Existence of solutions to gas expansion problem through a sharp corner for 2-D Euler equations with general equation of state

In this article, we study the gas expansion problem by turning a sharp corner into a vacuum for the two-dimensional (2-D) pseudosteady compressible Euler equations with a convex equation of state. This problem can be considered as the interaction of a centered simple wave with a planar rarefaction wave. To obtain the global existence of a solution up to the vacuum boundary of the corresponding 2-D Riemann problem, we consider several Goursat-type boundary value problems for 2-D self-similar Euler equations and use the ideas of characteristic decomposition and bootstrap method. Further, we formulate 2-D-modified shallow water equations newly and solve a dam-break-type problem for them as an application of this work. Moreover, we also recover the results from the available literature for certain equations of states that provide a check that the results obtained in this article are actually correct.     

一阶中立型泛函微分方程解的渐近性与振动性

本文研究具有变系数和多偏差的一阶中立型微分方程，讨论了方程的非振动解的渐近性，得到了方程振动的充分判据，其中有些是变号系数的情形．     

OSCILLATION AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS WITH IMPULSES

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ASYMPTOTICS AND OSCILLATION FOR FIRST ORDER MIXED NEUTRAL DIFFERENTIAL EQUATIONS

1IntroductionConsidertheneutraldifferentialequationswherepiERfori=l,2,...,m;o相似文献   

二阶泛函微分方程的振动性质

在本文中，我们研究了一类较广泛的二阶非线性泛函微分方程的振动性质。文中指出，在一定条件下，方程的非振动解仅有两类，而且给出了每一类非振动解存在的必要条件，同时也建立了方程振动的若干充分判据。     

A boundary layer problem in nonflat domains with measurable viscous coefficients

We study the convergence of weak solutions of the Navier–Stokes equations with vanishing measurable viscous coefficients in domains with nonflat boundaries. Sufficient anisotropic conditions on the vanishing rates of the viscous coefficients are found to prove the convergence of Leray–Hopf weak solutions of the Navier–Stokes equations to solutions of the corresponding Euler equations. As the domains are not flat, we apply a change of variables to flatten the domains. We then construct explicit boundary layers for the system of Navier–Stokes equations in the upper-half space with measurable viscous coefficients. The result is new even when the viscous coefficients are constant, and it recovers the classical results when domains are flat and with constant viscous coefficients.     

Oscillatory and Asymptotic Behavior for Second Order Quasilinear Difference Equations

1 IntroductionConsider the second order quasilinear difference equationA(g(Ay.--l)) + f(n,y.) = 0, for n E N(no), (l'l)where A is defined by Ay. = Vn+1--yn, n E N(no) = {no, no + 1,'' }, nO E N = {l, 2,'. }.The following hold throughout the paPer:(H0) (i) g: R-R is a continuous increasing fUnction with propertiessgng(y) = sgny) g(R) = R;(il) f: N(no) x R--+ R is continuous as a function of y E R;(iii) yf(n,y) > 0 for n E N and y / 0.By a solution of the equation (1.l) we mean a non…     

Oscillation and stability of nonlinear discrete models exhibiting the Allee effect

In this paper, we consider the discrete nonlinear delay population model exhibiting the Allee effect

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

The purpose of this paper is to describe the oscillatory properties of second‐order Euler‐type half‐linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non‐oscillation of the considered equations, including the so‐called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.     

OSCILLATION AND EXISTENCE OF POSITIVESOLUTIONS OF DIFFERENTIAL EQUATIONSWITH POSITIVE, NEGATIVE COEFFICIENTS

This paper is concerned with the existence of eventually positive solutions and oscillatory solutions of first-order neutral differential equation with positive and negative coefficients. The "sharp" conditions of oscillation for the equation have been obtained, i.e., the conditions are sufficient and necessary when all of the coefficients are constants. These conditions are distinguished from the previous criteria in the literatures, and also weaker than those existing results.     

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

This paper is mainly concerned with the long-term random dynamics for the nonautonomous 3D globally modified Navier–Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville-type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.     

