Global Uniqueness of Transonic Shocks in Two-Dimensional Steady Compressible Euler Flows

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C ¹ smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing a point where the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L ^∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of the shock polar applied on the (maybe curved) shock-fronts.     

Series solutions for steady unsaturated flow in irregular porous domains

For the most part, analytical solutions for steady unsaturated infiltration have been restricted to infinite and semi-infinite seepage geometries, using the quasi-linear approximation for the hydraulic conductivity. We provide analytical series methods to solve the steady quasi-linear flow equations, in finite irregular seepage geometries. Unlike the classical approach, the series method has been modified, to cater for arbitrary boundary geometry and surface recharge distributions. The matrix flux potential and the stream function both satisfy the same governing partial differential equation, and the stream function formulation is used to estimate the series coefficients. For a finite vadose zone, the stream function solution does not uniquely determine the matrix flux potential, when flux boundary conditions are used. Consequently, the stream function solution applies to a range of moisture distributions, for given infiltration and evapotranspiration rates through the surface.     

Nonlinear Bending of thin Elastic Rods

Exact solutions of the problem of nonlinear bending of thin rods under various fixing conditions and point dead loads are obtained. The solutions written in a unified parametric form and expressed in terms of the elliptic Jacobi functions are classified. These solutions depend on a single parameter — modulus of elliptic functions.     

A General Solution of the Duffing Equation

In this paper, we describe the application of the elliptic balance method (EBM) to obtain a general solution of the forced, damped Duffing equation by assuming that the modulus of the Jacobian elliptic functions are slowly varying as a function of time. From this solution, the maximum transient and steady-state amplitudes will be determined for large nonlinearities and positive damping. The amplitude–time response curves obtained from our elliptic balance approximate solution are in good agreement with those obtained from the numerical integration solution over the selected time interval.     

Analysis and synthesis of oscillator systems described by a perturbed double-well Duffing equation

This paper presents an investigation of limit cycles in oscillator systems described by a perturbed double-well Duffing equation. The analysis of limit cycles is made by the Melnikov theory. Expressing the solutions of the unperturbed Duffing equation by Jacobi elliptic functions allows us to calculate explicitly the Melnikov function, whereupon the final result is a function involving the complete elliptic integrals. The Melnikov function is analyzed with the aid of the Picard–Fuchs and Riccati equations. It has been proved that the considered oscillator system can have two small hyperbolic limit cycles located symmetrically with respect to the y-axis, or one large hyperbolic limit cycle, or two large hyperbolic limit cycles, or one large limit cycle of multiplicity 2. Moreover, we have obtained the conditions under which each of these limit cycles arises. The present work gives the conditions for the arising of limit cycles around the homoclinic trajectory. In this connection, an alternative approach is proposed for obtaining a series expansion of the Melnikov function near the homoclinic trajectory. This approach uses the series expansion of the complete elliptic integrals as the elliptic modulus tends to 1. It is shown that a jumping phenomenon may occur between limit cycles in the analyzed oscillator system. The conditions for the occurrence of this jumping phenomenon are given. A method for the synthesis of an oscillator system with a preliminary assigned limit cycle is also presented in the article. The obtained analytical results are illustrated and confirmed by numerical simulations.     

Subsonic Flow for the Multidimensional Euler–Poisson System

We establish the existence and stability of subsonic potential flow for the steady Euler–Poisson system in a multidimensional nozzle of a finite length when prescribing the electric potential difference on a non-insulated boundary from a fixed point at the exit, and prescribing the pressure at the exit of the nozzle. The Euler–Poisson system for subsonic potential flow can be reduced to a nonlinear elliptic system of second order. In this paper, we develop a technique to achieve a priori \({C^{1,\alpha}}\) estimates of solutions to a quasi-linear second order elliptic system with mixed boundary conditions in a multidimensional domain enclosed by a Lipschitz continuous boundary. In particular, we discovered a special structure of the Euler–Poisson system which enables us to obtain \({C^{1,\alpha}}\) estimates of the velocity potential and the electric potential functions, and this leads us to establish structural stability of subsonic flows for the Euler–Poisson system under perturbations of various data.     

A PRIORI ESTIMATE FOR LOWER SOLUTIONS OF ACLASS QUASILINEAR ELLIPTIC EQUATIONS

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Blow-Up Estimates for a Non-Newtonian Filtration System

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＴｈｅｓｔｒｕｃｔｕｒｅｏｆｐｏｓｉｔｉｖｅｓｏｌｕｔｉｏｎｓｆｏｒｑｕａｓｉ＿ｌｉｎｅａｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｓｙｓｔｅｍｓ (ｎｏｎｌｉｎｅａｒＮｅｗｔｏｎｉａｎｆｉｌｔｒａｔｉｏｎｓｙｓｔｅｍｓ)ａｎｄｓｅｍｉ＿ｌｉｎｅａｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｓｙｓｔｅｍｓ (Ｎｅｗｔｏｎｉａｎｆｉｌｔｒａｔｉｏｎｓｙｓｔｅｍｓ)ｉｓａｈｏｔｔｏｐｉｃｉｎｔｈｅｓｔｕｄｙｏｆｎｏｎｌｉｎｅａｒｄ…     

On solutions of backward stochastic differential equations with jumps, with unbounded stopping times as terminal and with non-lipschitz coefficients, and probabilistic interpretation of quasi-linear elliptic type integro-differential equations

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non-Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi-linear elliptic type integro-differential equations is obtained. Foundation item: the National Natural Science Foundation of China (79790130); the Foundation of Zhongshan University Front Research Biography: Situ Rong (1935 −)     

Analytic solutions of simple waves

B.Riemann furnished the general solution of simple waves in1860.But it is difficult to find out the exact forms of the arbitrary function contained in the general solution which must satisfy boundary or initial conditions.For this reason it is inconvenient to probe into the characteristics of concrete problems.In this paper the analytic solutions of simple waves are afforded according to the geometric theory of quasi-linear partial differential equation,and they are determined with boundary or initial conditions.By using these solutions the specific properties of certain flows are discussed and novel results are obtained.     

