The proof constructs solutions by mollifying the nonlinearity, solving the resulting ODE in $H^s_\sigma$ via the [Banach Fixed Point Theorem](/theorems/289), extracting uniform-in-$\varepsilon$ bounds from the energy structure, and passing to the limit. The main analytical ingredients are the [Kato Ponce Commutator Estimate](/theorems/904), the [Mollifier Approximation In Sobolev Spaces](/theorems/905), and the [Gronwall Inequality](/theorems/872). The proof proceeds as follows: Step 1 eliminates the pressure; Step 2 derives the key a priori $H^s$ energy estimate; Steps 3-4 construct and uniformly bound the mollified solutions; Step 5 shows convergence; Step 6 identifies the limit; Step 7 proves uniqueness.

Throughout, write $\langle\nabla\rangle^s$ for the Fourier multiplier with symbol $\langle\xi\rangle^s := (1 + |\xi|^2)^{s/2}$, so that $\|f\|_{H^s} = \|\langle\nabla\rangle^s f\|_{L^2}$. Write $\mathbb{P}$ for the Leray projector from the [Helmholtz Decomposition](/theorems/654).

**Step 1: Pressure-free formulation.** Applying $\mathbb{P}$ to the Euler equations and using $\mathbb{P}\nabla p = 0$ (since $\nabla p \in L^2_\sigma(\mathbb{R}^n)^\perp$ by the [Helmholtz Decomposition](/theorems/654)) and $\mathbb{P}u = u$ (since $u \in L^2_\sigma$), the system reduces to \begin{align*} \partial_t u &= -\mathbb{P}[(u \cdot \nabla)u], \quad u(0) = u_0. \end{align*}

**Step 2: A priori $H^s$ energy estimate.**

[claim:Divergence Free Cancellation] If $u \in H^s(\mathbb{R}^n; \mathbb{R}^n)$ with $s > n/2 + 1$ satisfies $\nabla \cdot u = 0$, then $\int_{\mathbb{R}^n} \langle\nabla\rangle^s u \cdot (u \cdot \nabla)\langle\nabla\rangle^s u\,d\mathcal{L}^n = 0$. [/claim]

[proof] Write $v := \langle\nabla\rangle^s u$. The integral equals $\sum_{i,j} \int_{\mathbb{R}^n} v_i\,u_j\,\partial_j v_i\,d\mathcal{L}^n$. Since $u_j\,\partial_j v_i = \partial_j(u_j v_i) - v_i\,\partial_j u_j$ and $\sum_j \partial_j u_j = \nabla \cdot u = 0$, this becomes $\sum_{i,j} \int v_i\,\partial_j(u_j v_i)\,d\mathcal{L}^n$. Integration by parts (justified by $u, v \in H^s \hookrightarrow L^\infty$ via the [Sobolev Embedding Into Continuous Functions](/theorems/226)) gives \begin{align*} \sum_{i,j} \int_{\mathbb{R}^n} v_i\,\partial_j(u_j v_i)\,d\mathcal{L}^n &= -\sum_{i,j} \int_{\mathbb{R}^n} \partial_j v_i \cdot u_j v_i\,d\mathcal{L}^n = -\int_{\mathbb{R}^n} \langle\nabla\rangle^s u \cdot (u \cdot \nabla)\langle\nabla\rangle^s u\,d\mathcal{L}^n. \end{align*} Hence the integral equals its own negation and is therefore zero. [/proof]

[claim:A Priori Energy Estimate] There exists $C_s > 0$ such that any sufficiently smooth divergence-free solution of $\partial_t u = -\mathbb{P}[(u \cdot \nabla)u]$ satisfies $\frac{d}{dt}\|u(t)\|_{H^s} \le C_s\,\|u(t)\|_{H^s}^2$. [/claim]

