Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier–Stokes equations in an exterior domain $\varOmega \subset \mathbb{R }^3 $ with smooth boundary. For every prescribed constant vector $u_{\infty } \ne 0$ and every external force $f \in \dot{H}_2^{-1} (\varOmega )$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $u $ with $\nabla u \in L_2 (\varOmega )$ and $u - u_{\infty } \in L_6(\varOmega )$ . Here $\dot{H}^{-1}_2 (\varOmega )$ denotes the dual space of the homogeneous Sobolev space $\dot{H}^1_{2}(\varOmega ) $ . We prove that the weak solution $u$ fulfills the additional regularity property $u- u_{\infty } \in L_4(\varOmega )$ and $u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$ without any restriction on $f$ except for $f \in \dot{H}_2^{-1} (\varOmega )$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$ and $|u_{\infty }|$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $u_{\infty } \rightarrow 0$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $u_{\infty } \ne 0$ .     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .     

Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

We prove a global implicit function theorem. In particular we show that any Lipschitz map ${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$ (with n-dim. image) can be precomposed with a bi-Lipschitz map ${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$ such that ${f \circ \bar{g}}$ will satisfy, when we restrict to a large portion of the domain ${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$ , that ${f \circ \bar{g}}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map ${\bar{g}}$ distorts ${\mathbb{R}^{n+m}}$ in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman’s 1979 construction of a C ¹ map from [0, 1]³ onto [0, 1]² with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function ${f : [0,1]^{n} \rightarrow \mathbb{R}^{D}}$ gives rise to a large (in an appropriate sense) subset ${E \subset [0,1]^{n}}$ such that ${f|_E}$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of ${\mathbb{R}^{n}}$ . This extends results of Jones and David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors–David regular spaces lying in ${\mathbb{R}^{D}}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in ${\mathbb{R}^{D}}$ . This was previously known only for D ≥ 2n?+?1 by a result of David and Semmes.     

Hurwitz-type bound, knot surgery, and smooth {\mathbb{S }}^1-four-manifolds

In this paper we prove several related results concerning smooth $\mathbb{Z }_p$ or $\mathbb{S }^1$ actions on $4$ -manifolds. We show that there exists an infinite sequence of smooth $4$ -manifolds $X_n$ , $n\ge 2$ , which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each $X_n$ supports no smooth $\mathbb{S }^1$ -actions but admits a smooth $\mathbb{Z }_n$ -action. In order to construct such manifolds, we devise a method for annihilating smooth $\mathbb{S }^1$ -actions on $4$ -manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting. Finally, the method for annihilating smooth $\mathbb{S }^1$ -actions relies on a new obstruction we derived in this paper for existence of smooth $\mathbb{S }^1$ -actions on a $4$ -manifold: the fundamental group of a smooth $\mathbb{S }^1$ -four-manifold with nonzero Seiberg-Witten invariant must have infinite center. We also include a discussion on various analogous or related results in the literature, including locally linear actions or smooth actions in dimensions other than four.     

The second order pullback equation

Let $f,g$ be two closed $k$ -forms over $\mathbb{R }^{n}.$ The pullback equation studies the existence of a diffeomorphism $\varphi :\mathbb{R }^{n} \rightarrow \mathbb{R }^{n}$ such that $$\begin{aligned} \varphi ^{*}(g)=f. \end{aligned}$$ We prove two types of results. The first one sharpens some of the existing regularity results. The second one discusses the possibility of choosing the map $\varphi $ as the gradient of a function $\Phi :\mathbb{R }^{n} \rightarrow \mathbb R .$ We show that this is a very rare event unless the two forms are constant.     

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

On Radial Functions and Distributions and Their Fourier Transforms

We give formulas relating the Fourier transform of a radial function in $\mathbb{R}^{n}$ and the Fourier transform of the same function in $\mathbb{R}^{n+1}$ , completing the analysis of Grafakos and Teschl (J. Fourier Anal. Appl. 19:167–179, 2013) where the case of $\mathbb{R}^{n}$ and $\mathbb{R}^{n+2}$ was considered.     

The Lagrangian Radon Transform and the Weil Representation

We consider the operator $\mathcal {R}$ , which sends a function on ${\mathbb {R}}^{2n}$ to its integrals over all affine Lagrangian subspaces in ${\mathbb {R}}^{2n}$ . We discuss properties of the operator $\mathcal {R}$ and of the representation of the affine symplectic group in several function spaces on ${\mathbb {R}}^{2n}$ .     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

On p-tuples of the Grassmann manifolds

We provide a matrix invariant for isometry classes of p-tuples of points in the Grassmann manifold ${G_{n}\left(\mathbb{K}^{d}\right) }$ ( ${\mathbb{K=\mathbb{R}}}$ or ${\mathbb{C}}$ ). This invariant fully characterizes the p-tuple. We use it to classify the regular p-tuples of ${G_{2}\left(\mathbb{R}^{d}\right) }$ , ${G_{3}\left( \mathbb{R}^{d}\right) }$ and ${G_{2}\left( \mathbb{C}^{d}\right) }$ .     

On nonlinear elliptic equations with Hardy potential and L^{1}-data

We consider a class of nonlinear elliptic equations involving the Hardy potential and lower order terms whose simplest model is $$\begin{aligned} -\Delta u +b(|u|)|\nabla u|^{2}+\nu |u|^{s-1}u=\lambda \frac{u}{|x|^{2}}+f \end{aligned}$$ in a bounded open $\varOmega $ of $\mathbf{R }^{N}, N\ge 3,$ containing the origin, $s>\frac{N}{N-2}, \nu $ and $\lambda $ are positive real numbers. We prove that the presence of the term $\nu |u|^{s-1}u$ has an effect on the existence of solutions when $f\in L^{1}(\varOmega )$ assuming only that $b\in L^{1}(\mathbf{R })$ without any sign condition (i.e. $b(s)s\ge 0$ ).