For averages over parallelepipeds of unitary actions of the groups R d \mathbb {R}^d and Z d {\mathbb {Z}^d} , a criterion for a power rate of convergence in norm with all possible exponents is obtained. The proof is based on the asymptotics of the integrals of product of cardinal sines.
Let S-k, 1 < k < m, be pairwise disjoint segments, S-k =[a(k), b(k)], 1 < pk < infinity. Suppose that functions f(k) are defined on Sk, fk belongs to C(S-k) and f(k) are belongs to L-k(P) (S-k). It is shown that for n = 1, 2, ... there are polynomials Pn, deg Pn < n, that approximate all functions f(k) in the metric of L-k(p) with weights tending to infinity near the points a(k), b(k).
. The Polyak and Brandenbursky invariants are applied to estimate the crossing number of (closed) braids and extend the previously known minimality criteria for diagrams of positive and alternating braids to homogeneous ones. In particular, it is proved that a diagram of a homogeneous braid is minimal if and only if this diagram is homogeneous. These results lay the groundwork for a potential solution to the recognition problem for homogeneous knots and links. The approach developed here is conceptually similar to recognizing alternating links on the basis of the Tait conjectures.
. A well-posed initial-boundary value problem is stated for the wave equation with variable propagation speed in a half-plane on a semi-infinite time interval. Certain conditions on the boundary data are indicated that ensure the existence and uniqueness of the solution as well as its stability under small changes of the boundary data in a certain function class. The problem statement is motivated by the need to justify the Poincare wavelet-based integral representation of the wave equation solution in terms of localized solutions, in particular, quasiphotons.
. The article provides a systematic presentation of an operator pattern that forms the basis of a certain approach to inverse problems in mathematical physics - the boundary control method. The pattern is based on the triangular factorization of operators. Not only solves it inverse problems but also provides functional models for an important class of symmetric semibounded operators. These models are constructed by using an evolutionary dynamical system determined by the operator.
. It has been known for over a century that certain large classes of functions defined on a compact nowhere dense subset X of the complex plane, and obtained as limits of analytic functions in various metrics, can sometimes inherit the property of unique continuation characteristic of the approximating family. The first example of the transfer of the uniqueness property in this way to R(X), the space of functions that can be uniformly approximated on X by a sequence of rational functions whose poles lie outside of X, was obtained by M. V. Keldysh around 1940, but apparently never published. Years later in 1975 A. A. Gonchar exhibited a qualitatively definitive improvement of Keldysh's example, and our goal here is to extend that result to Rp(X, dA), p >= 2, the evidently larger space obtained as the closure of the rational functions in Lp(X, dA), where dA denotes 2-dimensional Lebesgue, or area, measure.
. This paper is devoted to the study of the relationship between the fundamental algebraic, analytic, and topological invariants of Artinian local algebras. Among other things, it is shown that the length of the modules of derivations and K & auml;hler differentials of every local Artinian Gorenstein algebra is not greater than the length of the algebra reduced by one. The proof relies on the theory of duality in the cotangent complex of analytic algebras, the basic properties of faithful modules over Artinian rings, and the structure of annihilators and socles of the modules of derivations and K & auml;hler differentials. As a result, it turns out that the Tyurina number of a smoothable zero-dimensional Gorenstein singularity cannot be less than its Milnor number, so the inequality tau >= & micro; holds.
For averages over parallelepipeds of unitary actions of the groups Rd and Zd, a criterion for a power rate of convergence in norm with all possible exponents is obtained. The proof is based on the asymptotics of the integrals of product of cardinal sines.
. Consider a homogeneous Markov process with continuous time on the phase space Z+ = {0,1, 2, ... }, which is interpreted as the movement of a particle. The particle can only move to neighboring points of Z+, i.e., with each change in position, its coordinate changes by one. The process is equipped with a branching mechanism. Branching sources can be located at each point in Z+. At the moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) in accordance with the same laws as the initial particle. At each time t, we have a random field on Z+ consisting of particles present in the system at that moment. Functionals E(mj ,mk) Phi(mj, mk) of this field are considered, where the sum is taken over all ordered pairs (mj, mk) of different particles in the field. The asymptotic behavior of the mean value of this functional as t -> +infinity is studied.
The paper suggests a characterization of a positive function f such that f (0 infinity) is integrable for any limit value 0 infinity of the martingale transform for an indicator function. The characterizing condition is that a version of the Lipschitz majorant of f is integrable with respect to an exponential weight. The arguments are based on the calculation of particular minimal biconcave functions on a strip.
. The paper is devoted to the ill-posedness of the linearized hyperbolic Prandtl system around a shear flow. As for the classical Prandtl system (see Gerard- Varet and Dormy, J. Amer. Math. Soc., 23 (2010), no. 2, 591-609), the hyperbolic Prandtl system with initial data that does not satisfy monotonicity condition is ill posed at least in a Sobolev space. As a byproduct, it is deduced that the optimal Gevrey index for the well-poseness of the hyperbolic Prandtl system is 2.