[proof] Compute \begin{align*} \frac{1}{2}\frac{d}{dt}\|u\|_{H^s}^2 &= -\int_{\mathbb{R}^n} \langle\nabla\rangle^s u \cdot \langle\nabla\rangle^s \mathbb{P}[(u \cdot \nabla)u]\,d\mathcal{L}^n. \end{align*} Since $\mathbb{P}$ is self-adjoint on $L^2$ and $\mathbb{P}\langle\nabla\rangle^s u = \langle\nabla\rangle^s u$ (because $\langle\nabla\rangle^s$ commutes with $\mathbb{P}$ as both are Fourier multipliers, and $u \in L^2_\sigma$), this equals $-\int \langle\nabla\rangle^s u \cdot \langle\nabla\rangle^s[(u \cdot \nabla)u]\,d\mathcal{L}^n$. Writing \begin{align*} \langle\nabla\rangle^s[(u \cdot \nabla)u] &= (u \cdot \nabla)\langle\nabla\rangle^s u + [\langle\nabla\rangle^s, u \cdot \nabla]u \end{align*} and using the Divergence Free Cancellation claim to eliminate the first term: \begin{align*} \frac{1}{2}\frac{d}{dt}\|u\|_{H^s}^2 &= -\int_{\mathbb{R}^n} \langle\nabla\rangle^s u \cdot [\langle\nabla\rangle^s, u \cdot \nabla]u\,d\mathcal{L}^n. \end{align*} By the [Cauchy-Schwarz Inequality](/theorems/432), the right-hand side is bounded by $\|u\|_{H^s}\,\|[\langle\nabla\rangle^s, u \cdot \nabla]u\|_{L^2}$. The commutator equals $\sum_{j=1}^n [\langle\nabla\rangle^s, u_j]\,\partial_j u$. Applying the [Kato Ponce Commutator Estimate](/theorems/904) to each term: \begin{align*} \|[\langle\nabla\rangle^s, u_j]\,\partial_j u\|_{L^2} &\le C_s\bigl(\|\nabla u_j\|_{L^\infty}\,\|\partial_j u\|_{H^{s-1}} + \|u_j\|_{H^s}\,\|\partial_j u\|_{L^\infty}\bigr) \le C_s\,\|\nabla u\|_{L^\infty}\,\|u\|_{H^s}. \end{align*} Summing over $j$ and using the [Sobolev Embedding Into Continuous Functions](/theorems/226) ($\|\nabla u\|_{L^\infty} \le C\|u\|_{H^s}$ since $s - 1 > n/2$) gives $\|[\langle\nabla\rangle^s, u \cdot \nabla]u\|_{L^2} \le C_s\,\|u\|_{H^s}^2$. Therefore $\frac{1}{2}\frac{d}{dt}\|u\|_{H^s}^2 \le C_s\,\|u\|_{H^s}^3$, giving $\frac{d}{dt}\|u\|_{H^s} \le C_s\,\|u\|_{H^s}^2$. [/proof]

**Step 3: Mollified system.** Let $J_\varepsilon$ be the mollification operator from the [Mollifier Approximation In Sobolev Spaces](/theorems/905). Define \begin{align*} F_\varepsilon: H^s_\sigma(\mathbb{R}^n) &\to H^s_\sigma(\mathbb{R}^n) \\ u &\mapsto -J_\varepsilon\,\mathbb{P}\,\nabla \cdot (J_\varepsilon u \otimes J_\varepsilon u), \end{align*} and consider $\frac{du^\varepsilon}{dt} = F_\varepsilon(u^\varepsilon)$, $u^\varepsilon(0) = J_\varepsilon u_0$.

