7
$\begingroup$

I am studying conservation laws and reviewing the papers I get a doubt. Consider

$$u_t+f(u)_x=0$$ with $f$ smooth a conservation law and take the characteristics

$$x(t)\,\, ; \,\, x'(t)=f'(u(x(t),t))\,\, ; x(0)=x_0$$

Are they always straight lines for any $f$?

I did the following calculus:

$$x''(t)=f''(u(x(t),t)\underbrace{(u_x x'(t)+u_t)}_{0}=0$$

I think I am wrong, because I've thinked that this assertion that characteristics are lines holds only for some cases, as Burgers' equations.

Many thanks.

$\endgroup$
8
$\begingroup$

Note that $u$ is constant along the characteristic curves $x'(t) = f'(u(x(t),t))$ parameterized by $t$. Indeed, according to the chain rule and the quasi-linear PDE $u_t + f'(u) u_x = 0$ itself, we have $$ \frac{\text d}{\text d t}u(x(t),t) = u_x x'(t) + u_t = 0\, . $$ Thus, those curves are straight lines in the $x$-$t$ plane.

$\endgroup$
1
  • $\begingroup$ That's incredible... many thanks !! $\endgroup$ – Quiet_waters May 14 '19 at 20:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.