In the book of Filho Lopes, Weak solutions for the equations of incompressible and inviscid Fuid dynamics. Page 59 They want to prove the following:
Take $w^{\epsilon}_0$ a $C^{\infty}_c(\mathbb{R})^2$ initial vorticity, then defining $u^{\epsilon}_0$ the initial velocity field by Biot Savart law, we have it in the space $C^{\infty}(\mathbb{R}^2)\cap L^2_{loc}$, so that we can apply to it a theorem of existence of classical solutions and get a velocity flow in $C^1([0;\infty) \times \mathbb{R}^2)$ that solves the 2-D Euler equations.
Now they want to prove that $ ||u^{\epsilon}(.,t)||_{L^{\infty}} \leq C(||w^{\epsilon}_0||_{L^{\infty}} + ||w^{\epsilon}_0||_{L^1})$ For this they argue with the following, they consider the representation of $u^{\epsilon}(.,t)$ by $w^{\epsilon}(.,t)$ through the biot savart law and are able to show that $||u^{\epsilon}(.,t)||_{L^{\infty}} \leq C(||w^{\epsilon}(.,t)||_{L^{\infty}} + ||w^{\epsilon}(.,t)||_{L^1})$ And then they directly claim that this is controlled by the same quantities but at time 0.
A little above the theorem they quickly say that since the vorticity satisfy the transport equation $\partial_tw + u\cdot \nabla w =0$ its $L^{\infty}$ and other $L^p$ norms are preserved in time. They say that this comes from the fact that $w(.,t)$ is given from $w(.,0)$ by means of particle trajectories.
My problem is that I don't see why would it be that at each point the particles trajectories exists globally and are smooth so that we could use them to show theses conservations. Especially that a little later they claim that the estimates that they prove in this theorem allow them to have smooth and global particles trajectories. The estimates they show are the following:
$||u^{\epsilon}(.,t)||_{L^{\infty}} \leq C(||w^{\epsilon}_0||_{L^{\infty}} + ||w^{\epsilon}_0||_{L^1})$
$|u^{\epsilon}(x,t)-u^{\epsilon}(y,t)| \leq C|x-y|(||w_0||_{L^{\infty}} + ||w_0||_{L^1} - ||w_0||_{L^{\infty}}log^-(|x-y|)$
Am I wrong to think that global existence of the particles trajectories and their smoothness come only from the fact that $u^{\epsilon}$ is smooth and from the first estimate, or is really the second estimate necessary ? And also how could we justify rigourously preservation of the $L^{\infty}$norm of the vorticity without having the global existence of particle trajectories.
Well, you can do that by standard energy estimates. Consider the equation $$ \partial_t w+u\cdot\nabla w=0, $$ where $u$ is divergence free. Then you have the following estimates $$ \frac{d}{dt}\frac{1}{1+p}\int w^{p+1}=\int w^p\partial_tw=-\int u\cdot \nabla w w^p=-\frac{1}{1+p}\int u\cdot \nabla(w^{p+1})=\frac{1}{1+p}\int \nabla\cdot u w^{p+1}. $$ The last integral vanish due to the divergence free condition. Consequently, integrating in time, we have $$ \|w(t)\|_{L^{p+1}}=\|w(0)\|_{L^{p+1}} $$ To recover the $L^\infty$ case, just take the appropriate limit in $p$. Is this helping?