[claim:Local Existence For The Mollified System] For each $\varepsilon > 0$, there exists $T_\varepsilon > 0$ and a unique $u^\varepsilon \in C^1([-T_\varepsilon, T_\varepsilon]; H^s_\sigma)$. [/claim]

[proof] By Part (ii) of the [Mollifier Approximation In Sobolev Spaces](/theorems/905), $\|J_\varepsilon f\|_{H^k} \le C_k\,\varepsilon^{-k}\,\|f\|_{L^2}$. Combined with the [Algebra Property Of Inhomogeneous Sobolev Spaces](/theorems/468), for $u, v$ in $B_R := \{w : \|w\|_{H^s} \le R\}$: \begin{align*} \|F_\varepsilon(u)\|_{H^s} &\le C\,\varepsilon^{-1}\,\|u\|_{H^s}^2, \\ \|F_\varepsilon(u) - F_\varepsilon(v)\|_{H^s} &\le C\,\varepsilon^{-1}\,R\,\|u - v\|_{H^s}. \end{align*} Set $R_0 := 2\|u_0\|_{H^s}$ and $T_\varepsilon := \varepsilon/(2CR_0)$. The Picard operator $(\mathcal{T} u)(t) := J_\varepsilon u_0 + \int_0^t F_\varepsilon(u(\tau))\,d\mathcal{L}^1(\tau)$ maps $C([-T_\varepsilon, T_\varepsilon]; B_{R_0})$ into itself (since $\|\mathcal{T}u(t)\|_{H^s} \le \|u_0\|_{H^s} + T_\varepsilon C\varepsilon^{-1}R_0^2 = R_0$) and is a $1/2$-contraction. The [Banach Fixed Point Theorem](/theorems/289) gives the unique fixed point. [/proof]

**Step 4: Uniform bounds.**

[claim:L2 Conservation For The Mollified System] $\|u^\varepsilon(t)\|_{L^2} = \|J_\varepsilon u_0\|_{L^2}$ for all $t$. [/claim]

[proof] Write $w := J_\varepsilon u^\varepsilon$. Since $J_\varepsilon$ is self-adjoint (Part (iii) of the [Mollifier Approximation In Sobolev Spaces](/theorems/905)), $\mathbb{P}$ is self-adjoint (by the [Helmholtz Decomposition](/theorems/654)), and $\mathbb{P}w = w$ (because $\nabla \cdot w = J_\varepsilon(\nabla \cdot u^\varepsilon) = 0$ by Part (iv) of the [Mollifier Approximation In Sobolev Spaces](/theorems/905)): \begin{align*} \frac{1}{2}\frac{d}{dt}\|u^\varepsilon\|_{L^2}^2 &= -\int_{\mathbb{R}^n} w \cdot \nabla \cdot (w \otimes w)\,d\mathcal{L}^n = -\sum_{i,j}\int_{\mathbb{R}^n} w_i\,\partial_j(w_j w_i)\,d\mathcal{L}^n. \end{align*} Expanding: $w_i\,\partial_j(w_j w_i) = w_i w_j\,\partial_j w_i + w_i^2\,\partial_j w_j$. The second term vanishes since $\nabla \cdot w = 0$. For the first: \begin{align*} \sum_{i,j}\int_{\mathbb{R}^n} w_i w_j\,\partial_j w_i\,d\mathcal{L}^n &= \frac{1}{2}\sum_j\int_{\mathbb{R}^n} w_j\,\partial_j|w|^2\,d\mathcal{L}^n = -\frac{1}{2}\int_{\mathbb{R}^n} |w|^2\,\nabla \cdot w\,d\mathcal{L}^n = 0. \end{align*} [/proof]

[claim:Uniform Hs Bound] Setting $T_0 := (2C_s\|u_0\|_{H^s})^{-1}$, we have $\sup_{|t| \le T_0} \|u^\varepsilon(t)\|_{H^s} \le 2\|u_0\|_{H^s}$ uniformly in $\varepsilon$. [/claim]

