1
$\begingroup$

In my book the "Global classical solutions for quasilinear hyperbolic systems" written by Li Ta-Tsien, following are written,

Local existence and uniqueness of $C^1$ function+ uniform a priori estimate for the $C^1$ norm of the solution

imply Global existence and uniqueness of $C^1$ solution.

More specipically,

Cauchy problem for first order quasilinear equations in two independent variables:

$$\partial_t u+\lambda(u)\partial_x u=0\ \ \ (t\geq0,-\infty<x<\infty)$$ $$t=0: u=\phi(x)\ \ \ (-\infty<x<\infty)$$

i) local existence and uniqueness of solution

for each $C^1$ function $\phi(x)$ there exists a positive constant $T_0$ only depending on the $C^1$ norm of $\phi(x)$ such that above Cauchy problem admits a unique $C^1$ solution $u=u(t,x)$ on the domain $$D(T_0)=\{(t,x)|0\leq t\leq T_0,-\infty<x<\infty\}$$.

ii) uniform a priori estimate for the $C^1$ norm of the solution:

For any fixed T>0, if above Cauchy problem admits a $C^1$ solution $u=u(t,x)$ on the domain $$D(T)=\{(t,x)|0\leq t\leq T,-\infty<x<\infty\}$$ then $C^0$ norm of $u(t,x)$ and $\partial_x u(t,x)$ has an upper bound independent of T.

if above Cauchy problem satisfy i),ii) then there exist a unique global $C^1$solution for above Cauchy problem.

But I don't have any idea how I can verify i)and ii) imply the Global existence and uniqueness of the equation.

Please give me some help, if you have some ideas or knowledges.

$\endgroup$

1 Answer 1

2
$\begingroup$

By $(1)$, you have a unique solution up to a certain time $T$. By $(2)$ you have an estimate on the $C^1$-norm of $u(\cdot, T)$. Then you solve the problem starting at time $T$. For that problem, by $(1)$ you have a unique local solution defined on $[T,T+T_1]$ with $C_1$-norm bounded by the same constant as before. Pasting together the previous solutions, you get a solution on $[0,T+T_1]$. Applying $(1)$ again, you can extend your solution up to $T+2T_1$ and so on. In this way, you define a global solution, defined on $[0,+\infty)$. The solution is unique because if you have two solutions that differ for the first time at some $t_0>0$, that would violate local uniqueness on the interval $[T+nT_1, T+(n+1)T_1]$ for some $n$.

$\endgroup$
5
  • $\begingroup$ thank you for your answer, but I wonder that in above argument, why the uniform upper bounded of $C^1$ norm of u imply, existence of constant $T_1$ , although constant $T_0$ is only depend on the $C_1$ norm of $\phi(x)$. But it is not imply that constant $T_0$ is only depend on the upperbound of $C_1$ norm of $\phi$. $\endgroup$
    – 백주상
    Aug 20, 2021 at 4:51
  • $\begingroup$ In other words, what we can guarantee in (2) is just uniform upper bound of C1 norm of $\phi$. but , In (1). the constant $T_0$ is not depend on the upper bound of C1 norm of $\phi$, but C1 norm of $\phi$. then how we can fix constant $T_1$ in this suggestion. $\endgroup$
    – 백주상
    Aug 20, 2021 at 5:11
  • $\begingroup$ $T_0$ in part $(1)$ depends on the $C^1$ norm of $\phi$ through an upper bound. $\endgroup$
    – GReyes
    Aug 20, 2021 at 14:34
  • $\begingroup$ Is there any reference to study $T_0$ is depend on the $C_1$ norm of $\phi$ through an upper bound? $\endgroup$
    – 백주상
    Oct 27, 2021 at 0:54
  • $\begingroup$ Typically those estimates follow from some general result (a fixed point for example) and the constants are pretty much unknown. $\endgroup$
    – GReyes
    Oct 27, 2021 at 3:10

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .