Showing papers in "Communications on Pure and Applied Mathematics in 1980"

Radiation boundary conditions for wave-like equations

Abstract: In the numerical computation of hyperbolic equations it is not practical to use infinite domains; instead, the domain is truncated with an artificial boundary. In the present study, a sequence of radiating boundary conditions is constructed for wave-like equations. It is proved that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r exp -m-1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature of the boundary conditions.

The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature

Abstract: The purpose of this paper is to study minimal surfaces in three-dimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian three-manifold N, then the condition that M be stable is expressed analytically by the requirement that o n any compact domain of M, the first eigenvalue of the operator A+Ric(v)+(AI* be positive. Here Ric (v) is the Ricci curvature of N in the normal direction to M and (A)’ is the square of the length of the second fundamental form of M. In the case that N is the flat R3, we prove that any complcte stable minimal surface M is a plane (Corollary 4). The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement is true provided the image of the Gauss map of M omits an open set on the sphere. The relationship of the stable regions on M to the area of their Gaussian image has been studied by Barbosa and do Carmo [l] (cf. Remark 5 ) . The methods of Schoen-Simon-Yau [ 113 give a proof of this result provided the area growth of a geodesic ball of radius r in M is not larger than r6. An interesting feature of our theorem is that it does not assume that M is of finite type topologically, or that the area growth of M is suitably small. The theorem for R3 is a special case of a classification theorem which we prove for stable surfaces in three-dimensional manifolds N having scalar curvature SZO. We use an observation of Schoen-Yau [8] to rearrange the stability operator so that S comes into play (see formula (12)). Using this, and the study of certain differential operators on the disc (Theorem 2), we are

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

Abstract: Inverse spectral theory is used to prescribe and study equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single-phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first-order terms.

The fluid dynamic limit of the nonlinear boltzmann equation

Abstract: Solutions of the nonlinear Boltzmann equation are constructed up to the first appearance of shocks in the corresponding fluid dynamics. This construction assumes the knowledge of solutions of the Euler equations for compressible gas flow. The Boltzmann solution is found as a truncated Hilbert expansion with a remainder, and the remainder term solves a weakly nonlinear equation which is solved by iteration. The solutions found have special initial values. They should serve as “outer expansions” to which initial layers, boundary layers and shock layers can be matched.

On subharmonic solutions of hamiltonian systems

Abstract: : Some existence results for T periodic solutions of (0.1) were presented in (1) for superquadratic Hamiltonian systems using finite dimensional minimax arguments together with estimates suitable to pass to a limit. An improved existence mechanism was introduced in (2) and applied to some of the super-quadratic problems of (1) as well as to several subquadratic cases. It will show here that these problems possess not only one T periodic solutions z sub 1 but infinitely many distinct subharmonic solutions z sub k. A word of caution must be entered at this point. Although z sub k has period kT, it may not be the case that z sub k has minimal (i.e. primitive) period kT. Indeed simple examples show that there may be an upper bound on the minimal period of z sub k.

Biow‐up of solutions of some nonlinear hyperbolic equations

Abstract: We prove that any nontrivial solution of certain nonlinear hyperbolic partial differential equations of second order blows up in a finite time if the initial data are localized, the initial velocity being on the average non-negative.

Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz

Abstract: A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation uu - uxx + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u∞ as |u|∞. The proof uses a modified form (PS)c of the Palais-Smale condition (PS).

Formation of singularities for wave equations including the nonlinear vibrating string

Abstract: Under very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small-amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small-amplitude time-periodic solutions.

On the cauchy problem for harmonic maps defined on two-dimensional Minkowski space

Abstract: Let R1+1 be two-dimensional Minkowski space and M a complete Riemannian manifold of dimension n. It is proved that the solution of the Cauchy problem for the harmonic map ϕ : R1+1 M exists globally. As an application to physics we conclude that the field function in a two-dimensional chiral field theory is regular for all time, if it is regular initially.

Local boundary regularity of holomorphic mappings

Abstract: Let f be a holomorphic mapping between two bounded domains D and D' in complex space ℂn. Suppose that D and D' contain smooth real hypersurfaces Γ and Γ′ as open subsets of their respective boundaries, which correspond under a continuous extension of f. We shall show that this extension is smooth, given certain restrictions on Γ, Γ, and f.

Finite difference schemes for conservation laws

Abstract: : We study finite difference approximations to weak solutions of the Cauchy problem for hyperbolic systems of conservation laws in one space dimension. We establish stability in the total variation norm and convergence for a class of hybridized schemes which employ the random choice scheme together with perturbations of classical conservative schemes. We also establish partial stability results for classical conservative schemes. Our approach is based on an analysis of finite difference operators on local and global wave configurations. (Author)

The similarity solution for a bore on a beach

Abstract: A similarity solution is found for the asymptotic behavior of a bore as it approaches the shoreline on a sloping beach. This gives direct confirmation of earlier results on the motion of the bore and adds details of the associated flow field. It also makes explicit the analogy with Guderley's implosion problem in gas dynamics; the solution is constructed closely following Guderley's arguments.