[proof] The same computation as in the A Priori Energy Estimate applies to the mollified equation: $J_\varepsilon$ preserves the divergence-free condition (Part (iv) of the [Mollifier Approximation In Sobolev Spaces](/theorems/905)) and $\|J_\varepsilon f\|_{H^s} \le \|f\|_{H^s}$ (since $|\hat\phi_\varepsilon(\xi)| \le 1$). The constant $C_s$ does not depend on $\varepsilon$. The inequality $\frac{d}{dt}\|u^\varepsilon\|_{H^s} \le C_s\|u^\varepsilon\|_{H^s}^2$ with $\|u^\varepsilon(0)\|_{H^s} \le \|u_0\|_{H^s}$ integrates by comparison with $y' = C_s y^2$, $y(0) = \|u_0\|_{H^s}$, whose solution is \begin{align*} y(t) &= \frac{\|u_0\|_{H^s}}{1 - C_s\|u_0\|_{H^s}\,t}, \end{align*} to give $\|u^\varepsilon(t)\|_{H^s} \le 2\|u_0\|_{H^s}$ for $|t| \le T_0$. [/proof]

**Step 5: Convergence.**

[claim:Cauchy Property In L2] $\sup_{|t| \le T_0} \|u^\varepsilon(t) - u^\delta(t)\|_{L^2} \le C\,\max\{\varepsilon, \delta\}\,\|u_0\|_{H^s}$. [/claim]

[proof] Define $w := u^\varepsilon - u^\delta$. Taking the $L^2$ inner product of $\partial_t w = F_\varepsilon(u^\varepsilon) - F_\delta(u^\delta)$ with $w$, the terms involving $(J_\delta u^\delta \cdot \nabla)J_\delta w$ cancel by the divergence-free mechanism from the $L^2$ Conservation claim. The remaining terms involve $J_\varepsilon - J_\delta$ applied to functions bounded in $H^s$ by Step 4. By Part (i) of the [Mollifier Approximation In Sobolev Spaces](/theorems/905), $\|(J_\varepsilon - I)f\|_{L^2} \le C\varepsilon\|f\|_{H^1}$. After collecting terms: \begin{align*} \frac{d}{dt}\|w(t)\|_{L^2} &\le C\,\|u_0\|_{H^s}\,\|w(t)\|_{L^2} + C\,\max\{\varepsilon, \delta\}\,\|u_0\|_{H^s}^2. \end{align*} Since $\|w(0)\|_{L^2} \le C\max\{\varepsilon,\delta\}\|u_0\|_{H^s}$ by Part (i) of the [Mollifier Approximation In Sobolev Spaces](/theorems/905), the [Gronwall Inequality](/theorems/872) gives the bound. [/proof]

Hence $\{u^\varepsilon\}$ is Cauchy in $C([-T_0, T_0]; L^2_\sigma)$, with limit $u$. By the Uniform $H^s$ Bound and the [Sobolev Interpolation Inequality](/theorems/906), \begin{align*} \|u^\varepsilon - u\|_{H^r} &\le \|u^\varepsilon - u\|_{L^2}^{1-r/s}\,(4\|u_0\|_{H^s})^{r/s} \to 0 \end{align*} for all $0 \le r < s$. Taking $r > n/2 + 1$, the [Sobolev Embedding Into Continuous Functions](/theorems/226) gives $u^\varepsilon \to u$ in $C_t W^{1,\infty}$.

**Step 6: Identification of the limit.** The mollified solutions satisfy \begin{align*} u^\varepsilon(t) &= J_\varepsilon u_0 - \int_0^t J_\varepsilon\,\mathbb{P}\,\nabla \cdot (J_\varepsilon u^\varepsilon \otimes J_\varepsilon u^\varepsilon)\,d\mathcal{L}^1(\tau). \end{align*} As $\varepsilon \to 0$: $J_\varepsilon u_0 \to u_0$ in $H^s$ by Part (i) of the [Mollifier Approximation In Sobolev Spaces](/theorems/905); $J_\varepsilon u^\varepsilon \to u$ in $C_t W^{1,\infty}$ by Step 5; and the outer $J_\varepsilon$ converges to the identity in $L^2$. The dominated convergence theorem (with the uniform $H^s$ bound) gives \begin{align*} u(t) &= u_0 - \int_0^t \mathbb{P}[(u \cdot \nabla)u]\,d\mathcal{L}^1(\tau). \end{align*} Since $u \in L^\infty_t H^s$ and the integrand is continuous in $H^{s-1}$ (by the [Algebra Property Of Inhomogeneous Sobolev Spaces](/theorems/468)), $u \in C^1_t H^{s-1}$ with $\partial_t u = -\mathbb{P}[(u \cdot \nabla)u]$. The pressure is $\nabla p = (I - \mathbb{P})[(u \cdot \nabla)u]$ by the [Helmholtz Decomposition](/theorems/654).

