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.     

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.     

Self-similar solutions of stationary Navier–Stokes equations

In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension $n=3,4$. For $n=3$, if the external force is axisymmetric, scaling invariant, $C^{1,\alpha }$ continuous away from the origin and small enough on the sphere $S^2$, we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class $C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$. Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular ($U\in C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$) axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in $L^{4/3,\infty }({\mathbb{R}}^4)$.     

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.     

Complicated asymptotic behavior of solutions for porous medium equation in unbounded space

In this paper, we find that the unbounded spaces $Y_{\sigma }({\mathbb{R}}^N)$$(0<\sigma <\frac{2}{m?1})$ can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted $L^1$–$L^{\infty }$ estimates for the solutions.     

Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation ${(?\mathrm{\Delta })}^{\alpha }$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta }$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q({\mathrm{BMO}}_x^{?\gamma })$ with $q>2$ and $\gamma \in (0,2\alpha ?1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.     

Construction and stability of blowup solutions for a non-variational semilinear parabolic systemConstruction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel

We consider the following parabolic system whose nonlinearity has no gradient structure:

$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+|v{|}^{p?1}v,\hfill & {?}_{t}v=\mu \mathrm{\Delta }v+|u{|}^{q?1}u,\hfill \\ u(?,0)=u_0,\hfill & v(?,0)=v_0,\hfill \end{array}$

in the whole space ${\mathbb{R}}^N$, where $p,q>1$ and $\mu >0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0,v_0)$ simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:

$\{\begin{array}{cc}\hfill & u(x,t)～\mathrm{\Gamma }{[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(p+1)}{pq?1}},\hfill \\ \hfill & v(x,t)～\gamma {[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(q+1)}{pq?1}},\hfill \end{array}$

with $b=b(p,q,\mu )>0$ and $(\mathrm{\Gamma },\gamma )=(\mathrm{\Gamma }(p,q),\gamma (p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $\mu =1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.     

Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of ${\mathbb{R}}^N$ ($N\ge 2$). In the case of non-degenerate diffusion, Cie?lak–Stinner [3], [4] proved that if $q>m+\frac{2}{N}$, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if $q>m+\frac{2}{N}$ (see Ishida–Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when $q>m+\frac{2}{N}$.     

Global well-posedness and long-time dynamics for a higher order quasi-geostrophic type equation

In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier–Stokes–Korteweg system with Coriolis force, when the Mach, Rossby and Weber numbers go to zero at the same rate.The scope of the present paper is twofold. First of all, we investigate well-posedness of such a model on the whole space ${\mathbb{R}}^2$: we prove that it is well-posed in $H^s$ for any $s\ge 3$, globally in time. Interestingly enough, we show that this equation owns two levels of energy estimates, for which one gets existence and uniqueness of weak solutions with different regularities (namely, $H^3$ and $H^4$ regularities); this fact can be viewed as a remainder of the so called BD-entropy structure of the original system.In the second part of the paper we investigate the long-time behavior of these solutions. We show that they converge to the solution of the corresponding linear parabolic type equation, with same initial datum and external force. Our proof is based on dispersive estimates both for the solutions to the linear and non-linear problems.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

A regularity theory for quasi-linear Stochastic PDEs in weighted Sobolev spaces

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$-domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and Hölder estimates of both the solution and its gradient.     

Global solvability of the Navier–Stokes equations with a free surface in the maximal Lp-Lq regularity class

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the N-dimensional Euclidean space for $N\ge 2$. The aim of this paper is to show the global solvability of the Navier–Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}{L}_q$ regularity class. Our approach is based on the maximal $L_p\text{-}{L}_q$ regularity with exponential stability for the linearized equations, and also it is proved that solutions to the original nonlinear problem are exponentially stable.     

The free energy of the random walk pinning model

We consider the random walk pinning model. This is a random walk on ${\mathbb{Z}}^d$ whose law is given as the Gibbs measure ${\mu }_{N,Y}^{\beta }$, where the polymer measure ${\mu }_{N,Y}^{\beta }$ is defined by using the collision local time with another simple symmetric random walk $Y$ on ${\mathbb{Z}}^d$ up to time $N$. Then, at least two definitions of the phase transitions are known, described in terms of the partition function and the free energy. In this paper, we will show that the two critical points coincide and give an explicit formula for the free energy in terms of a variational representation. Also, we will prove that if $\beta$ is smaller than the critical point, then $X$ under ${\mu }_{N,Y}^{\beta }$ satisfies the central limit theorem and the invariance principle $P_Y$-almost surely.     

Boundary regularity criteria for the 6D steady Navier–Stokes and MHD equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier–Stokes and MHD equations are Hölder continuous near boundary provided that either $r^{?3}{\int }_{B_r^{+}}|u(x){|}^{3}dx$ or $r^{?2}{\int }_{B_r^{+}}|\mathrm{?}u(x){|}^{2}dx$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong–Strain [5].     

On the asymptotic behavior of radial entire solutions for the equation (−Δ)3u = up in Rn

Our main task in this note is to prove the existence and to classify the exact growth at infinity of radial positive $C^6$-solutions of ${(?\mathrm{\Delta })}^{3}u=u^p$ in ${\mathbf{R}}^n$, where $n?15$ and p is bounded from below by the sixth-order Joseph–Lundgren exponent. Following the main work of Winkler, we introduce the sub- and super-solution method and comparison principle to conclude the asymptotic behavior of solutions.     

Quasi-periodic solutions to nonlinear beam equations on compact Lie groups with a multiplicative potential

The goal of this work is to study the existence of quasi-periodic solutions to nonlinear beam equations with a multiplicative potential. The nonlinearity is required to only finitely differentiable and the frequency is along a pre-assigned direction. The result holds on any compact Lie group or homogeneous manifold with respect to a compact Lie group, which includes standard torus ${\mathbb{T}}^d$, special orthogonal group $SO(d)$, special unitary group $SU(d)$, spheres ${\mathbb{S}}^d$ and the real and complex Grassmannians. The proof is based on a differentiable Nash–Moser iteration scheme.