Construction of characteristic functions for the system of Navier-Stokes-Voigt equations and the BBM equation

For the system of Navier-Stokes-Voigt equations $$\frac{{\partial \vec v}}{{\partial t}} - v\Delta \vec v - \aleph \frac{{\partial \Delta \vec v}}{{\partial t}} + v_\kappa \frac{{\partial \Delta \vec v}}{{\partial x_\kappa }} + grad \rho = 0, div \vec v = 0$$ and the BBM equation $$\frac{{\partial v}}{{\partial t}} + v\frac{{\partial \Delta v}}{{\partial x}} - \frac{{\partial ^3 v}}{{\partial t\partial x^2 }} = 0$$ characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) of the measure μ_t(ω)=μ(V _?1 ^t (ω)), describing the evolution in time of the probability measure μ(ω) defined on the set of initial conditions for the first initial boundary-value problem for system (1) or Eq. (2) are constructed and investigated. It is shown that the characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) constructed satisfy partial differential equations with an infinite number of independent variables (t; θ₁,θ₂,...) [the statistical equations of E. Hopf for the system (1) or Eq. (2)].     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

On the discrepancy function in arbitrary dimension, close to L 1

Let A _N to be N points in the unit cube in dimension d, and consider the discrepancy function $$ D_N (\vec x): = \sharp \left( {\mathcal{A}_N \cap \left[ {\vec 0,\vec x} \right)} \right) - N\left| {\left[ {\vec 0,\vec x} \right)} \right| $$ Here, $$ \vec x = \left( {\vec x,...,x_d } \right),\left[ {0,\vec x} \right) = \prod\limits_{t = 1}^d {\left[ {0,x_t } \right),} $$ and $ \left| {\left[ {0,\vec x} \right)} \right| $ denotes the Lebesgue measure of the rectangle. We show that necessarily $$ \left\| {D_N } \right\|_{L^1 (log L)^{(d - 2)/2} } \gtrsim \left( {log N} \right)^{\left( {d - 1} \right)/2} . $$ In dimension d = 2, the ‘log L’ term has power zero, which corresponds to a Theorem due to [11]. The power on log L in dimension d ≥ 3 appears to be new, and supports a well-known conjecture on the L ¹ norm of D _N . Comments on the discrepancy function in Hardy space also support the conjecture.     

On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration

Let Ω be an arbitrary open set in R _n , and let σ(x) and g _i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W _p ^r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g ₁(x), g ₂(x), ..., g _n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W _p ^r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.     

Inverse problem of finding n coefficients of lower derivatives in a parabolic equation

We consider the problem of reconstructing the vector function $\vec b(x) = (b_1 ,...,b_n )$ in the term $(\vec b,\nabla u)$ in a linear parabolic equation. This coefficient inverse problem is considered in a bounded domain Ω ? R ⁿ . To find the above-mentioned function $\vec b(x)$ , in addition to initial and boundary conditions we pose an integral observation of the form $\int_0^T {u(x,t)\vec \omega (t)dt = \vec \chi (x)} $ , where $\vec \omega (t) = (\omega _1 (t),...,\omega _n (t))$ is a given weight vector function. We derive sufficient existence and uniqueness conditions for the generalized solution of the inverse problem. We present an example of input data for which the assumptions of the theorems proved in the paper are necessarily satisfied.     

The Algebra of {{\mathbf{SL}}_3}\left( \mathbb{C} \right) Conformal Blocks

We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL₃( $ \mathbb{C} $ ) bundles on a smooth, marked curve (C, $ \vec{p} $ ): Elements of this algebra have a well known interpretation as conformal blocks, from the Wess-Zumino-Witten model of conformal field theory. For the genus 0; 1 cases we find the level of conformal blocks necessary to generate the algebra. In the genus 0 case we also find bounds on the degrees of relations required to present the algebra. As a consequence we obtain a toric degeneration for the projective coordinate ring of an effective divisor on the moduli $ {{\mathcal{M}}_{{C,\vec{p}}}}\left( {\mathrm{S}{{\mathrm{L}}_3}\left( \mathbb{C} \right)} \right) $ of quasi-parabolic principal SL₃( $ \mathbb{C} $ ) bundles on (C, $ \vec{p} $ ). Along the way we recover positive polyhedral rules for counting conformal blocks.     

