I think I have found a solution for a PDE of the form

  1. $u_t + g(u)u_x = 0$

  2. where $u(x, 0) = g^{-1}(x)$

The solution is $u(x,y) = g^{-1}\left(\frac x{t+1}\right)$

This solution satisfies 1 and 2 under the assumption that $\forall z, g\left(g(z)^{-1}\right) = z$

However I am worried about the effects of discontinuities in $g$ or its inverse, and issues where the function is not 1-1.

What sort of problems should I watch out for and how can I get around them?

  • 1
    $\begingroup$ The tag differential-equations is intended for questions about ordinary differential equations, there is a separate tag for pdes; see the tag-wiki and the tag-excerpt. (The tag-excerpt is also shown when you are adding a tag to a question.) $\endgroup$ – Alex R. Jan 13 '14 at 16:51
  • $\begingroup$ The original place I found it had $g(x) = e^x$ which gave an answer $u(x,y) = ln(\frac x{t+1})$ But as far as I know the solution would hold for any smooth function with an inverse, that is what I am trying to determine. $\endgroup$ – kleineg Jan 15 '14 at 19:39

This equation you are dealing with, although it looks simple, it is a very problematic one: it is a nonlinear conservation law. These equation DO NOT have smooth solutions for all $t>0$. Their solution break down after a short time and the develop "shock waves" - discontinuous weak solutions.

The simpler such equation is Burger's Equation ($g(u)=u$) $$ u_t+uu_x=0, $$ and there is a written about it as it is the simplest model for shock formation.


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