The main purpose of this paper is to establish probabilistic asymptotic convergence properties of the power sums D-n,D-k= Sigma Y-n (i=1)i(k) of the Dirichlet distribution Y = (Y-1,Y-2,& centerdot;& centerdot;& centerdot;, Y-n) similar to Dir(a) for positive integers k. We propose stability conditions on the parameters a = (a(1), a(2),& centerdot;& centerdot;& centerdot; , a(n)) is an element of Rn+ such that these power sums satisfy the almost sure convergence n(k-2-delta)D(n,k) -> 0 for all delta > 0 and their distributions are also asymptotically normal as n -> infinity. This extends previously known results for the special case of symmetric Dirichlet distributions, including especially the widely studied case of uniform spacings associated with the classical random division. As an application, we show that the semiperimeters and areas of random cyclic polygons generated from such Dirichlet distributions satisfy analogous probabilistic convergence and asymptotic normality estimates as n -> infinity. Additionally, we also present some extrapolation improvements with faster rates of convergence.
The present paper concerns the Taylor-Couette flow for 2-D isentropic compressible Navier-Stokes system in a bounded annular domain. We have proved the stability of Taylor-Couette flow provided the initial perturbation is small and the difference in rotational speeds between the inner and outer cylinders is small. Notably, we allow the rotational speeds of the inner and outer cylinders to be large, which relaxes the requirement of [24] on the rotational speeds.
This paper investigates the uniform regularity estimates of solutions to the initial boundary value problems of three-dimensional incompressible magnetohydrodynamics (MHD) equations in the half-space, where either the viscosity or the magnetic diffusion coefficient goes to zero. Under four kinds of combination forms of boundary conditions, we can establish the uniform energy estimates of solutions which are independent of the viscosity and the magnetic diffusion coefficients in the co-normal Sobolev space. Furthermore, the related partially vanishing dissipation limits are thus proved in L-infinity sense through the related compactness arguments based on these uniform regularity estimates of solutions.
For the scalar convex conservation laws, we show a nonlinear stability of shock waves to a large class of non-localized perturbations, which includes kinds of oscillatory functions such as the periodic and quasi-periodic ones. Besides, we give an explicit formula of the final shock locations affected by the periodic perturbations for the viscous conservation laws.
We present the main ideas behind the proofs of a regularity cri-terion for the free boundary problem of the incompressible ideal magnetohydro dynamics (MHD) equations with surface tension in a bounded domain, as given by the authors in [18]. Furthermore, we provide an overview of the latest advancements in this field.
In this paper, we consider the initidal-boundary value problem of the compressible Navier-Stokes equations with variable coefficients, which was first introduced by Kazhikhov-Vaigant[10] in 1995. By assuming that the shear viscosity is a constant and the bulk viscosity lambda = p(beta) with beta > 3, they established the global exsistence of strong solutions with arbitrary initial data under Navier-slip or periodic boundary conditions. Here we prove that the system will admit global radially symmetric strong solutions under the Dirichlet boundary conditions even in the endpoint case of /3 = 1 for arbitrary initial smooth data. This improves all the previous results [3, 4, 5, 7, 10].
In this paper, we study the problem of the vanishing viscosity limit of one-dimensional quasilinear viscous equations with a positive definite viscosity matrix in the presence of two non-interacting weak boundary layers. We investigate the existence and stability of the two weak boundary layers by discussing the nonlinear well-posedness of the inviscid flow with certain boundary conditions. By the method of matched asymptotic expansions and the energy estimates, we finally prove the L infinity asymptotic equivalence between the solutions of the viscous equations and the approximate solution, then we get the asymptotic limit to the corresponding inviscid equations away from the boundaries.
In this paper, we explore the motion of the gaseous stars as described by the Euler-Poisson system. At the end of a large star's evolution, stellar collapse occurs. Under given physical conditions, we prove that there exists a transonic shock solution to the EulerPoisson system. Furthermore, we establish the monotonic relationship between the density of the stellar core and the location of the transonic shock.
