A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation

We present the existence of the subsonic solution to a two-dimensional Riemann problem governed by a self-similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle. Thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle. By utilizing the barrier methods and iterative methods, we show the well-posedness of the transonic shock in the entire subsonic region and thus establish the global solution. This result does not rely on any smallness of Riemann data.     

Transonic shocks for steady Euler flows with cylindrical symmetry

In this paper, we prove the existence of transonic shocks adjacent to a uniform one for the full Euler system for steady compressible fluids with cylindrical symmetry in a cylinder, and consequently show the stability of such uniform transonic shocks. Mathematically we solve a free boundary problem for a quasi-linear elliptic–hyperbolic composite system. This reveals that the boundary conditions and equations interact in a subtle way. The key point is to “separate” in a suitable way the elliptic and hyperbolic parts of the system. The approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full Euler system.     

The wave equation on asymptotically de Sitter-like spaces

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X^○,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y_±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y₊ to Y₋.     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.     

Uniqueness of transonic shock solutions in a duct for steady potential flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.     

High-resolution numerical simulation of 2D nonlinear wave structures in electromagnetic fluids with absorbing boundary conditions

Here we show how the full set of governing equations for the dynamics of charged-particle fluids in an electromagnetic field may be solved numerically in order to model nonlinear wave structures propagating in two dimensions. We employ a source-term adaptation and two-fluid extension of the second-order high-resolution central scheme of Balbas et al. (2004) [1]. The model employed is a 2D extension of that used by Baboolal and Bharuthram (2007) [5] in studies of 1D shocks and solitons in a two-fluid plasma under 3D electromagnetic fields. Further, we outline the use of free-flow boundary conditions to obtain stable wave structures over sufficiently long modelling times. As illustrative results, we examine the formation and evolution of shock-like and soliton structures of the magnetosonic mode.     

Periodic solutions for an asymptotically linear wave equation with resonance

In this paper, we study the existence and multiplicity of nontrivial periodic solutions for an asymptotically linear wave equation with resonance, both at infinity and at zero. The main features are using Morse theory for the strongly indefinite functional and the precise computation of critical groups under conditions which are more general.     

Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.     

From the dynamics of gaseous stars to the incompressible Euler equations

A model for the dynamics of gaseous stars is introduced and formulated by the Navier-Stokes-Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier-Stokes-Poisson system in the torus Tn is investigated. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Existence and continuous dependence of large solutions for the magnetohydrodynamic equations

Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general large initial data are investigated. First the existence and uniqueness of global solutions are established with large initial data in H ¹. It is shown that neither shock waves nor vacuum and concentration are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon the initial data is proved. The equivalence between the well-posedness problems of the system in Euler and Lagrangian coordinates is also showed.     

Tutorial on analytical methods as a complement to numerical computing

The importance of using analytical methods is taught through the discussion of an example, where the analytical treatment of a partial differential equation provides not only a suitable time scale and an asymptotic solution, but also information important for the accuracy of the numerical solution.Dedicated to Peter Naur on the occasion of his 60th birthday     

Error bounds and parameter choice strategies for simplified regularization in Hilbert scales

Simplified regularization in the setting of Hilbert scales has been considered for obtaining stable approximate solutions for ill-posed operator equations. The derived error estimates using an a posteriori as well as an a priori parameter choice strategy are shown to be of optimal order with respect to certain natural assumptions on the ill-posedness of the equation.The work of M. Thamban Nair is partially supported by IC&SR, I.I.T., Madras     

The ignition problem for a scalar nonconvex combustion model

The ignition problem for the scalar Chapman-Jouguet combustion model without convexity is considered. Under the pointwise and global entropy conditions, we constructively obtain the existence and uniqueness of the solution and show that the unburnt state is stable (unstable) when the binding energy is small (large), which is the desired property for a combustion model. The transitions between deflagration and detonation are shown, which do not appear in the convex case.     

The capillary problem for an infinite trough

Consider an infinite trough (or wedge) with dihedral angle 2, 0 < < and a quantity of fluid inside contacting the edge. In equilibrium the free interface of the fluid will be a surface of constant mean curvature meeting the planar walls at a constant angle determined from physical considerations. One obvious configuration is for the free surface to be a section of a round circular cylinder parallel to the axis of the wedge whose position is determined by the angles and . For + > /2 the cylinder configuration is unstable and bifurcation occurs. We exhibit the full family of bifurcating solutions starting with the round cylinder solution and proceeding through a beading up process into a series of spherical sections suitably positioned. Furthermore, if the edge of the wedge is a re-entrant corner ( > /2) then there are further bifurcating families. One is a secondary bifurcation from the family initially constructed while the other is a primary bifurcation from the cylinder which are less symmetric than the initial families.This paper was completed while the author was on a subbatical from the University of Toledo and was a visitor at Stanford University and also with SFB 256, University of Bonn.     

Second order parallel algorithms for piecewise smooth displacement kernels

Second order parallel algorithms for Fredholm integral equations with piecewise smooth displacement kernels are derived. One is based on a difference scheme of Runge-Kutta type for an unusual partial differential equations for continuous functions of two variables. The other is based on the trapezoidal quadrature rule applied to a modified integral equations. It is found that the Runge-Kutta type algorithm exhibits certain advantages.The work of these authors was supported in part by the NSF Grant DMS-9007030The work of this author was supported in part by a grant from the National Science and Engineering Research Council of Canada     

Sharp decay estimates for hyperbolic balance laws

We consider strictly hyperbolic and genuinely nonlinear systems of hyperbolic balance laws in one-space dimension. Sharp decay estimates are derived for the positive waves in an entropy weak solution. The result is obtained by introducing a partial ordering within the family of positive Radon measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources as well as lower semicontinuity properties of continuous Glimm-type functionals.     

Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow

We establish global solutions of nonconcave hyperbolic equations with relaxation arising from traffic flow. One of the characteristic fields of the system is neither linearly degenerate nor genuinely nonlinear. Furthermore, there is no dissipative mechanism in the relaxation system. Characteristics travel no faster than traffic. The global existence and uniqueness of the solution to the Cauchy problem are established by means of a finite difference approximation. To deal with the nonconcavity, we use a modified argument of Oleinik (Amer. Math. Soc. Translations 26 (1963) 95). It is also shown that the zero relaxation limit of the solutions exists and is the unique entropy solution of the equilibrium equation.     

On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \mathbb{R}^{N}

In this paper we establish a Serrin’s type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equations in It is proved that if the gradient of pressure belongs to L^{α, γ} with then the weak solution actually is regular and unique. Received: May 4, 2004