λ-Central BMO estimates for commutators of multilinear maximal operators

The authors establish λ-central BMO estimates for commutators of maximal multilinear Calderón-Zygmund operators $T_{\Pi \vec b}^*$ and multilinear fractional operators $I_{\alpha \vec b}$ on central Morrey spaces respectively. Similar results still hold for $T_{\vec b} ,T_{\vec b}^*$ and $I_{\alpha ,\vec b}^*$ .     

Three-dimensional solenoidal vectors

We consider the problem of the possibility of approximating solenoidal vectors from the Sobolev spaces \(\mathop W\limits^o{_\rho}{^1} (\Omega )\) by finite solenoidal vectors. The answer is positive if ΩcRⁿ, n=2,3, is a strictly Lipschitz domain. We give examples of domains with noncompact boundaries for which such an approximation is not possible. We consider the auxiliary problem \(div \vec u = \varphi ,\vec u \in \mathop W\limits^o{_\rho}{^1} (\Omega )\) if the function τ ε L_p(Ω) is given.     

The Cauchy Problem for the Average Field Equation Describing a Model of a Magnetic Solid

We study a model of a magnetic solid treated as a system of particles with mechanical moment $\vec s,\vec s \in S^2$ , and magnetic moment $\vec \mu ,\vec \mu = \vec s$ , interacting with one another via the magnetic field, which determines variations in the mechanical moment of each particle. We study the system of integro-differential equations describing the evolution of the one-particle distribution function for this system of particles. We prove existence and uniqueness theorems for the generalized and the classical solution of the Cauchy problem for this system of equations. We also prove that the generalized solution continuously depends on the initial conditions.     

О факториэации и о нулях преобраэовании Фуряе бесконечно дифференцируемых функции с компактным носителем

LetF denote the class of Fourier transforms of infinitely differentiable functions on the real line with compact support. We prove that if each zero of a functionF ∈ $F \in \mathcal{F}$ lies in the union of a horizontal strip with a finite number of semistrips, them a factorizationF=F ₁ F ₂ holds, where $F_1 ,F_2 \in \mathcal{F}$ . We give estimates of |F ₁(z)/F ₂(z)| from above and from below. The zero sets of functions fromF are described in terms of integral sequences.     

An estimate of the solutions of a stationary problem for the Navier-Stokes equations

It is shown that the solutions of a nonlinear stationary problem for the Navier-Stokes equations in a bounded domain Ω ? ?³ with boundary conditions $\vec \upsilon \left| {_{\partial \Omega } } \right. = \vec a(x)$ satisfy the inequality $\left. {_{x \in \Omega }^{\sup } } \right|\left. {\vec v(x)} \right| \leqslant c\left( {\left. {_{x \in \partial \Omega }^{\sup } } \right|\left. {\vec a(x)} \right|} \right)$ for arbitrary Reynolds numbers. Bibliography: 9 titles.     

Uniqueness of generalized p-area minimizers and integrability of a horizontal normal in the Heisenberg group

We study the uniqueness of generalized \(p\) -minimal surfaces in the Heisenberg group. The generalized \(p\) -area of a graph defined by \(u\) reads \(\int |\nabla u+\vec {F}|+Hu.\) If \(u\) and \(v\) are two minimizers for the generalized \(p\) -area satisfying the same Dirichlet boundary condition, then we can only get \(N_{\vec {F}}(u) = N_{\vec {F}}(v)\) (on the nonsingular set) where \(N_{\vec {F}}(w) := \frac{\nabla w+\vec {F}}{|\nabla w+\vec {F}|}.\) To conclude \(u = v\) (or \(\nabla u = \nabla v)\) , it is not straightforward as in the Riemannian case, but requires some special argument in general. In this paper, we prove that \(N_{\vec {F}}(u) = N_{ \vec {F}}(v)\) implies \(\nabla u = \nabla v\) in dimension \(\ge \) 3 under some rank condition on derivatives of \(\vec {F}\) or the nonintegrability condition of contact form associated to \(u\) or \(v\) . Note that in dimension 2 ( \(n=1),\) the above statement is no longer true. Inspired by an equation for the horizontal normal \(N_{\vec {F}}(u),\) we study the integrability for a unit vector to be the horizontal normal of a graph. We find a Codazzi-like equation together with this equation to form an integrability condition.     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

