摘要:

AbstractWe study the large time behavior of solutions of scalar conservation laws in one and two space dimensions with periodic initial data. Under a very weak nonlinearity condition, we prove that the solutions converge to constants as time goes to infinity. Even in one space dimension our results improve the earlier ones since we only require the fluxes to be nonlinear in a neighborhood of the mean value of the initial data. We then use these results to study the homogenization problem for scalar conservation laws with oscillatory initial data.

ISSN:0010-3640    

DOI:10.1002/cpa.3160460102

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1993

数据来源: WILEY

摘要:

AbstractThe inhomogeneous mean‐field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions. The results apply to various physical models of translation invariant plasmas, gravitating systems, as well as to planar fluid vortex motion. For attractive interactions a critical behavior occurs which can be classified as an extreme case of a second‐order phase transition. To include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic‐geometric mean inequality. The method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions. In addition, it also provides us with a new proof of the Trudinger‐Moser inequality known from differential geometry – in its s

ISSN:0010-3640    

DOI:10.1002/cpa.3160460103

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1993

数据来源: WILEY

摘要:

AbstractThe problem of establishing whether there are sets satisfying a formula in the first order set theoretic language ℒ︁ϵbased on =,ϵ, which involves only restricted quantifiers and has an equivalent ∀∃‐prenex form ((∀∃)0‐formula), is neither decidable nor semidecidable. In fact, given any ω‐model ofZF – FA, whereFAdenotes the Foundation Axiom, the set of existential closures of (∀∃)0‐formulae true in the model is a productive set. Undecidability arises even when dealing with restricted universal quantifiers only, provided a predicateis_a_pair(x), meaning thatxis a pair of distinct sets, is added to ℒ︁ϵ. If satisfiability refers to ω‐models ofZF – FAin which a form of Boffa's antifoundation axiom holds, then semidecidability fails as well; in fact, given any such model, the set of existential closures of formulae involving only restricted quantifiers and the predicateis_a_pairwhich are true in it, is a productive set. These results are all proved by making use of appropriate codings of Turing machine comp

ISSN:0010-3640    

DOI:10.1002/cpa.3160460104

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1993

数据来源: WILEY

摘要:

AbstractWe study the large time behavior of solutions of the one‐dimensional equations of elasticity, typically a nonconvex system of conservation laws. We show that solutions with compact initial support converge inL1to a superposition ofN‐waves, at algebraic rate. Likewise, we show that the perturbation expansion of weakly nonlinear geometric optics is uniformly valid to second order. Our analysis uses approximate scalar laws, together withL1stability of scalar solutions and decay in total variation of the wave sp

ISSN:0010-3640    

DOI:10.1002/cpa.3160460105

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1993

数据来源: WILEY

摘要:

AbstractWe consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins‐Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all timet≧ 0. In addition, we establish estimates of the rate at which solutions tend to

ISSN:0010-3640    

DOI:10.1002/cpa.3160460106

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1993

数据来源: WILEY

ISSN:0010-3640    

DOI:10.1002/cpa.3160460101

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1993

数据来源: WILEY