18.306 Lecture 18 - Thu 2021 04 29 - Virtual
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Finish #033b Characteristics as locus of weak singularities.
             Second order scalar equations.
Start  #044  First order quasi-linear 1-D systems of equations.

Note:
SKIP #034--#038
#039--#040 ... already covered.
#041 Will be in a PSet.
#042-#043 Students should read.
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 % -------------------------------------------------------------------- #033b
 Characteristics as locus of possible weak singularities in solution.
 Elliptic as equations that do not accept any singularities.

 Connection with the "principal symbol" of the equation: write
   equation by replacing derivatives \partial x_j by k_j. This
   yields a polynomial in the x_j.

 Example: Definition of hyperbolic and elliptic for scalar second
          order pde in 2-D using the symbol.
      a*u_xx + b*u_xy + c*u_yy + lower order terms = 0       [A]
      Symbol:  S = a*k1^2 + b*k1*k2 + c*k2^2.
 Elliptic:    S \neq 0 for all k1 and k2 not zero.
              Laplace:       S = k1^2 + k2^2.
 Hyperbolic:  S can be factored as two linear polynomials.
              Wave equation: S = k1^2 - k2^2.

 See note 3 below.

 Sketch of argument for weak singularities:
 Let the curve, G, along which a weak singularity in the solution to [A]
 can occur be given by the equation phi(x, y) = 0, where phi is some C^2
 function defined near G.   Let psi(x, y) be a C^2 function defined near
 G as well, such grad phi and grad psi are not co-lineal.   [see note 1]

 For [A] a "weak singularity" means that u is C^1 near G, C2 away from G,
 with the 2nd derivatives, possibly, having simple discontinuities across
 G.                                                         [see note 2]
 
 Let us now re-write the equation in terms of the local coordinate
 system provided by phi and psi. Then it can be seen that the equation
 takes the form

    (a*phi_x^2 + b*phi_x*phi_y + c*phi_y^2)*u_{phi,phi} = C

 where C involves u, u_phi, u_psi, u_{psi, psi}, and u_{phi, psi}, but
 not u_{phi, phi}. The weak singularity assumptions then tell us that C
 is continuous, while u_{phi,phi} has a simple discontinuity across G.
 This can only happen if  
                 a*phi_x^2 + b*phi_x*phi_y + c*phi_y^2 = 0.

 Note 1: if the curve is smooth enough, one can always define such a phi
    (at least near the curve) by: phi is the (signed) distance to the
    curve from the point (x, y).  Then let s be the arc-length along the
    curve, and define psi = s along the straight line segment normal to
    the curve, and crossing it, at the point s.

 Note 2: the idea of a "weak singularity" is that it is a failure in the
    smoothness of the solution, which is weak enough that it does not
    destroy the notion of being a solution in the classical sense. It
    turns out that the exact details of how this is implemented do not
    matter much. For example: if we assume that the lack of smoothness
    happens in the third order derivatives, the constraint on the curves
    across which this can happen is the same.

 Note 3: Point out relationship of classification names [elliptic,
    hyperbolic] with the level curves of the symbol. In the particular
    case of parabolic, note that the symbol [upon a rotation change of
    variables] corresponds to an ode: a*k_1^2; a \neq 0. Thus the lower
    order terms matter and one should look at  a*k_1^2 + b*k_1 + c*k_2.
    This is not an ode only if c \neq 0, in which case the level curves
    are parabolas.
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 % -------------------------------------------------------------------- #044
 First order quasi-linear 1-D systems of equations. Classification.
 Hyperbolic systems and characteristics.
 Domains of dependence and influence. Examples.

 First order quasilinear systems of equations
            u_t + A*u_x = F(x, t, u), where A = A(u, x, t).

 Characteristics and characteristic form of the equations.
 Example: linear, constant coefficients, no sources, case.

 Hyperbolic if A is real diagonalizable.
 Example: general solution for a hyperbolic system where A is
          constant and F = 0.
 In general, F \neq 0, characteristics couple.

 Domains of dependence and influence.

 Examples:
   Linear Gas Dynamics (acoustics). Sound waves, general solution.
   Wave equation. Reduce to form above.
   Klein Gordon equation. Characteristic form.
      Domains of dependence and influence.
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 Examples of first order 1-D hyperbolic systems. Linear acoustics.
 Example:  Linear acoustics in 1-D.
           Exact solution by characteristics.
           System equivalent to wave equation.
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 % -------------------------------------------------------------------- #046
 Systems of the form Y_t + A*Y_x = 0.
 Show ill-posed if complex eigenvalue exists.

 Motivation and physical meaning:
 When A is not a constant (particularly if A = A(u)) the system can
 change type, switching from well posed to ill posed.

 Example: stratified flows.
 --- Interpretation of the Richardson # criteria as a change of
     type (not an instability issue, as often portrayed).
 --- Physical reason: mixing occurs, stratification assumption fails.
     Ad-hoc fix: add vertical diffusivity in mass equation (mixing).
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 Example:  Wave equation.
 Solution of the initial value problem: D'Alembert.
 Domains of dependence and influence (Note that they involve the full
 wedge, not just the characteristics at the edge, for data u and u_t).

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