Estimates of the Number of Eigenvalues of Self-Adjoint Operator Functions

We consider an operator function F defined on the interval $\user2{[}\sigma \user2{,}\tau \user2{]} \subset \mathbb{R}$ whose values are semibounded self-adjoint operators in the Hilbert space $\mathfrak{H}$ . To the operator function F we assign quantities $\mathcal{N}_\user1{F}$ and ν _F (λ) that are, respectively, the number of eigenvalues of the operator function F on the half-interval [σ,τ) and the number of negative eigenvalues of the operator F(λ) for an arbitrary λ ∈ [σ,τ]. We present conditions under which the estimate $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ holds. We also establish conditions for the relation $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.     

Theory of nonstationary flows of Kelvin-Voigt fluids

One proves the global unique solvability in class \(W_\infty ^1 (0,T;C^{2,d} (\bar \Omega ) \cap H(\Omega ))\) of the initial-boundary-value problem for the quasilinear system $$\frac{{\partial \vec \upsilon }}{{\partial t}} + \upsilon _k \frac{{\partial \vec \upsilon }}{{\partial x_k }} - \mu _1 \frac{{\partial \Delta \vec \upsilon }}{{\partial t}} - \int\limits_0^t {K(t - \tau )\Delta \vec \upsilon (\tau )d\tau + grad p = \vec f,di\upsilon \bar \upsilon = 0,\upsilon , > 0.}$$ This system described the nonstationary flows of the elastic-viscous Kelvin-Voigt fluids with defining relation $$\left( {1 + \sum\limits_{\ell = 1}^L {\lambda _\ell } \frac{{\partial ^\ell }}{{\partial t^\ell }}} \right)\sigma = 2\left( {v + \sum\limits_{m = 1}^{L + 1} {\user2{\ae }_m } \frac{{\partial ^m }}{{\partial t^m }}} \right)D,L = 0,1,2,...;\lambda _L ,\user2{\ae }_{L + 1} > 0.$$      

The invertibility of the isoparametric mappings for triangular quadratic Lagrange finite elements

A reference triangular quadratic Lagrange finite element consists of a right triangle $\hat K$ with unit legs S ₁, S ₂, a local space $\hat L$ of quadratic polynomials on $\hat K$ and of parameters relating the values in the vertices and midpoints of sides of $\hat K$ to every function from $\hat L$ . Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping ${F_h} = ({F_1},{F_2}) \in \hat L \times \hat L$ . We explicitly describe such invertible isoparametric mappings F _h for which the images F _h (S ₁), F _h (S ₂) of the segments S ₁, S ₂ are segments, too. In this way we extend the well-known result going back to W.B. Jordan, 1970, characterizing those invertible isoparametric mappings whose restrictions to the segments S ₁ and S ₂ are linear.     

Metrics on tiling spaces, local isomorphism and an application of Brown’s lemma

We give an application of a topological dynamics version of multidimensional Brown’s lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points $F$ , one can find a patch such that, for each scale factor $\lambda $ , there is a vector $\vec {t}_\lambda $ so that copies of this patch appear in the tilling “nearly” centered on $\lambda F+\vec {t}_\lambda $ once we allow “bounded perturbations” in the structure of the homothetic copies of $F$ . Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group “actions” on patches and we discuss the local isomorphism property of tilings within this setting.     

The polynomial multidimensional Szemerédi Theorem along shifted primes

If $\vec q_1 ,...,\vec q_m $ : ? → ? ^? are polynomials with zero constant terms and E ? ? ^? has positive upper Banach density, then we show that the set E ∩ (E ? $\vec q_1 $ (p ? 1)) ∩ … ∩ (E ? $\vec q_m $ (p ? 1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.     

Cosymplectic quasi-Sasakian pseudo-Riemannian manifolds and coisotropic foliations

Si considera una varietà neutra \(\tilde M\) di dimensione 2m munita di una struttura conforme simplettica \(CS_p \left( {2m; R} \right) = \left( {\tilde \Omega , \tilde \upsilon } \right)\) . Vengono studiati i differenti problemi concernenti gli automorfismi infinitesimali della 2-forma quasi simplettica \(\tilde \Omega \) . Inoltre vengono formulate alcune proprietà di un fogliettamento con isotropoF _c su \(\tilde M\) .