Time Periodic Traveling Curved Fronts in the Periodic Lotka–Volterra Competition–Diffusion System

This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$

where \(\Delta \) denotes \(\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }\), \(x,y\in {\mathbb {R}}\) and \(d>0\) is a constant, the functions \(r_i(t),a_i(t)\) and \(b_i(t)\) are T-periodic and Hölder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution \(\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) \) connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.

     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

The effect of a singular perturbation on nonconvex variational problems

We study the effect of a singular perturbation on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation parameter 0. The technique utilizes the notion of -convergence of variational problems developed by De Giorgi. The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics.     

Small parameter method for determining motion of a viscous incompressible fluid in a thrust bearing

We consider the plane stationary motion of a viscous incompressible fluid between two surfaces. The fixed surface is given by the equation y=h[1+f(x/h)], where the functionf(x/h=h) characterizes the deviation of the fixed surface from the plane y=h(h and , are constants). The moving surface is a plane which moves with constant velocity along the x axis and remains parallel to the plane y=h. The small parameter method is used to solve the problem. The problem formulation is presented in the first section, the solvability of the linear equations obtained using the small parameter method is investigated in the second section, and the third section studies the convergence of the method and finds the radius of convergence of the constructed series.     

Piecewise Implicit Differential Systems

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$\begin{aligned} \dot{x}=1,\quad \left( \dot{y}\right) ^2=\left\{ \begin{array}{lll} g_1(x,y) \quad \text{ if }\quad \varphi (x,y)\ge 0 \\ g_2(x,y) \quad \text{ if }\quad \varphi (x,y)\le 0 \end{array},\right. \end{aligned}$$

where \(g_1,g_2,\varphi :U\rightarrow \mathbb {R}\) are smooth functions and \(U\subseteq \mathbb {R}^2\) is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form

$$\begin{aligned} \dot{x}= f(x,y,\varepsilon ) ,\quad (\varepsilon \dot{ y})^2=g ( x,y,\varepsilon ) \end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with \((x, y) \in U\) , \(\varepsilon \ge 0\), and f, g smooth in all variables.

     

On exact solutions of a class of fractional Euler–Lagrange equations

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where _a ^c D _t ^α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange

A three-dimensional variational problem in the gasdynamic theory of a lubricant

In [1–3] optimal forms of the gap were found for one-dimensional aerodynamic sliding bearings. The coefficient of the bearing capacity is optimized under the condition that the one-dimensional Reynolds equation of a gas lubricant is used to determine the pressure in the bearing. In the present article the three-dimensional problem of finding the optimal profile of an aerodynamic sliding bearing in the case of small compressibility numbers is considered. The problem is solved by the methods of variational calculation. A qualitative investigation is made of the form of the optimal profile, the results of which are confirmed by a numerical solution of a system of Euler-Lagrange equations. The results of the calculations are given for different elongations of the bearing. On the basis of the profiles obtained, optimal profiles with a rectangular pocket, which are more practical to fabricate, are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 34–39, September–October, 1975.     

A Widder’s Type Theorem for the Heat Equation with Nonlocal Diffusion

The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$ where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$ and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

Study of convection in a liquid layer with vibrating forces

We consider convection in a gravity force field in a liquid enclosed in a vessel which is vibrating along the vertical axis according to the lawa/ sin t ( ).Time-averaged convection equations describing the basic motion in first approximation are derived in [1]. In addition, criteria are introduced in [1] which determine the initiation of convection in this case, and a model problem is considered: the case of spatially periodic disturbances. It is found that in this case high frequency vibrations stabilize the state of relative rest.In the present paper we consider convection in a liquid layer between two horizontal planes whose equations are z=±l/2.The temperatures T₁ and T₂ are specified on the upper and lower surfaces, respectively. It is shown (§1) that for small values of the vibrational criterion the principle of stability variation is satisfied. In this case there is a variational principle (§2) from which it follows that high frequency vibrations prevent the occurrence of convection in the horizontal liquid layer. With the aid of the variational principle a calculation is made of the dependence of the values of Rayleigh criterion on the vibrational criterion.The author wishes to thank I. B. Sinomenko and V. I. Yudovich for their continued interest in this study.     

On mass transfer in an acoustic field

Chemical processes governed by the laws of diffusion kinetics can be intensified by elastic oscillations. It is also known that the rate of combustion of liquid and solid fuels changes markedly with the onset of acoustic vibrations in the combustion chamber. Despite the extensive application of vibrational processes in technology, the mechanisms of heat and mass transfer in the presence of vibrations are not well known. The aim of this research was to analyze the mass transfer from a sphere in an acoustic field.Notation angular frequency of oscillations - wavelength - R characteristic dimension of axisymmetric body - s amplitude of displacement of fluid particles in a plane acoustic wave - B amplitude of oscillation velocity - x, y longitudinal and transverse coordinates - u, v longitudinal and transverse velocity components - v kinematic viscosity - U — A(x) cos t velocity of potential flow - ⁺ thickness of momentum boundary layer - ^– thickness of diffusion boundary layer - m dimensionless concentration - m_* concentration of diffusing component at surface of vaporization - t time - D diffusion coefficient - average density of mixture - erf error function - r radius of axisymmetric body - R Reynolds number - P diffusion Prandtl number - time average - N, N_d Nusselt numbers based on radius and diameter respectively - pulsating component of velocity or concentration - o stationary component of velocity or concentration In conclusion, the authors wish to thank S. S. Kutateladze and I. A. Yavorskii, who supervised the present work.     

Improvement on stability and convergence of A. D. I. schemes

IntroductionAlternatingdirectionimplicit(A.D.I.)schemeswhichwasdiscoveredin1950',hasbecomeoneofthemostimportantmethodsintheapproximationofthesolutionsofparabolicpartialdifferentialequationsinmulti-dimensionalspace.Someofresultsaboutstabilityandconvergencearetooweakandincomplete,we'lltrytoimprovetheminthispaper.Considerinitial-boundaryvalueproblemintwospacevariablesLetΩhbeauniformrectangularmeshofO.h>0isthespacestepinxandydireehon*ProjectsupPOI'tedbytheNahonalNaturalScienceFoundahonofChi…     

SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＦＯＲＡＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＦＩＲＳＴＯＲＤＥＲＳＹＳＴＥＭＣｈｅｎＳｏｎｇｌｉｎ（陈松林）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ８,１９８４;ＲｅｖｉｓｅｄＡｐｒｉｌ１５,１９...     

The Stability of the Equilibrium of Planar Hamiltonian and Reversible Systems

In this paper, we study the stability of the equilibrium of planar systems