A Mini-course on EllipticâHyperbolic Equations - ICMS

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The linear theoryWhy bother with the linear theory of elliptic–hyperbolic equations?Two arguments against emphasizing on the linear theory:“The linear theory of elliptic equations extends in a qualitativesense to much of the nonlinear theory, but nonlinear hyperbolicequations are much more singular than in the linear case, so thelinear elliptic–hyperbolic theory is not much of a guide to nonlinearequations, which are the ones of real interest.”“In the important applications, linear elliptic–hyperbolic equationsare only associated with small initial data.”Five reasons for developing a sound linear theory:
1. Elliptic–hyperbolic continuity equations tend to arise in specialforms which allow linearization by the hodograph method. Theseequations are more ubiquitous than people think (see below), andthe hodograph method is more useful than people think (see, forexample, the work of Yuxi Zheng)2. In, e.g., plasma contexts, linear elliptic–hyperbolic models arisepartly from certain physical hypotheses, for example:◮ magnetically dominated low-beta fields (“force-free models”)which arise in models of the solar corona, ball lightning, andintergalactic jets;◮ fields in which small-amplitude electromagnetic wavespropagate with a velocity much greater than the thermaloscillations of the particles (“cold plasma models”), whicharise in certain models of tokamak plasmas;◮ equilibrium models, in which a steady plasma is surrounded bya vacuum [R. Temam, Arch. Rat. Mech. Anal. 60, (1975)].
Page 1 and 2: A Mini-cou
Page 3 and 4: Part I. Orientation◮ Why study th
Page 5 and 6: But, for example:◮ Separating var
Page 7 and 8: ◮ the governing equations for ide
Page 9 and 10: ◮ the Laplace-Beltrami equation o
Page 11 and 12: ◮ In certain models for plasma fl
Page 13 and 14: ◮ The relevant equation for the e
Page 15 and 16: Adopting the “differential polyno
Page 17 and 18: Example of a first-order elliptic-h
Page 19 and 20: We obtain from the first-order syst
Page 21 and 22: Geometric approach: replace the cla
Page 23: Four features of the geometric appr
Page 27 and 28: Denote by Ω a domain of R n and co
Page 29 and 30: Given a function K (x, y) defined i
Page 31 and 32: For most of the past century, the s
Page 33 and 34: Tricomi’s assistant Cinquini-Cibr
Page 35 and 36: M.V. Keldysh had nothing to do with
Page 37 and 38: Equations of Keldysh type can be ch
Page 39 and 40: Why do equations of Keldysh type ha
Page 41 and 42: Let ξ = K (x, y) and η = λ (x, y
Page 43 and 44: Moreover the Implicit Function Theo
Page 45 and 46: If a system of equations has the fo
Page 47 and 48: The coordinate systems (p, q) and (
Page 49 and 50: Making the substitutions p = θ x ,
Page 51 and 52: The partial hodograph transformatio
Page 53 and 54: Example for an elliptic-hyperbolic
Page 55 and 56: Because the nonlinear Lavrent’ev-
Page 57 and 58: In order to fix the boundary, we no
Page 59 and 60: Example 2: Find an L 2 solution to
Page 61 and 62: Open boundary conditions: Data are
Page 63 and 64: Geometries of canonical domains:Geo
Page 65 and 66: Ingredients:1. A one-parameter fami
Page 67 and 68: Moreover, whenever a domain is star
Page 69 and 70: Example: the cold plasma model(x
Page 71 and 72: TheoremLet Ω be star-shaped with r
Page 73 and 74: Idea of the proof: Integral variant
Page 75 and 76:
If we find such a, b, and c, then w
Page 77 and 78:
Suppose instead that we prove the f
Page 79:
Instead of the L 2 solution which r
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A Mini-course on EllipticâHyperbolic Equations - ICMS

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A Mini-course on EllipticâHyperbolic Equations - ICMS