Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

Convergence of the nonisentropic Euler-Poisson equations to incompressible type Euler equations

In this paper, we investigate a multidimensional nonisentropic hydrodynamic (Euler-Poisson) model for semiconductors. We study the convergence of the nonisentropic Euler-Poisson equation to the incompressible nonisentropic Euler type equation via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and energy methods are used to rigorously justify the convergence of the limit.     

Asymptotics of Stationary Navier Stokes Equations in Higher Dimensions

We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.     

Universal Models of Soliton Hierarchies

We consider commutativity equations and a related new model similar to the Kadomtsev–Petviashvili equation in the Sato theory. Integration of this model equation with three independent variables is based on a generalization of the Dubrovin equations and the recently developed theory of transformations of spectral problems. We give examples of equations with a fractional-power dispersion law that can be linearized in this theory.     

Shape Optimization for Semi-Linear Elliptic Equations Based on an Embedding Domain Method

We study a class of shape optimization problems for semi-linear elliptic equations with Dirichlet boundary conditions in smooth domains in ℝ². A part of the boundary of the domain is variable as the graph of a smooth function. The problem is equivalently reformulated on a fixed domain. Continuity of the solution to the state equation with respect to domain variations is shown. This is used to obtain differentiability in the general case, and moreover a useful formula for the gradient of the cost functional in the case where the principal part of the differential operator is the Laplacian. Online publication 23 January 2004.     

On Taylor-series expansion methods for the second kind integral equations

In this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.     

On the convergence of a fourth order evolution equation to the Allen-Cahn equation

We construct special sequences of solutions to a fourth order nonlinear parabolic equation of Cahn-Hilliard/Allen-Cahn type, converging to the second order Allen-Cahn equation. We consider the evolution equation without boundary, as well as the stationary case on domains with Dirichlet boundary conditions. The proofs exploit the equivalence of the fourth order equation with a system of two second order elliptic equations with “good signs”.     

Hugoniót--Maslov Conditions for Vortex Singular Solutions of the Shallow Water Equations

For the phase of vortex singular solutions of the shallow water equations we justify the Hamilton--Jacobi equation corresponding to the hydrodynamical mode of perturbation propagation. We also obtain the next correction to the Cauchy--Riemann conditions describing how the singular part of the solution affects the smooth background.     

Weak approximations. A Malliavin calculus approach

We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore, if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable, then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions.

     

FINITE ELEMENT NONLINEAR GALERKIN COUPLING METHOD FOR THE EXTERIOR STEADY NAVIER-STOKES PROBLEM

1.IntroductionNonlinearGalerkinmethodsaremultilevelschemesforthedissipativeevolutionpartialdifferentialequations.Theycorrespondtothesplittingsoftheunknownu:u=y z)wherethecomponentsareofdifferentorderofmagnitudewithrespecttoaparameterrelatedtothespati...     

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L^∞ norm and weighted L² norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.     

Multiple scattering and modified Green's functions

Exterior boundary-value problems for the Helmholtz equation can be reduced to boundary integral equations. It is known that the simplest of these fail to be uniquely solvable at certain ‘irregular frequencies.’ For a single smooth scatterer, it is also known that irregular frequencies can be eliminated by using a modified fundamental solution, one that has additional singularities inside the scatterer. This approach is extended to treat the three-dimensional exterior Neumann problem for any finite number of disjoint smooth scatterers, using a fundamental solution that has additional singularities inside every scatterer.     

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper,upper bounds of the L2-decay rate for the Boussinesq equations are considered.Using the L2 decay rate of solutions for the heat equation,and assuming that the solutions of the Boussinesq equations are smooth,we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data.The decay results may then be obtained by passing to the limit of approximating sequences of solutions.The main tool is the Fourier splitting method.     

Higher nonlocal problems with bounded potential

The aim of this paper is to study a class of nonlocal fractional Laplacian equations depending on two real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci, we establish the existence of three weak solutions for nonlocal fractional problems exploiting an abstract critical point result for smooth functionals. We emphasize that the dependence of the underlying equation from one of the real parameters is not necessarily of affine type.     

Navier-Stokes方程的奇异性对大气运动方程组的影响

综述了大气运动基本方程组在光滑函数类中的稳定性和Navier-Stokes方程的不稳定性的若干结论．在此基础上,以大气运动方程组的Boussinesq近似为例,阐述了Navier-Stokes方程的不稳定性导致的大气运动基本方程组的某些简化模式的不稳定性,从而得到在简化基本方程过程中应该遵守的一个原则,以保证简化方程的稳定性．     

Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations

We study a forward-backward system of stochastic differential equations in an infinite-dimensional framework and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction--diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black--Scholes or Hamilton-Jacobi-Bellman type.     

Decay and growth estimates for solutions of second-order and third-order differential-operator equations

We obtained decay and growth estimates for solutions of second-order and third-order differential-operator equations in a Hilbert space. Applications to initial–boundary value problems for linear and nonlinear non-stationary partial differential equations modeling the strongly damped nonlinear improved Boussinesq equation, the dual-phase-lag heat conduction equations, the equation describing wave propagation in relaxing media, and the Moore–Gibson–Thompson equation are given.     

Using restrictions to accept or reject solutions of radical equations

The standard technique for solving equations with radicals is to square both sides of the equation as many times as necessary to eliminate all radicals. Because the procedure violates logical equivalence, it results in extraneous solutions that do not satisfy the original equation, making it necessary to check all solutions against the original equation. We propose alternative solution procedures that are rigorous and simple to execute where the extraneous solutions can be identified without verification against the original equation. In this article, we review previous literature, establish and illustrate rigorous solution procedures for radical equations of depth 1 (i.e. equations where all radicals can be eliminated in one step), and deal with an ambiguity concerning the definition of real-valued solutions to radical equations. An application to defining the inverse function, resulting in a parametric radical equation, is also explained.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable “Coefficients” without Convexity Assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.