Stability of contact discontinuity for the Boltzmann equation

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and Navier-Stokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation corresponding to the contact discontinuities for the Euler equations, we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate. In our analysis, an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates.     

A numerical method for the first-order wave equation with discontinuous initial data

We introduce a method, constructed such that numerical solutions of the wave equation are well behaved when the solutions also contain discontinuities. The wave equation serves as a model problem for the Euler equations when the solution contains a contact discontinuity. Numerical computations of linear equations and the Euler equations in one and two dimensions are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 353–365, 1998     

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.     

Functions of two variables continuous along straigt lines

For a functionf(x, y), the setsJ _a of all its discontinuity points with a jump ofa or more (that is, such that the oscillation of the function in the neighborhood of any point fromJ _a is not smaller thana) are studied. Two cases are considered: (1)f is continuous along any straight line; (2)f is continuous along lines parallel to thex- andy-axes. In the first case, conditions that must be met by the setJ _a are given. In the second case, it is shown that a (closed) setF can be the setJ _a for a certain function if and only if the projections ofF on the coordinate axes nowhere dense. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 306–311, August, 1997. Translated by V. N. Dubrovsky     

Pointwise Stability of Contact Discontinuity for Viscous Conservation Laws with General Perturbations

The large time asymptotic behavior towards viscous contact waves for a class of systems of viscous conservation laws is studied in this paper for general initial perturbations. The high order deviation of the viscous solutions from the leading order ansatz is estimated pointwisely via the approximate Green function approach. The structural constraint on the left eigenvector belonging to the principal linearly degenerate family used in [13 Liu , T. , Xin , Z. ( 1988 ). Pointwise decay to contact discontinuities for systems of viscous conservation laws . Asian J. Math. 1 : 34 – 84 . [Google Scholar]] is removed so that our results hold, in particular, for the one-dimensional compressible Navier–Stokes equations of gas dynamics in both Lagrangian and Eulerian coordinates.     

A Theorem on Discrete,Torsion Free Subgroups of Isom H <Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let H ⁿ be the hyperbolic n-space with n⩾ 2. Suppose that Γ H ⁿ is a discrete, torsion free subgroup and a is a point in the domain of discontinuity Ω (Γ). Let p be the projection map from H ⁿ to the quotient manifold M=H ⁿ/Γ. In this paper we prove that there exists an open neighborhood U of a in H ⁿ∪ Ω (Γ) such that p is an isometry on U∩ H ⁿ.Mathematics Subject Classification (2000). primary: 51M10; secondary: 22E40, 57M50.     

Similarity analysis of modified shallow water equations and evolution of weak waves

In this paper, we obtain exact solutions to the nonlinear system of partial differential equations (PDEs), describing the one dimensional modified shallow water equations, using invariance group properties of the governing system. Lie group of point symmetries with commuting infinitesimal operators, are presented. The symmetry generators are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities.     

具有不连续系数的奇异摄动拟线性边值问题

研究一类具有不连续系数的奇异摄动二阶拟线性边值问题,其解因一阶导数的不连续性而出现内部层.用合成展开法和上下解定理得到所提问题内部层解的存在性和渐近估计.所得结果应用到由Farrell等（Farrell P A,O＇Riordan E,Shishkin G.A class of singularly perturbed quasilineax differential equations with interiors layers.Mathematics of Computation,2009,78：103-127）所提出的一个特殊拟线性问题.     

Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations

In this paper, we study the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the Navier-Stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of as the heat-conductivity coefficient κ tends to zero, provided that the viscosity μ is of higher order than the heat-conductivity κ. Without loss of generality, we set μ≡0. Here we have no need to restrict the strength of the contact discontinuity to be small.     

Functional solutions of the problem of discontinuity disintegration

A definition is given of functional solutions of the problem of discontinuity disintegration, and an example of such a solution is presented for the isoentropic system of gas dynamics. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 280–288, February, 1998.     

Contact discontinuity with general perturbations for gas motions

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the Navier-Stokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the Navier-Stokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate , it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of . Thus, this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the L^∞ norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.     

Numeric treatment of contact discontinuity with multi-gases

In this paper we work with a finite-volume-procedure for the computation of the one-dimensional Euler equations for the treatment of multi-gases and the problem of the correct treatment of the discontinuity was more nearly investigated. A suggested equation was embedded in the finite volumes context and implemented in our code accordingly and regarded as validated.     

高速扩展平面应力裂纹尖端的理想塑性场

在裂纹尖端的理想塑性应力分量都只是θ的函数的条件下,利用Mises屈服条件、定常运动方程及弹塑性本构方程,我们导出了高速扩展平面应力裂纹尖端的理想塑性场的一般解析表达式.将这些一般解析表达式用于具体裂纹,我们就得到高速扩展平面应力Ⅰ型和Ⅱ型裂纹的尖端的理想塑性场.     

On the well-posedness of the vacuum Einstein��s equations

The Cauchy problem of the vacuum Einstein’s equations aims to find a semi-metric g _αβ of a spacetime with vanishing Ricci curvature R _α,β and prescribed initial data. Under the harmonic gauge condition, the equations R _α,β  = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric h _ab and a second fundamental form K _ab . A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h _ab , K _ab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of the present article is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.     

Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P ₁ and P ₂ equations and special cases of the P ₃ and P ₅ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.     

On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative—either a solution exists for all t ≥ 0 or it goes to infinity in a finite time t = T (blowup mode). Sufficient conditions for the existence of a blowup mode are given.     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Linear one-dimensional integrodifferential problems

Linear one-dimensional integrodifferentiat equations of second order with constant limits of integration are considered. Methods are developed for transforming these equations into equivalent Fredholm integral equations of second order. Thus we show that in order to correctly pose problems for this class of equations it is necessary to prescribe a number of linearly independent side conditions (initial, boundary, etc.) equal to the total order of the free differential expression. Theorems are formulated for the existence of solutions and the uniqueness of the solution to correctly posed problems. These problems are also briefly examined for other classes of integrodiffential equations (weighted equations, certain classes of nonlinear equation, etc.) Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 8–16, 1989.     

On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+|x| ^p ) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O(1+|x| ^p ) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.