0

Consider the Burgers Equation $$\left\{\begin{array}{ccl} u_t + [e^u]_x & = & 0;\\ u(x,0) & = & u_0(x). \end{array}\right.$$

a) Study the behavior of the characteristic curves in the cases:

$\bullet$ $u_0$ increasing;

$\bullet$ $u_0$ decreasing;

$\bullet$ $u_0$ compact support soft.

b) Find an expression for $T^*$, the maximum lifetime of the solution.

c) Is it possible to guarantee in some of the cases of item (a) that the maximum lifetime of the solution is $T^* = +\infty$? That is, that the solution found by the method of characteristics is globally defined?

PS. When trying to solve, I found the characteristic curves that go through $(x_0,0)$ being given by $$x = x_0 + te^{u_0(x_0)}.$$ Then I couldn't develop further... Help me, please! :)

EditPiAf
  • 21,328
Santos
  • 823

1 Answers1

2

From $x = x_0 + t e^{u}$ and $u = u_0(x_0)$, we get $u = u_0(x-te^u)$ in implicit form. The rest of the resolution is a direct consequence of the steps described in this post.

EditPiAf
  • 21,328