# An identity for the continuous integral solution of the conservation law

This is an exercise in Evans, Partial Differential Equations (1st edition), page 164, problem 13.

## Problem

## Proof with some annotations

**$u$ may not be differentiable anymore**, thinking all those blows up in time, and shock waves in space!), the trick in (1) is not applicable anymore, here the way to prove this is to choose proper test function $v$. Think $u(x,\tau)$ for any $\tau>0$, and consider the problem when the time starts at $\tau$: $$ \left\{ \begin{aligned} u_t + F(u)_x &= 0 &\text{ in } \mathbb{R} \times (\tau,\infty), \\ u&=u(x,\tau) &\text{ at } \mathbb{R} \times \left\{t=\tau\right\} , \end{aligned} \right.\tag{$\star$} $$ The weak solution to $(\star)$ coincides with the original IVP

**if we assume the solution is unique**. However, unfortunately this is not true for the integral solution, that's why we solve for an entropy solution by artificially adding a diffusion perturbation, which is the so-called vanishing viscosity method. We can see $u$ satisfies: $$ \int^{\infty}_{\tau} \int^{\infty}_{-\infty} \Big(u v_t + F(u) v_x\Big)\,dxdt + \int^{\infty}_{-\infty} uv\,dx \big|_{t=\tau}= 0,\tag{3} $$ for $v\in C^{\infty}_c(\mathbb{R}\times [\tau,\infty))$ which can be easily extended to the whole time domain smoothly. The difference between (2) and (3) is $$ \int^{\tau}_{0} \int^{\infty}_{-\infty} \Big(u \color{cyan}{v_t} + F(u) \color{cyan}{v_x}\Big)\,dxdt + \int^{\infty}_{-\infty} gv\,dx \big|_{t=0} - \int^{\infty}_{-\infty} uv\,dx \big|_{t=\tau}= 0.\tag{4} $$ Now we can either argue by (A) the test function makes (2) and (3) together must vanish on $[0,\tau)$, hence the blue terms are gone, or (B) choosing $v = 1$ on a set containing the support of $u$ from $t=0$ to $t=\tau$, and the support of $F(u)$ (notice $F\big(u(x\to \infty,t)\big) = F(0) = 0$, the compactly supportedness of $u$ implies that $F(u)$ is compactly supported), then blue terms are gone as well. Hence we have by (4): for any $\tau>0$ $$\int^{\infty}_{-\infty} gv \,dx \big|_{t=0} =\int^{\infty}_{-\infty} uv\,dx \big|_{t=\tau}.$$ By the choice of the test function $v$ above in (B), we have $$\int^{\infty}_{-\infty} g(x) \,dx =\int^{\infty}_{-\infty} u(x,\tau)\,dx.$$

## Comments