 Solution

For each $\nu >0$, choose the solution from parts (a)--(c). The maximum estimate gives a $\nu$-independent boundThus a sequence ${\nu }_j\downarrow 0$ has ${\omega }_{{\nu }_j}{\rightharpoonup }^{\ast }\omega$ in $L^{\infty }$. The periodic elliptic estimates forbound ${\Psi }_{{\nu }_j}$ in $W^{2,p}$ and $u_{{\nu }_j}$ in $W^{1,p}$ for every finite $p$. Taking $p>2$ and using compact sobolev embedding gives, after a further subsequence, uniform convergence $u_{{\nu }_j}\to u$. It follows that $u_{{\nu }_j}{\omega }_{{\nu }_j}\rightharpoonup u\omega$ distributionally.

The first energy estimate givesTherefore the viscous term vanishes in $H^{-1}$, and the weak formulation passes to the limit asThe elliptic relations pass to the limit as well and giveSince $\omega \in L^{\infty }\subset L^2$, periodic elliptic regularity gives $\Psi \in H_{\mathrm{per}}^2$ and $u\in H_{\mathrm{per}}^1\cap H$. Moreover $u\omega \in L^2$, so its divergence lies in $H_{\mathrm{per}}^{-1}$. This constructs the required weak solution of the damped-driven Euler system by a vanishing-viscosity limit.

Solved by gpt-5.6-sol high.