Pseudo-differential Operators and the Nash–Moser Theorem

Abstract

Building bridges between classical results and contemporary nonstandard problems, Mathematical Bridges embraces important topics in analysis and algebra from a problem-solving perspective. The book is structured to assist the reader in formulating and proving conjectures, as well as devising solutions to important mathematical problems by making connections between various concepts and ideas from different areas of mathematics. Instructors, motivated mathematics students from high school juniors to college seniors, and students interested in mathematical competitions should have This volume-the first comprehensive treatment of all the major types of systems models-emphasizes both theory and applications throughout the text; the applicability of the developed theory is demonstrated by means of many specific examples and applications to important classes of systems in areas such as power and energy, feedback control, artifical neural networks, digital signal processing and control, manufacturing, computer networks, and socioeconomics. The book may be used as a graduate textbook or as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, physics, chemistry, biology, and economics.

The strong relaxation limit of the multidimensional isothermal Euler equations

“…The velocity of the fluid is u := m/ρ. The aim of this paper is to construct global smooth solutions to (1) with initial data that are independent of the relaxation time τ , and to show that, in an appropriate time scaling, the density converges towards the solution to the heat equation as τ tends to 0. The sound speed a is always kept constant.…”

Introduction to Pseudodifferential Operators

“…Definition 3.2. We shall say a is a symbol of order m if a ∈ C ∞ (R n × R k ) and |D α x D β θ a(x, θ)| ≤ C α,β,K < θ > m−|β| , for x in K compact, all θ where < θ >= (1 + |θ| 2 ) 1 2 . The class is denoted S m (R n ; R k ) or just S m .…”

“…To check that the formal derivatives and the actual ones agree we evaluate the quotient -wlog we consider differentiation in x 1 P u(x + he 1 ) − P u(x) h = e i<x+he 1 ,ξ> a(x + he j , y, ξ) − e i<x,ξ> a(x, y, ξ) h e −i<y,ξ> u(y)dξdy = ∂ ∂s 1 e i(<s 1 ,ξ 1 >+<x ′′ ,ξ ′′ > a(s 1 , x ′′ , y, ξ) |s=x 1 +ǫh e −i<y,ξ> u(y)dξdy with 0 ≤ ǫ ≤ 1. This is just the value of the formal derivative at x 1 + ǫh and as h → 0 this must converge to the value of the formal derivative at x and so the result follows.…”

“…To make sense of this idea we will introduce the algebra of pseudo-differential operators. Our starting point is (1) -p(x, ξ) is a polynomial in ξ and smooth in x so we can think of it a finite sum of homogeneous functions p m−j (x, ξ) where p m−j (x, λξ) = λ m−j p m−j (x, ξ) with j running from 0 to m. We therefore define a (classical) pseudodifferential operator, of order m, to be an operator of the form (1) where p(x, ξ) is an infinite sum (in a sense, we will later make precise) of terms p m−j (x, ξ) which are smooth in ξ = 0 and p m−j (x, λξ) = λ m−j p m−j (x, ξ) λ > 0.…”

Global regularity for the minimal surface equation in Minkowskian geometry

“…@ t / 2 . In particular, this will follow, if we require that kr k L 1 1, which we will do henceforth.…”