deletions | additions       diff --git a/paragraph_2_Mimic_the_derivation__.tex b/paragraph_2_Mimic_the_derivation__.tex  index 4757c9b..8aec383 100644  --- a/paragraph_2_Mimic_the_derivation__.tex  +++ b/paragraph_2_Mimic_the_derivation__.tex 
...
Then it is a weak solution satisfying.   $$\int_{0}^{\infty}\int_{-\infty}^{\infty} \phi_{t}u + \phi_{x}f(u) dxdt + \int_{-\infty}^{\infty} \phi(x,0)u_{0}(x)dx = 0   $$ $$\\ \forall \phi \in C_{0}^{1}(\mathbb{R}^2 $$ If u is a weak solution, x =\varepsilon (t), then u must satisfy the condition  \frac{f(u^{-} - f(u^{+}}{u^{-} - u^{+}} = \varepsilon'(t)