The purpose of this paper is to present recent progress on the study of classical solutions for degenerate compressible Navier-Stokes equations (CNS) with vacuum. We will introduce theories for both isentropic and non-isentropic system on well-posedness, singularity formation, inviscid limit and so on. Moreover, some related open problems of high mathematical interest for this system will be mentioned.
This paper investigates the global existence and stability of steady weak oblique shock waves in the context of perturbed hypersonic Euler flows around a wedge. The flow is modeled as polytropic and governed by the 2D steady full Euler equations. By employing the Lagrange transformation and quasi-Riemann invariants, and utilizing the method of characteristics, we demonstrate the global existence of the shock wave and the continuity of the flow between the shock and the wedge.
In this paper, we study two-phase flow models consisting of the compressible isothermal Euler (or Navier-Stokes) equations coupled with the compressible non-isentropic Navier-Stokes equations through a drag forcing term. For the 3-D Cauchy problem, the existence and uniqueness of global smooth solutions is proved in perturbation framework, for more general gases including ideal polytropic gas. Moreover, by a pure elementary energy method, we show the time decay rates for higher-order spatial derivatives of global smooth solutions.
In this note, we investigate partial regularity of weak solutions of the three dimensional chemotaxis-Navier-Stokes equations, and obtain the $\frac53$-dimensional Hausdorff measure of the possible singular set is vanishing at the first blow-up time. The new ingredients are to establish certain type of local energy inequality and deal with the non-scaling invariant quantity of $n\ln n$, which seems to be the first description for the singular set of weak solutions of the chemotaxis-fluid model, which is motivated by Caffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}.
Expanding on my former work along with the more recent work of Kasuya and Takase, we demonstrate that for a given link L subset of M which is null-homologous in H1(M) and for any smooth oriented 2-plane field eta over L there exists a smooth embedding F : M (sic) C-3 so that the set of complex tangents to the embedding is exactly L and at each x is an element of L the holomorphic tangent space is exactly eta(x). Furthermore, we demonstrate how the analyticity of a complex tangent, as given by the Bishop invariant, may be determined exactly from the angle formed between the holomorphic complex line and the the curve of complex tangents.
The scheme concerned in this study is a non-homogeneous extension of an optimal second order SSP (strong stability-preserving) Runge-Kutta method for scalar convex conservation laws with source terms. The homogeneous counterpart (HCP) of this scheme belongs to a class of SSP high-order time discretization methods that was first introduced by Shu [22] and by Shu and Osher [23] for solving time-dependent partial differential equations (PDEs), especially for hyperbolic conservation laws. The entropy convergence of the HCP of this scheme, when the spatial discretization is constructed by Sweby's flux limiter method [25], was previously established by the author [15]. The goal of this paper is to show that the extended scheme, with the same spatial discretization as its HCP in [15], also possesses the entropy consistency for non-homogeneous scalar convex conservation laws by using author's earlier results [14] on the extended Yang's wave tracing theory [28].
In this paper, we study the discrete cohomological equation (e) : f - f omicron gamma = g where gamma is a hyperbolic automorphism of the torus T-d = R-d/Z(d). More precisely, if we denote by E the Frechet space of all C-infinity functions on T-d and by delta the operator defined on E by delta(f) = f - f omicron gamma, we show that the space delta(E) is a closed vector subspace of E as the intersection of a countable family of closed hyperplanes of E and we determine a continuous linear operator L : delta(E) -> E such that for every element g of delta(E), the function f = L(g) is a solution of the equation (e).
In this paper, we discuss the asymptotic behaviour of the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear Laplacian viscosity. Firstly, we obtain the existence, uniqueness and regularity of solutions when the initial data u(0 )is an element of(C1)(R-N)boolean AND W-1,infinity(R-N).Secondly, whenu0is periodic, we prove the time-decay rate of the periodic solution and its gradient. At last, we study the long-time behaviour of perturbed solution to the Cauchy problem, in which the initial data is a N-dimensional periodic perturbation around a planar rarefaction wave and obtain the time-decay rate of the perturbed solution approaching approximate planar rarefaction wave. The proof is given by technical energy methods and iteration technique.
