logistic damping on boundedness for the well-known minimal Keller–Segel model with logistic source:

$\{\begin{array}{ccc}\hfill & u_t=\mathrm{?}?(\mathrm{?}u?\chi u\mathrm{?}v)+u?\mu u^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^2$ with $\chi ,\mu >0$, nonnegative initial data $u_0$, $v_0$, and homogeneous Neumann boundary data. It is well known that this model allows only for global and uniform-in-time bounded solutions for any $\chi ,\mu >0$. Here, we carefully employ a simple and new method to regain its boundedness, with particular attention to how upper bounds of solutions qualitatively depend on χ and μ. More, precisely, it is shown that there exists $C=C(u_0,v_0,\mathrm{\Omega })>0$ such that

${6u(?,t)6}_{L^{\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\chi K(\chi ,\mu )N(\chi ,\mu )]$

and

${6v(?,t)6}_{W^{1,\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\frac{{\chi }^{\frac{8}{3}}}{\mu }{K}^{\frac{8}{3}}(\chi ,\mu )]=:CN(\chi ,\mu )$

uniformly on $[0,\infty )$, where

$K(\chi ,\mu )=M(\chi ,\mu )E(\chi ,\mu ),M(\chi ,\mu )=1+\frac{1}{\mu }+\sqrt{\chi }(1+\frac{1}{{\mu }^2})$

and

$E(\chi ,\mu )=\exp ?\left[\frac{\chi C_{GN}^2}{2\min ?\{1,\frac{2}{\chi }\}}\right(\frac{4}{\mu }{6u_{0}6}_{L^1(\mathrm{\Omega })}+\frac{13}{2{\mu }^2}|\mathrm{\Omega }|+{6\mathrm{?}{v}_{0}6}_{L^2(\mathrm{\Omega })}^2\left)\right].$

We notice that these upper bounds are increasing in χ, decreasing in μ, and have only one singularity at $\mu =0$, where the corresponding minimal model (removing the term $u?\mu u^2$ in the first equation) is widely known to possess blow-ups for large initial data.