In this paper, we prove that the vorticity belongs to (L^{infty}(0,T;L^2(Omega))) for 3D incompressible Navier–Stokes equation with space-periodic boundary conditions, then the existence of a global smooth solution is obtained. Our approach is to construct a set of auxiliary systems to approximate the original system of vorticity equation.
This paper deals with the study of initial and final value theorems by means of fractional Hankel wavelet transform function and afterwards tempered distributions.
This paper is concerned to the study of global existence of weak solutions to a class of compressible non-Newtonian fluids in three-dimensional bounded domain. More precisely, we consider an isentropic compressible non-Newtonian fluid with adiabatic constant (gamma>frac{3}{2}). We study the global existence of an initial boundary value problem with nonhomogeneous Dirichlet boundary conditions by constructing an approximation scheme, energy estimates, and a weak convergence method.
In 2008, Spivey found a recurrence relation for the Bell numbers (phi_{n}). We consider the probabilistic (r)-Bell polynomials associated with (Y), (phi_{n,r}^{Y}(x)), which are a probabilistic extension of the (r)-Bell polynomials. Here (Y) is a random variable whose moment generating function exists in some neighborhood of the origin and (phi_{n}=phi_{n,0}^{1}(1)). The aim of this paper is to generalize the relation for the Bell numbers to that for the probabilistic (r)-Bell polynomials associated with (Y).
The vortical Whitham equation is modeled with quadratic and cubic nonlinearity, satisfying the unidirectional dispersion relation used to describe the propagation of nonlinear waves in the presence of a vertically sheared current of constant vorticity. In this article, we neglect the quadratic nonlinearity to numerically investigate solitary wave interactions. We show that the geometric Lax categorization is satisfied; however, an algebraic categorization based on the ratio of the initial solitary wave amplitudes is not possible. Specifically, our numerical simulations indicate that for solitary waves with large amplitudes, the interactions maintain two well-separated crests. Additionally, for solitary waves of different polarities, we find that wave-breaking may occur.
The problem of a uniformly rotating disk with slightly perturbed surface immersed in a viscous fluid is considered for large Reynolds numbers. The asymptotic solutions with double-deck structure of the boundary layer are constructed for a nonsymmetric irregularity localized on the disk surface. The results of numerical simulation of the flow near the surface are presented. The differences between the problem under consideration and the case of an irregularity symmetric with respect to the disk axis of rotation are shown.
In this paper, we obtain two Lichnerowicz type formulas for the generalized Zhang’s operator. And we give the proof of the Kastler–Kalau–Walze type theorem for the generalized Zhang’s operator on 4-dimensional oriented compact manifolds with (respectively, without) boundary.
We consider a Schrödinger operator on the segment ((0,1)) subject to the Dirichlet condition and perturb it by a delta-potential concentrated at the point (x= varepsilon ), where ( varepsilon ) is a small positive parameter. We show that the perturbed operator converges to the unperturbed one in the norm resolvent sense and this also implies the convergence of the spectrum. However, the latter convergence is true only inside each compact set on the complex plane and it does not characterize the behavior of the total ensemble of the eigenvalues under the perturbation. Our main result is the spectral asymptotics for the eigenvalues of the perturbed operator with an estimate for the error term uniform in the small parameter. This asymptotics involves an additional nonstandard term, which allows us to describe a global behavior of the total ensemble of the eigenvalues under the perturbation.
DOI 10.1134/S1061920824020018
In this paper, we present a presumably new representation of the Bernoulli numbers. We also give an elementary proof of the Akiyama-Tanigawa algorithm for calculating the Bernoulli numbers.
We study the problem on semiclassical asymptotics for (pseudo)differential equations with singularities on a stratified manifold of a special form—the orbit space (X) of a smooth action of a compact Lie group (G) on a smooth manifold (M). The operators under consideration are obtained as the restriction of (G)-invariant operators with smooth coefficients on (M) to the subspace of (G)-invariant functions, naturally identified with functions on (X), and have singularities on strata of positive codimension. The asymptotics are associated with Lagrangian manifolds in the phase space defined by the Marsden–Weinstein symplectic reduction of the cotangent bundle (T^*M) under the action of the group (G); rapidly oscillating integrals defining the Maslov canonical operator on such manifolds contain exponentials as well as special functions related to representations of the group (G). For the simplest stratified manifold—a manifold with boundary obtained as the orbit space of a semi-free action of the group ( mathbb{S} ^1) on a closed manifold—the corresponding construction of semiclassical asymptotics was realized earlier. Note that, in this case, the class of equations under consideration on manifolds with boundary includes the linearized shallow water equations in a basin with a sloping beach. The present paper deals with the general case.
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