Stability of stationary solutions to the Navier–Stokes equations in the Besov space

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space ${\mathbb{R}}^n$ for $n\ge 3$. It is clarified that if w is small in ${\dot{B}}_{p_{*},q^{\prime }}^{-1+\frac{n}{p_{*}}}$ for and , then for every small initial disturbance $a\in {\dot{B}}_{p_0,q}^{-1+\frac{n}{p_0}}$ with and ($1/q+1/q^{\prime }=1$), there exists a unique solution $v(t)$ of the nonstationary Navier–Stokes equations on (0, ∞) with $v(0)=w+a$ such that ${\parallel v(t)-w\parallel }_{L^r}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{r})})$ and ${\parallel v(t)-w\parallel }_{{\dot{B}}_{p,q}^s}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{p})-\frac{s}{2}})$ as $t\to \infty$, for , , and small $s>0$.     

Estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the space of quasi-continuous functions

We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions $B_{p,\theta }^{\mathit{r}}({\mathbb{T}}^d)$ with mixed smoothness in the metric of the space of quasi-continuous functions $QC({\mathbb{T}}^d)$. We also showed that for $2\le p\le \infty$, , $r_1>\frac{1}{2}$, $d\ge 2$, the estimate of the corresponding asymptotic characteristic is exact in order.     

On the n-th linear polarization constant of Rn$\mathbb {R}^n$

We prove that given any set of n unit vectors ${\{v_i\}}_{i=1}^n\subset {\mathbb{R}}^n$, the inequality $$\underset{{\parallel x\parallel }_{{\mathbb{R}}^n}=1}{sup}|⟨x,v_1⟩\cdots ⟨x,v_n⟩|\ge n^{-n/2}$$ holds for $n\le 14$. Moreover, the equality is attained if and only if ${\{v_i\}}_{i=1}^n$ is an orthonormal system.     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.     

The two-dimensional stationary Navier–Stokes equations in toroidal Besov spaces

We consider the stationary Navier–Stokes equations in the two-dimensional torus ${\mathbb{T}}^2$. For any $\epsilon >0$, we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces ${\dot{B}}_{p+\epsilon ,q}^{-1+\frac{2}{p}}({\mathbb{T}}^2)$ for given small external forces in ${\dot{B}}_{p+\epsilon ,q}^{-3+\frac{2}{p}}({\mathbb{T}}^2)$ when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well-posedness is proved by using the embedding property and the para-product estimate in homogeneous Besov spaces. In addition, for the case $(p,q)\in (\{2\}\times (2,\infty ])\cup ((2,\infty ]\times [1,\infty ])$, we can show the ill-posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill-posedness by showing the discontinuity of a certain solution map from ${\dot{B}}_{p,q}^{-3+\frac{2}{p}}({\mathbb{T}}^2)$ to ${\dot{B}}_{p,q}^{-1+\frac{2}{p}}({\mathbb{T}}^2)$.     

On fractional semidiscrete Dirac operators of Lévy–Leblond type

In this paper, we introduce a wide class of space-fractional and time-fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space-time lattice $h{\mathbb{Z}}^n\times [0,\infty )$ ($h>0$), resembling to fractional semidiscrete counterparts of the so-called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup ${\left\{\exp (-t{e}^{i\theta }{(-{\mathrm{\Delta }}_h)}^{\alpha })\right\}}_{t\ge 0}$, carrying the parameter constraints and $|\theta |\le \frac{\alpha \pi }{2}$. The results obtained involve the study of Cauchy problems on $h{\mathbb{Z}}^n\times [0,\infty )$.     

On the fixed part of pluricanonical systems for surfaces

We show that $|m{K}_X|$ defines a birational map and has no fixed part for some bounded positive integer m for any $\frac{1}{2}$-lc surface X such that $K_X$ is big and nef. For every positive integer $n\ge 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, $\mathrm{mld}(X_{n,i})>\frac{1}{n}$ for every i, ${\lim }_{i\to +\infty }\mathrm{mld}(X_{n,i})=\frac{1}{n}$, and for any positive integer m, there exists i such that $|m{K}_{X_{n,i}}|$ has nonzero fixed part. These results answer the surface case of a question of Xu.     

Vanishing of higher order Alexander-type invariants of plane curves

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves $C^{\prime }$ and $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of $C^{\prime }$ and $C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ${\mathrm{\Delta }}_C^{multi}$ is a power of $(t-1)$, and we characterize when ${\mathrm{\Delta }}_C^{multi}=1$ in terms of the defining equations of $C^{\prime }$ and $C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.     

On the interpolation constants for variable Lebesgue spaces

For $\theta \in (0,1)$ and variable exponents $p_0(·),q_0(·)$ and $p_1(·),q_1(·)$ with values in [1, ∞], let the variable exponents $p_{\theta }(·),q_{\theta }(·)$ be defined by $$1/p_{\theta }(·):=(1-\theta )/p_0(·)+\theta /p_1(·),1/q_{\theta }(·):=(1-\theta )/q_0(·)+\theta /q_1(·).$$ The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space $L^{p_j(·)}$ to the variable Lebesgue space $L^{q_j(·)}$ for $j=0,1$, then $${\parallel T\parallel }_{L^{p_{\theta }(·)}\to L^{q_{\theta }(·)}}\le {C\parallel T\parallel }_{L^{p_0(·)}\to L^{q_0(·)}}^{1-\theta }{\parallel T\parallel }_{L^{p_1(·)}\to L^{q_1(·)}}^{\theta },$$ where C is an interpolation constant independent of T. We consider two different modulars ${\varrho }^{\max }(·)$ and ${\varrho }^{\mathrm{sum}}(·)$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants C_max and C_sum, which imply that $C_{\max }\le 2$ and $C_{\mathrm{sum}}\le 4$, as well as, lead to sufficient conditions for $C_{\max }=1$ and $C_{\mathrm{sum}}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that $p_j(·)=q_j(·)$, $j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ${\varrho }^{\max }(·)={\varrho }^{\mathrm{sum}}(·)$).     

Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$-actions and the locus of Gorenstein stable surfaces

In this note, the geography of minimal surfaces of general type admitting ${\mathbb{Z}}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ contains surfaces admitting a ${\mathbb{Z}}_2^2$-action for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$ or $K^2=8\chi$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification ${\overline{\mathfrak{M}}}_{K^2,\chi }$ of Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$.     

Regularity of powers of Stanley–Reisner ideals of one-dimensional simplicial complexes

Let Δ be a one-dimensional simplicial complex. Let $I_{\mathrm{\Delta }}$ be the Stanley–Reisner ideal of Δ. We prove that for all $s\ge 1$ and all intermediate ideals J generated by $I_{\mathrm{\Delta }}^s$ and some minimal generators of $I_{\mathrm{\Delta }}^{(s)}$, we have $$regJ=reg{I}_{\mathrm{\Delta }}^s=reg{I}_{\mathrm{\Delta }}^{(s)}.$$     

Totally real flat minimal surfaces in the hyperquadric

In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both $Q_{N-2}$ and $\mathbb{C}{P}^{N-1}$, we determine them for $N=4,5,6$, and give a classification theorem when they are Clifford solutions.