A Mini-course on EllipticâHyperbolic Equations - ICMS

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This is equivalent to the systemw t = v x , v t = σ(w) x .We can impose initial values w (x, 0) = w 0 (x) and v (x, 0) = v 0 (x).Equating mixed partial derivatives of v yieldsw tt = σ(w) xx ,and making the substitutions w = u x and v = u t , yieldsu tt = σ (u x ) x,which is eq. (1) of the preceding frame.[Pego and Serre, SIAM J. Numer. Anal. 25 (1988); Boonkasameand Milewski, Studies in Appl. Math. 128 (2011); Bialy andMironov, arXiv e-print, December 2012]
We obtain from the first-order system the matrix equation( ) ( ) ( )w 0 1 w=v σ ′ .(w) 0 vtThis is of <strong>course</strong> a matrix equation of the form X t = AX x . Nowadopting the “eigenvalue approach” to classification, we computethe eigenvalues of the matrix −A by solving the scalar equation|A + λI | =∣Because the eigenvaluesλ 1σ ′ (w) λλ = ± √ σ ′ (w)∣ = λ2 − σ ′ (w) = 0.are real-valued and nondegenerate when σ ′ is positive, in that casethe characteristics are real and we take this to be the region ofhyperbolicity for the system.Remark: classical solutions of this equation tend to havefinite-time singularities even in the strictly hyperbolic regime, so weusually look for weak solutions.x
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: Example of a first-order elliptic-h
Page 21 and 22: Geometric approach: replace the cla
Page 23 and 24: Four features of the geometric appr
Page 25 and 26: 1. Elliptic-hyperbolic continuity e
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