# Local existence and uniqueness and uniform a priori estimate for the C1 norm of the solution imply existence and uniqueness of global solution

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:

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

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 $$D(T)=\{(t,x)|0\leq t\leq T,-\infty then $$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.

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 $$n$$.

• 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 $\phi$.
– 백주상
Aug 20, 2021 at 4:51
• 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.
– 백주상
Aug 20, 2021 at 5:11
• $T_0$ in part $(1)$ depends on the $C^1$ norm of $\phi$ through an upper bound. Aug 20, 2021 at 14:34
• Is there any reference to study $T_0$ is depend on the $C_1$ norm of $\phi$ through an upper bound?
– 백주상
Oct 27, 2021 at 0:54
• Typically those estimates follow from some general result (a fixed point for example) and the constants are pretty much unknown. Oct 27, 2021 at 3:10