I am studying conservation laws, particularly the Burger's equation and I am dealing with the proposition that states that the solutions of the Burger's equation with viscosity converges to a solution (weak) of the conservation law (unviscous).
Well, when we do the integration by parts to transpose the derivatives to the test function $\phi$, because of the viscosity $\epsilon u^\epsilon_{xx}$ term, we must do $\phi_{xx}$ . This means, I must have $\phi\in C_0^2$ (twice derivative).
So, we do all the calculus and by end the integration with the term $\phi_{xx}$ will be lost (because $\epsilon\rightarrow 0 $).
My doubt is this:
At the proof that the limit is solution of the conservation law we use $\phi \in C_0^2$ and prove that the integrals are valid for THESE test functions. But the integrals must be valid for all $\phi \in C_0^1$ functions…
Thanks.
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Edit
$u^\epsilon_t+f(u^\epsilon)_x=\epsilon u^\epsilon_{xx}\qquad (i)$
Let be $\phi$ a test function with $\phi(x,t)=0$ if $x\not\in (a,b)$ or $t\geq T$ (*). Define $\phi_0(x):=\phi (x,0)$ and $u_0^\epsilon(x):=u^\epsilon(x,0)$.
Integrating $(i)$:
$\int_a^b\int_0^T u^\epsilon_t\phi dtdx+\int_0^T \int_a^b f(u^\epsilon)_x\phi dxdt=\epsilon\int_a^b\int_0^T u^\epsilon_{xx}\phi dtdx\qquad (ii)$
Integrating by parts at left member and doing the cancels from (*):
$-\int_a^b\int_0^T [u^\epsilon\phi_t +f(u^\epsilon)\phi_x] dtdx+\int_a^b u^\epsilon_0\phi_0 dx=\epsilon\int_a^b\int_0^T u^\epsilon_{xx}\phi dxdt\qquad (iii)$
Integrating by parts at right member and doing the cancels from (*):
$-\int_a^b\int_0^T [u^\epsilon\phi_t +f(u^\epsilon)\phi_x] dtdx+\int_a^b u^\epsilon_0\phi_0 dx=-\epsilon\int_0^T\int_a^b u^\epsilon_{x}\phi_x dxdt\qquad (iv)$
So, at paper I am studying, the autor derives one more time at the right member, considering $\phi \in C_0^2$. Once I have to prove for any $\phi \in C_0^1$ by definition of weak solution, I have this doubt.
Thanks.
Best Answer
Equation (iv) obviously does not hold for test functions in $C^1_0$ because these don't have second derivatives . But once you take the limit as $\varepsilon \to 0$, an identity results that requires only $\phi \in C_0^1$. And you have proved it for all $\phi \in C_0^2$. Therefore you can now obtain it also for all $\phi \in C_0^1$ by approximation.
The main issue is to show that $u^\epsilon$ converges to some function $u$ and that $f(u^\epsilon)$ converges to $f(u)$, in some suitable sense. That is done with entropy conditions (e.g.).