Uniqueness of smooth periodic solutions and continuous dependence in $H^{s-1}$ on the common local interval

For either smooth solution $z=u$ or $z=v$, put $Y(t)=\mathrm{\parallel }z(t){\mathrm{\parallel }}_{H^s}$. Pair its equation with ${\mathrm{\Lambda }}^{2s}z$, using periodic integration by parts and the commuting orthogonal Leray projection. The dissipative term is nonpositive since $\nu \ge 0$, while claims 2-3 of The transport term is orthogonal to itself and the tame commutator estimate $\mathrm{\parallel }[{\mathrm{\Lambda }}^m,u\cdot \mathrm{\nabla }]u{\mathrm{\parallel }}_{L^2}\le C_m\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$ give $\frac{1}{2}(Y^2{)}^{\mathrm{\prime }}\le C_s{Y}^3\mathrm{.}$ With $Y_{\epsilon }=(Y^2+{\epsilon }^2{)}^{1\mathrm{/}2}$ this implies $Y_{\epsilon }^{\mathrm{\prime }}\le C_s{Y}_{\epsilon }^2$, hence $Y_{\epsilon }(t)\le Y_{\epsilon }(0)\mathrm{/}(1-C_s{Y}_{\epsilon }(0)t)$ while the denominator is positive, by differentiating $1\mathrm{/}{Y}_{\epsilon }$. Letting $\epsilon \downarrow 0$ gives $Y(t)\le Y(0)\mathrm{/}(1-C_{s}Y(0)t)\le 2B$ for $t\le T=(2C_{s}B{)}^{-1}$; if $Y(0)=0$ the same limit gives $Y=0$. This establishes the bound required in [F3] for both solutions, without substituting $2B$ for the weaker exponential bound in the existence supplier.

Subtracting the equations for $u$ and $v$ and writing $u=v+w$, we get $(u\cdot \mathrm{\nabla })u-(v\cdot \mathrm{\nabla })v=(v\cdot \mathrm{\nabla })w+(w\cdot \mathrm{\nabla })v+(w\cdot \mathrm{\nabla })w=(u\cdot \mathrm{\nabla })w+(w\cdot \mathrm{\nabla })v$, so $w$ solves $w^{\mathrm{\prime }}=\nu \mathrm{\Delta }w-P((u\cdot \mathrm{\nabla })w+(w\cdot \mathrm{\nabla })v)$; $w$ is smooth, real and divergence-free.

Pairing the equation for $w$ with $w$ and using that $P$ is self-adjoint and fixes divergence-free fields, $\frac{1}{2}\frac{d}{dt}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}^2=⟨w^{\mathrm{\prime }},w⟩=-\nu \mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2}^2-⟨(u\cdot \mathrm{\nabla })w,w⟩-⟨(w\cdot \mathrm{\nabla })v,w⟩$. By [F2] the middle term vanishes, and the pointwise bound $\mathrm{\mid }{\sum }_j{w}_j({\mathrm{\partial }}_{j}v\cdot w)\mathrm{\mid }\le \mathrm{\mid }\mathrm{\nabla }v\mathrm{\mid }\mathrm{\mid }w{\mathrm{\mid }}^2$ gives $\mid ⟨(w\cdot \mathrm{\nabla })v,w⟩\mid \le \mathrm{\parallel }\mathrm{\mid }\mathrm{\nabla }v\mathrm{\mid }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}^2$; hence $\frac{1}{2}\frac{d}{dt}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}^2\le \mathrm{\Theta }\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}^2$. Integrating this differential inequality gives $\mathrm{\parallel }w(t){\mathrm{\parallel }}_{L^2}^2\le \mathrm{\parallel }w(0){\mathrm{\parallel }}_{L^2}^2+2\mathrm{\Theta }{\int }_0^t\mathrm{\parallel }w(\sigma ){\mathrm{\parallel }}_{L^2}^{2}d\sigma$, and Gronwall [F5] yields the stability estimate of claim 2.

$\mathrm{\Theta }\le 6W_{s-1}B$ by [F3]: pointwise $\mathrm{\mid }\mathrm{\nabla }v\mathrm{\mid }\le {\sum }_{j=1}^3\mathrm{\mid }{\mathrm{\partial }}_{j}v\mathrm{\mid }$ so $\mathrm{\parallel }\mathrm{\mid }\mathrm{\nabla }v(t)\mathrm{\mid }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le {\sum }_{j=1}^3\mathrm{\parallel }{\mathrm{\partial }}_{j}v(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le 3W_{s-1}\mathrm{\parallel }v(t){\mathrm{\parallel }}_{H^s}\le 3W_{s-1}\cdot 2B$.

Uniqueness: if $u(0)=v(0)$ then $\mathrm{\parallel }w(0){\mathrm{\parallel }}_{L^2}=0$, so $\mathrm{\parallel }w(t){\mathrm{\parallel }}_{L^2}=0$ for all $t\le T$ by step 2.1; hence $w=0$ almost everywhere, and since $w$ is continuous, $w=0$ everywhere by [F6]; this is claim 1.

Continuous dependence: by [F4] applied to $w(t)$ and by the bounds for both solutions in step 1.1, $\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^{s-1}}\le \mathrm{\parallel }w(t){\mathrm{\parallel }}_{L^2}^{1\mathrm{/}s}\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^s}^{(s-1)\mathrm{/}s}\le (e^{\mathrm{\Theta }T}\mathrm{\parallel }w(0){\mathrm{\parallel }}_{L^2}{)}^{1\mathrm{/}s}(4B{)}^{(s-1)\mathrm{/}s}$. Taking the supremum over $t\in [0,T]$ and using $\mathrm{\Theta }\le 6W_{s-1}B$ gives claim 3 with the uniform constant $K:=e^{6W_{s-1}BT\mathrm{/}s}(4B{)}^{(s-1)\mathrm{/}s}$, which is finite and independent of the data and $\nu$.

Claims 1-3 are established by steps 3.2, 2.1 and 4.1. Existence and uniqueness give a well-defined solution map on ${\mathcal{D}}_{s,B}$, and step 4.1 makes its output distance tend to zero whenever its input $L^2$ distance tends to zero, proving the stated continuity.