To upgrade $u \in L^\infty_t H^s$ to $u \in C_t H^s$: for $t_k \to t$, the sequence $u(t_k)$ is bounded in $H^s$ and converges in $L^2$. By the [Sobolev Interpolation Inequality](/theorems/906), $u(t_k) \to u(t)$ in $H^r$ for $r < s$. Banach-Alaoglu gives a weakly convergent subsequence in $H^s$ with limit $u(t)$, so $\|u(t)\|_{H^s} \le \liminf_k \|u(t_k)\|_{H^s}$. The reverse inequality follows from the energy estimate applied backwards. Weak convergence plus norm convergence gives strong convergence in the Hilbert space $H^s$.

**Step 7: Uniqueness.**

[claim:Uniqueness Of Solutions] If $u, \tilde{u} \in C([-T_0, T_0]; H^s_\sigma)$ both solve $\partial_t u = -\mathbb{P}[(u \cdot \nabla)u]$ with the same initial data, then $u = \tilde{u}$. [/claim]

[proof] Define $w := u - \tilde{u}$. Then $\partial_t w = -\mathbb{P}[(w \cdot \nabla)u] - \mathbb{P}[(\tilde{u} \cdot \nabla)w]$. Taking the $L^2$ inner product with $w$: \begin{align*} \frac{1}{2}\frac{d}{dt}\|w\|_{L^2}^2 &= -\int_{\mathbb{R}^n} w \cdot \mathbb{P}[(w \cdot \nabla)u]\,d\mathcal{L}^n - \int_{\mathbb{R}^n} w \cdot \mathbb{P}[(\tilde{u} \cdot \nabla)w]\,d\mathcal{L}^n. \end{align*} For the second integral, since $\mathbb{P}$ is self-adjoint and $\mathbb{P}w = w$: \begin{align*} \int_{\mathbb{R}^n} w \cdot (\tilde{u} \cdot \nabla)w\,d\mathcal{L}^n &= \frac{1}{2}\int_{\mathbb{R}^n} \tilde{u} \cdot \nabla|w|^2\,d\mathcal{L}^n = -\frac{1}{2}\int_{\mathbb{R}^n} |w|^2\,\nabla \cdot \tilde{u}\,d\mathcal{L}^n = 0, \end{align*} by integration by parts and $\nabla \cdot \tilde{u} = 0$. For the first, by the [Cauchy-Schwarz Inequality](/theorems/432) and $\|\mathbb{P}\|_{\mathcal{L}(L^2)} \le 1$: \begin{align*} \Bigl|\int_{\mathbb{R}^n} w \cdot \mathbb{P}[(w \cdot \nabla)u]\,d\mathcal{L}^n\Bigr| &\le \|w\|_{L^2}\,\|(w \cdot \nabla)u\|_{L^2} \le \|\nabla u\|_{L^\infty}\,\|w\|_{L^2}^2. \end{align*} Hence $\frac{d}{dt}\|w\|_{L^2} \le \|\nabla u\|_{L^\infty}\,\|w\|_{L^2}$. Since $\|\nabla u\|_{L^\infty} \le C\|u\|_{H^s}$ by the [Sobolev Embedding Into Continuous Functions](/theorems/226), the [Gronwall Inequality](/theorems/872) with $w(0) = 0$ gives $w \equiv 0$. [/proof]

Setting $T := T_0$ completes the proof. Continuous dependence follows from the same Gronwall argument with nearby initial data. Higher regularity persistence follows by repeating the argument with $s$ replaced by $s'$.