Two-Scale Difference Equations. I. Existence and Global Regularity of Solutions

1991 ◽  
Vol 22 (5) ◽  
pp. 1388-1410 ◽  
Author(s):  
Ingrid Daubechies ◽  
Jeffrey C. Lagarias
Keyword(s):  
Difference Equations ◽  
Global Regularity ◽  
Regularity Of Solutions ◽  
Scale Difference

REGULARITY OF SOLUTIONS FOR SINGULAR SCHRÖDINGER EQUATIONS

1992 ◽  
Vol 04 (01) ◽  
pp. 95-161 ◽  
Author(s):  
ANDREAS M. HINZ
Keyword(s):  
Global Regularity ◽  
Fundamental Property ◽  
Mean Value ◽  
Borderline Case ◽  
Spectral Theorem ◽  
Regularity Of Solutions ◽  
Local Boundedness ◽  
Kato Class ◽  
Regularity Properties ◽  
Mean Value Inequalities

Local and global regularity properties of weak solutions of the Schrödinger equation −Δu+qu=λu play an important role in the spectral theory of the corresponding operator [Formula: see text]. Central among these properties is local boundedness of the solutions u, which is derived in an elementary way for potentials q whose negative parts q− lie in the local Kato class K loc . The method also provides mean value inequalities for and, in case q+ is in K loc too, continuity of u. To employ these mean value inequalities for bounds on eigenfunctions of T in a fixed direction, classes Kρ are introduced which reflect the behavior of q at infinity. A couple of examples allow to compare these classes with more conservative ones like the Stummel class Q and the global Kato class K. The fundamental property of local boundedness of solutions also serves as a base for a very short proof of the self-adjointness of T if the operator is bounded from below and q−∈K loc . If q(x) is permitted to go to −∞, as |x|→∞, a large class K ρ which guarantees self-adjointness of T is derived and contains the case q−(x)= O (|x|2). The Spectral Theorem then allows to deduce rapidly decaying bounds on eigenfunctions for discrete eigenvalues, at least if q−(x)= o (|x|2). This is also the condition under which the existence of a bounded solution is sufficient to guarantee λ∈σ(T). Here q−(x)= O (|x|2) appears as a borderline case and is discussed at some length by means of an explicit example. The class of admissible operators extending to these borderline cases with potentials singular locally and at infinity, the regularity results for solutions being mostly optimal, as demonstrated by numerous examples, yet the proofs being shorter and more straightforward than those to be found in literature for smaller classes and weaker results, the sets Kρ under consideration and the methods employed appear to be quite natural.

scholarly journals Recent advances on the global regularity for irrotational water waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170089 ◽  
Author(s):  
A. D. Ionescu ◽  
F. Pusateri
Keyword(s):  
Water Waves ◽  
Global Regularity ◽  
Local Existence ◽  
Three Dimensions ◽  
Regularity Of Solutions ◽  
Unified Framework ◽  
Nonlinear Water Waves ◽  
Eulerian Formulation ◽  
The Cauchy Problem ◽  
Dimensional Gravity

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach. We then describe in a unified framework, using the Eulerian formulation, global existence results for three- and two-dimensional gravity waves, and our joint result (with Deng and Pausader) on global regularity for the gravity–capillary model in three dimensions. We conclude this review with a short discussion about the formation of singularities and give a few additional references to other interesting topics in the theory. This article is part of the theme issue ‘Nonlinear water waves’.

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