Results 1 to 2 of 2

Math Help - jump discontinuity proof for PDE

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    jump discontinuity proof for PDE

    So here is the problem statement:

    Consider the equation

    (6.12) u_y+uu_x=0.

    Let u be a C^1 solution of (6.12) in each of two regions separated by a curve x=\xi(y). Let u be continuous, but u_x have a jump discontinuity along the curve. Prove that

    \frac{d\xi}{dy}=u

    and hence that the curve is characteristic.
    In addition, the following "hint" is given:

    Hint: By (6.12)

    (u_y^+-u_y^-)+u(u_x^+-u_x^-)=0.

    Moreover, u(\xi(y),y) and (d/dy)u(\xi(y),y) are continuous on the curve.
    I believe u_y^+,u_y^- denote the limits of u_y from the right and left, respectively, and similarly for u_x^+,u_x^-.

    This is all from Fritz John's Partial Differential Equations, exercise 1.6.3, p19.

    I'm pretty lost on this one. Any help would be much appreciated. Thanks !
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    Re: jump discontinuity proof for PDE

    I've been working on this for a bit, and I think I have an idea. However, I still need some help making the idea work.

    According to the textbook (Fritz John, p17), if we choose R(u),S(u) satisfying S'(u)=uR'(u), then we have the following "conservation law":

    \displaystyle 0=\frac{d}{dy}\int_a^b R(u(x,y))\;dx+S(u(b,y))-S(u(a,y))

    and therefore if we choose a<\xi(y)<b then

    \displaystyle 0=S(u(b,y))-S(u(a,y))+\frac{d}{dy}\left(\int_a^{\xi(y)} R(u)\;dx+\int_{\xi(y)}^b R(u)\;dx\right)

    Now, if we choose R(u)=u and S(u)=u^2/2 then this will satisfy the conservation law, giving us

    (1) \displaystyle 0=\frac{1}{2}\left[u(b,y)^2-u(a,y)^2\right]+\frac{d}{dy}\left(\int_a^{\xi(y)} u\;dx+\int_{\xi(y)}^b u\;dx\right)

    If I can show that this implies

    (2) \displaystyle 0=\frac{1}{2}\left[u(b,y)^2-u(a,y)^2\right]+\frac{d\xi}{dy}[u(a,y)-u(b,y)]

    then it will follow that

    \displaystyle \frac{1}{2}\left[u(b,y)+u(a,y)\right]=\frac{d\xi}{dy}

    and taking limits as a,b\to\xi(y) will give us the desired result u=\frac{d\xi}{dy}.

    But the problem is, I don't know how to show that (1) implies (2). Any suggestions ?

    Thanks again guys !
    Last edited by hatsoff; October 2nd 2011 at 10:41 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. IVT and Jump discontinuity
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 11th 2011, 05:45 PM
  2. [SOLVED] Derivative with finite jump discontinuity?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 2nd 2010, 08:16 PM
  3. How did he jump from this to that
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 28th 2010, 01:29 PM
  4. Jump discontinuity, where?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 9th 2009, 08:29 AM
  5. Replies: 2
    Last Post: November 27th 2008, 05:44 PM

Search Tags


/mathhelpforum @mathhelpforum