The proof exploits the conservation of vorticity along particle trajectories in two dimensions, which provides a uniform-in-time $L^\infty$ bound on the vorticity $\omega$. Combined with the [Beale-Kato-Majda Criterion](/theorems/658), this rules out finite-time blow-up. **Step 1: Conservation of vorticity in $L^\infty$.** In two dimensions, the vorticity equation reduces to the pure transport equation $\partial_t \omega + u \cdot \nabla \omega = 0$ (the vortex-stretching term $(\omega \cdot \nabla)u$ vanishes). The vorticity is therefore constant along particle trajectories: $\omega(t, \Phi_t(x)) = \omega_0(x)$ for all $t \ge 0$ and $x \in \mathbb{R}^2$. Since $\Phi_t$ is a volume-preserving bijection, this immediately gives \begin{align*} \|\omega(t)\|_{L^\infty} &= \|\omega_0\|_{L^\infty}, \quad \forall\, t \ge 0. \end{align*} Alternatively, one may take the $L^p$ inner product of $\partial_t \omega + u \cdot \nabla \omega = 0$ with $\omega|\omega|^{p-2}$ and use the incompressibility $\nabla \cdot u = 0$ to obtain $\frac{d}{dt}\|\omega\|_{L^p} = 0$ for all $p \ge 2$, then pass to the [limit](/page/Limit) $p \to \infty$. **Step 2: Application of the BKM criterion.** The conservation $\|\omega(t)\|_{L^\infty} = \|\omega_0\|_{L^\infty}$ yields \begin{align*} \int_0^T \|\omega(\tau)\|_{L^\infty}\,d\tau &= T\|\omega_0\|_{L^\infty} < \infty \end{align*} for every finite $T > 0$. By the [Beale-Kato-Majda Criterion](/theorems/658), the solution persists globally: $u \in L^\infty([0,T]; H^s_\sigma(\mathbb{R}^2))$ for all $T > 0$. **Step 3: Growth estimate.** Combining the vorticity conservation with the differential inequality from the [Lipschitz Continuation Criterion For The Euler Equations](/theorems/656) and the logarithmic estimate for the Biot–Savart singular [integral](/page/Integral) yields the double-exponential growth bound \begin{align*} \|u(t)\|_{H^s} &\le (1 + \|u_0\|_{H^s})^{\exp(Ct\|\omega_0\|_{L^\infty})} \cdot \exp\bigl(2Ct(1 + \|u_0\|_{L^2} + \|\omega_0\|_{L^\infty})\bigr), \end{align*} where $C = C(s) > 0$.