Analysis of a two-grid method for semiconductor device problem

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin method.We estimate the error of the numerical solutions in the sense of the Lqnorm.To linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton iteration.The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid.Error estimation for the two-grid solutions is analyzed in detail.It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.     

Existence and nonexistence of positive solutions of semilinear elliptic equation with inhomogeneous strong Allee effect

In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.     

The method of successive integration of the linear inhomogeneous wave equations in the theory of transient wave propagation in non-linear hereditary elastic media

A moderate distortion of the initial pulse form which takes place when a one-dimensional longitudinal pulse propagates through a sufficiently small distance in a non-linear hereditary clastic medium is considered. The governing equation is a quasi-linear integro-differential equation. Its first- and second-order asymptotic solutions arc derived with the aid of a method of successive integration of the linear inhomogeneous wave equations. Besides the constants which define the wave speed and the non-linear properties of the medium, the asymptotic solutions suggested in this paper contain two arbitrary functions whose properties are restricted only by certain smoothness conditions. One of them is the kernel function which defines the hereditary properties of the medium. and the other is the function which defines the initial form (shape) of the pulse. An example of the use of the asymptotic solutions is presented in which these two functions are given explicitly.     

A general nonlocal nonlinear Schrödinger equation with shifted parity,charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a \(\beta \)-plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.     

On a class of quasilinear Schrodinger equations

A type of quasilinear Schrodinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sutficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.     

Solids containing spherical nano-inclusions with interface stresses: Effective properties and thermal–mechanical connections

This work examines the overall thermoelastic behavior of solids containing spherical inclusions with surface effects. Elastic response is evaluated as a superposition of separate solutions for isotropic and deviatoric overall loads. Using a variational approach, we construct the Euler–Lagrange equation together with the natural transition (jump) conditions at the interface. The overall bulk modulus is derived in a simple form, based on the construction of neutral composite sphere. The transverse shear modulus estimate is derived using the generalized self-consistent method. Further, we show that there exists an exact connection between effective thermal expansion and bulk modulus. This connection is valid not only for a composite sphere, but also for a matrix-based composite reinforced by many randomly distributed spheres of the same size, and can be viewed as an analog of Levin’s formula for composites with surface effects.     

The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity

Theoretical analysis of the nonlinear vibration effects on the sound absorption of a panel absorber and sound transmission loss of a panel backed by a rectangular cavity is herein presented. The harmonic balance method is employed to derive a structural acoustic formulation from two-coupled partial differential equations representing the nonlinear structural forced vibration and induced acoustic pressure; one is the well-known von Karman??s plate equation and the other is the homogeneous wave equation. This method has been used in a previous study of nonlinear structural vibration, in which its results agreed well with the elliptic solution. To date, very few classical solutions for this nonlinear structural-acoustic problem have been developed, although there are many for nonlinear plate or linear structural-acoustic problems. Thus, for verification purposes, an approach based on the numerical integration method is also developed to solve the nonlinear structural-acoustic problem. The solutions obtained with the two methods agree well with each other. In the parametric study, the panel displacement amplitude converges with increases in the number of harmonic terms and acoustic and structural modes. The effects of excitation level, cavity depth, boundary condition, and damping factor are also examined. The main findings include the following: (1)?the well-known ??jump phenomenon?? in nonlinear vibration is seen in the sound absorption and transmission loss curves; (2)?the absorption peak and transmission loss dip due to the nonlinear resonance are significantly wider than those in the linear case because of the wider resonant bandwidth; and (3)?nonlinear vibration has the positive effect of widening the absorption bandwidth, but it also degrades the transmission loss at the resonant frequency.     

Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

Estimates of azimuthal numbers associated with elementary elliptic cylinder wave functions

The paper deals with issues related to the construction of solutions, 2 π-periodic in the angular variable, of the Mathieu differential equation for the circular elliptic cylinder harmonics, the associated characteristic values, and the azimuthal numbers needed to form the elementary elliptic cylinder wave functions. A superposition of the latter is one possible form for representing the analytic solution of the thermoelastic wave propagation problem in long waveguides with elliptic cross-section contour. The classical Sturm-Liouville problem for the Mathieu equation is reduced to a spectral problem for a linear self-adjoint operator in the Hilbert space of infinite square summable two-sided sequences. An approach is proposed that permits one to derive rather simple algorithms for computing the characteristic values of the angular Mathieu equation with real parameters and the corresponding eigenfunctions. Priority is given to the application of the most symmetric forms and equations that have not yet been used in the theory of the Mathieu equation. These algorithms amount to constructing a matrix diagonalizing an infinite symmetric pentadiagonal matrix. The problem of generalizing the notion of azimuthal number of a wave propagating in a cylindrical waveguide to the case of elliptic geometry is considered. Two-sided mutually refining estimates are constructed for the spectral values of the Mathieu differential operator with periodic and half-periodic (antiperiodic) boundary conditions.     

On the solvability of one class of problems of optimal control by coefficients in the principal part of a nonlinear elliptic operator

We consider the problem of the existence of solutions of an optimal-control problem for a nonlinear elliptic equation with Dirichlet conditions on the boundary in the case where the control functions are the coefficients in the principal part of the differential operator. It is shown that this problem has an optimal solutions in the class of generalized solenoidal matrices. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 59–72, January–March, 2009.