. The series of papers denoted by Part 1, Part 2, and Part 3 is devoted to motions of axially symmetric solutions to the Navier-Stokes equations in a cylindrical domain. Part 1 and Part 2 deal with motions such that psi 1 = psi/r, where psi is the stream function and r is the radius, vanishes on the axis of symmetry. The boundary of the cylinder equals S1 boolean OR S2, where S1 is parallel to the axis of symmetry and S2 is perpendicular to it. It is clear that S2 has two parts. In Part 1 it was assumed that the normal component of velocity, angular components of velocity, and vorticity vanish on S1 and the periodic boundary conditions were assumed on S2. In Part 2 the same boundary conditions were imposed on S1 as in Part 1 and it was assumed that the normal component of velocity, angular component of vorticity, and the normal derivative of the angular component of velocity vanish on S2. In this paper, Part 3, the same boundary conditions as in Part 2 are imposed but the restriction psi 1 = 0 on the axis of symmetry is dropped. This is done by looking for solutions to the problem under study in the form v = v' + v1 + 2v, p = p' + p1 + 2p, where (1v, 1p) is the solution from Part 2 and (2v, 2p) is the solution without swirl proved by O. A. Ladyzhenskaya. Finally, (v', p') is added because otherwise (v, p) cannot satisfy the Navier-Stokes equations.
The subject of this paper is a two-dimensional evolution free boundary problem for a viscous incompressible fluid that partially fills a container. The purpose is to prove the time-local unique solvability of the problem for the Navier-Stokes system in the Sobolev-Slobodetsky spaces and to estimate the obtained solution in these spaces. This result is achieved for small wetting angles.
The paper is devoted to the study of the two-phase free boundary problem for nonlinear partial differential equations describing the evolution of a foam drainage in the one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of a liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. In a series of papers, the authors have already solved the same problems without free boundary and with free boundary situated at the lower and the upper parts in the foam column, respectively. In this paper it is shown that the free boundary problem for the foam drainage equations with a sharp interface between dry and wet foams admits a unique global-in-time classical solution; this is done by a standard classical mathematical method, the maximum principle, and the comparison theorem. Moreover, the existence of the steady solution and its stability are shown.
. Asymptotics is constructed for eigenvalues and eigenfunctions of the biharmonic equation with the Neumann conditions perturbed by the spectral Winkler- Steklov conditions at small parts of the plate's edge. The zero eigenvalue has multiplicity three and the corresponding eigenfunctions are linear. Asymptotic expansions of positive eigenvalues differ essentially in the mid- and high-frequency ranges of the spectrum. In particular, eigenfunctions at low and medium frequencies are distributed along the entire domain while eigenfunctions at high frequencies are concentrated in the vicinity of the edge perturbations.
. The nonlinear monotone H1-energy stability of laminar flows in a layer between two parallel planes filled with a Navier-Stokes-Voigt fluid is studied. It is proved that the critical Reynolds numbers for monotone H1-energy stability for the Couette and Poiseuille flows of the zero-order Navier-Stokes-Voigt fluid are the same as those found by Orr for Newtonian fluids. However, the exponential decay coefficient depends on the Kelvin-Voigt parameter Lambda. Furthermore, a Squire theorem holds in the nonlinear case: the least stabilizing perturbations in H1-energy are the two-dimensional spanwise perturbations.
. Integral sufficient conditions on u, del u, and mixed, to guarantee the energy equality, EE, for solutions of the Navier-Stokes equations are discussed. An innovative interpretation of some main parameters is revisited, which makes it possible to overcome some apparent incongruence, and also present a simpler, self-contained, proof of the main related result. Furthermore, the above results are used to exhibit in the last section below a typical condition on the direction of vorticity that implies the energy equality.
The paper is devoted to the Navier-Stokes equations in a planar smooth bounded domain Omega under the no-slip boundary condition with initial velocity u(0) of finite kinetic energy, i.e., u(0) is an element of L-2. The existence is proved for a unique weak solution u is an element of L-2 (0, T; L-infinity(Omega)) satisfying the estimate ||u||(2)(L)((0,T;L infinity(Omega))) <= c (1 + ||u(0)||(L2)) ||u(0)||(L2) with some constant c depending only on Omega. Note that H-1(Omega) is not embedded into L-infinity(Omega) so that the argument is not based on the energy identity valid for u, but on a generalized Marcinkiewicz interpolation theorem. This estimate is extended to mild solutions of Serrin's class in n dimensions provided that u(0) is in a solenoidal L-n space.
. The paper deals with a solution of a stationary problem with unknown boundaries for the Navier-Stokes equations corresponding to the slow rigid rotation of a viscous two-phase drop consisting of compressible and incompressible embedded fluids. In this case, the internal fluid is incompressible. It is bounded by a closed interface that does not intersect the outer free surface. It is assumed that the compressible fluid is barotropic. Surface tension forces act at the boundaries. The existence of a family of equilibrium figures close to embedded balls is proved. The proof is carried out in Holder spaces by means of the implicit function theorem.