Hyperbolicity study of the modulation equations for the Benjamin–Bona–Mahony equation

A complete study of the modulation equations for the Benjamin–Bona–Mahony equation is performed. In particular, the boundary between the hyperbolic and elliptic regions of the modulation equations is found. When the wave amplitude is small, this boundary is approximately defined by $k=\sqrt{3}$, where k is the wave number. This particular value corresponds to the inflection point of the linear dispersion relation for the BBM equation. Numerical results are presented showing the appearance of the Benjamin–Feir instability when the periodic solutions are inside the ellipticity region.     

Singularity formation to the 2D Cauchy problem of nonbarotropic magnetohydrodynamic equations without heat conductivity

The question of whether the two-dimensional (2D) nonbarotropic compressible magnetohydrodynamic (MHD) equations with zero heat conduction can develop a finite-time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. Such a problem is interesting in studying global well-posedness of solutions. In this paper, we proved that, for the initial density allowing vacuum states, the strong solution exists globally if the density and the pressure are bounded from above. Our method relies on weighted energy estimates and a Hardy-type inequality.     

一类一阶中立型方程解的渐近性与振动性

讨论具有连续分布滞量的一阶中立型微分方程 ,(1)给出了非振动解当t→∞时收敛于零的充分条件和方程(1)所有解振动的判据。     

高阶泛函微分方程的振动性质*

本文借助于Lebesgue测度等工具研究了一类高阶非线性泛函微分方程的振动性质.文中指出.在一定条件下,方程的非振动解仅有两类,而且给出了每一类非振动解存在的必要条件,同时也建立了方程振动的若干充分判据.     

Oscillation of second‐order perturbed differential equations

We give constructive proof of the existence of vanishing at infinity oscillatory solutions for a second‐order perturbed nonlinear differential equation. In contrast to most results reported in the literature, we do not require oscillatory character of the associated unperturbed equation, monotonicity of nonlinearity, and we establish global existence of oscillatory solutions without assuming it a priori. Furthermore, as our example demonstrates, existence of bounded oscillatory solutions does not exclude existence of unbounded nonoscillatory solutions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Nonoscillatory Behavior of Solutions of a Second Order Linear Differential Equation

In this paper, wo improve the Sturm comparison theorem and two nonoscillation criteria of Leighton and Wintner, and establish two variants of a Wintner' s nonoscillatory criterion of the second order linear differential equation where r, c : t₀,∞) →, R > 0 a. e. on t₀,∞) and 1/r, c ε L^l(t₀,b) for each b ∞ (t₀,>) for some t₀ > 0. Using these two criteria, we improve some nonoscillation criteria of Hartman. Hille. Moore. Potter. WintnEr, and Willett. These proofs are more elegant and concise than those of theirs.     

One class of MHD equations: Conservation laws and exact solutions

The paper analyzes one of the models of equations of magnetohydrodynamics (MHD) derived earlier. The model was obtained as a result of group classification of the MHD equations in mass Lagrangian coordinates, where all dependent variables in Eulerian coordinates depend on time and two spatial coordinates. The use of Lagrangian coordinates made it possible to solve four equations, which led to the form of reduced equations containing four arbitrary functions: entropy and a three-dimensional vector associated with the magnetic field. The objective of this work is to develop conservation laws and exact solutions for the model. Conservation laws are obtained using Noether's theorem, while exact solutions are obtained either explicitly or by solving a system of ordinary or partial differential equations with two independent variables. Numerical methods are employed for the latter solutions.     

Chernoff approximations of Feller semigroups in Riemannian manifolds

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.     

LINEARIZED OSCILLATIONS FOR ODD-ORDERNEUTRAL DELAY DIFFERENTIAL EQUATIONSWITH ASYMPTOTIC PERIODIC COEFFICIENTS

1IntroductionSince1987,thelinearizedoscillationtheoryofnonlineardelaydifferentialequations(DDEs)hasabsorbedmanymathematicians(see[1,2]),itisparalleltotheso-calledlinearizedstabilitytheory,andhasbecomeamainstudyofoscillationtheoryofDDEs.ByusingthenecessaryandsufficientconditionsfortheoscillationoflinearDDEswithconstantcoefficients,theliteraturesconcentratedtoprovethatcertainnonlinearDDEhasthesameoscillatorycharacterasanassociatedlinearizedDDEwithconstantcoefficients.However)whenthelineari…