Smooth incompressible and symmetric-system stability

Let the two initial velocities be $u_{0,i}$, $i=1,2$, and put $M_i=\mathrm{\parallel }{u}_{0,i}{\mathrm{\parallel }}_{H^s}$. By [F1], for each datum there is an existence time $T_i>0$ depending only on $d,s,M_i$, independent of its viscosity. For any of these smooth solutions, pair its projected equation after applying ${\mathrm{\Lambda }}^m$ with ${\mathrm{\Lambda }}^{m}u$, for an integer $m\ge s$. By [F4], the projections disappear; the term $\int u\cdot \mathrm{\nabla }\mathrm{\mid }{\mathrm{\Lambda }}^{m}u{\mathrm{\mid }}^2\mathrm{/}2$ vanishes by periodicity and incompressibility. The componentwise commutator estimate [F5] bounds the remaining transport pairing by $C_m\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}^2$, and viscosity contributes $-\nu \mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{H^m}^2\le 0$. Thus $\frac{1}{2}(\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}^2{)}^{\mathrm{\prime }}\le C_m\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}^2$, with constants independent of $\nu$. At $m=s$, regularizing the norm if necessary and integrating the reciprocal inequality gives $\mathrm{\parallel }{u}_i(t){\mathrm{\parallel }}_{H^s}\le 2M_i$ up to $[2C_s\max (1,M_i){]}^{-1}$ within its existence interval; if $M_i=0$, uniqueness identifies the solution with zero. Choose $T$ as the minimum of the two $T_i$ and these two reciprocal bounds. For each $m\ge s$, multiplication of the squared-norm inequality by $e^{-4C_m{M}_{i}t}$ now gives $\mathrm{\parallel }{u}_i(t){\mathrm{\parallel }}_{H^m}\le \mathrm{\parallel }{u}_{0,i}{\mathrm{\parallel }}_{H^m}{e}^{2C_m{M}_{i}t}$ on $[0,T]$. Both $T$ and these all-order bounds are independent of the two viscosities.

For possibly different viscosities, subtract the two projected equations. The identity $u^{{\nu }_1}\cdot \mathrm{\nabla }{u}^{{\nu }_1}-u^{{\nu }_2}\cdot \mathrm{\nabla }{u}^{{\nu }_2}=u^{{\nu }_1}\cdot \mathrm{\nabla }w+w\cdot \mathrm{\nabla }{u}^{{\nu }_2}$ gives $w_t+P(u^{{\nu }_1}\cdot \mathrm{\nabla }w+w\cdot \mathrm{\nabla }{u}^{{\nu }_2})={\nu }_1\mathrm{\Delta }w+({\nu }_1-{\nu }_2)\mathrm{\Delta }{u}^{{\nu }_2}$. Pair with $w$. Since $Pw=w$ and $P$ is self-adjoint, it disappears from the inner product. Periodicity and $\mathrm{div}{u}^{{\nu }_1}=0$ cancel the first transport term; the second is bounded by $L(t)\mathrm{\parallel }w{\mathrm{\parallel }}_2^2$; and ${\nu }_1\int w\cdot \mathrm{\Delta }w=-{\nu }_1\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2^2\le 0$. Thus, with $R(t)=\mathrm{\mid }{\nu }_1-{\nu }_2\mathrm{\mid }\mathrm{\parallel }\mathrm{\Delta }{u}^{{\nu }_2}(t){\mathrm{\parallel }}_2$, $\frac{1}{2}(\mathrm{\parallel }w{\mathrm{\parallel }}_2^2{)}^{\mathrm{\prime }}\le L(t)\mathrm{\parallel }w{\mathrm{\parallel }}_2^2+R(t)\mathrm{\parallel }w{\mathrm{\parallel }}_2$.

In the polytropic branch, [F2] places the two given solutions in the symmetric system with fixed linear symmetric matrices. The symmetric part of [F3] therefore applies directly and gives the displayed squared $L^2$ estimate with exactly its stated gradient integral.

In the incompressible case with equal viscosity, apply the integrated estimate [F3] to the solutions from step 1.1 and take square roots. This gives the asserted norm bound with factor $\exp ({\int }_0^{t}L(\tau )d\tau )$.

Replace $\mathrm{\parallel }w{\mathrm{\parallel }}_2$ by $y_{ϵ}=(\mathrm{\parallel }w{\mathrm{\parallel }}_2^2+{ϵ}^2{)}^{1\mathrm{/}2}$, differentiate, integrate the scalar inequality from step 1.2 with factor $\exp (-{\int }_0^{t}L)$, and let $ϵ\downarrow 0$. This yields the displayed residual estimate, including times at which $w=0$.

For any $s\ge 1$, weighted coefficient Holder gives $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^{s-1}}\le \mathrm{\parallel }w{\mathrm{\parallel }}_2^{1\mathrm{/}s}\mathrm{\parallel }w{\mathrm{\parallel }}_{H^s}^{(s-1)\mathrm{/}s}$. Using the common $H^s$ bound $M$ from step 1.1 yields the stated factor $(2M{)}^{(s-1)\mathrm{/}s}$ for either solution pair.

Apply the coefficient Holder estimate from step 3.1 to the two symmetric states. Their common $H^s$ bound gives the same $H^{s-1}$ continuity estimate.