The Bair category method for intermittent convex integration (Q6061159)

The authors consider the transport equation \(\partial_{t}\rho +v\cdot \nabla \rho =0\), \(divv=0\), posed in \([0,T]\times \mathbb{T}^{d}\), \(v\) being the divergence-free velocity field, \(\rho \) the density and \(\mathbb{T}^{d}\) the \(d\)-dimensional torus. The purpose of the paper is to prove that under appropriate hypotheses, the set of solutions to the transport equations is generic in the Baire sense. The authors recall that if \(Y_{0}\) is a set of `good objects' equipped with a metric \(d\) and \(Y=\overline{Y_{0}}\) is the completion of \(Y_{0}\) with respect to this metric, a functional \(F\) on \(Y\) is of Baire class 1, if it is pointwise the limit of functionals which are continuous with respect to \(d\)., and if \(F\) satisfies: if \(F(y)=0\) for some \(y\in Y\), then \(y\) has the desired properties, and if \(F(y)\neq 0\) then \(F\) is discontinuous at \(y\). The points of continuity of a map of Baire class 1 form a residual set. Let \(e:[0,T]\rightarrow \mathbb{R}\) be a positive and smooth function and \(M\geq 1\) a constant. A subsolution to the above problem is a smooth tuple of functions \((\rho ,u):[0,T]\times \mathbb{T} ^{d}\rightarrow \mathbb{R}\times \mathbb{R}^{d}\) such that there exists a smooth vector field \(R:[0,T]\times \mathbb{T}^{d}\rightarrow \mathbb{R}^{d}\) such that the triple \((\rho ,u,R)\) solves the transport-defect equation \(\partial_{t}\rho +u\cdot \nabla \rho =-R\), \(divu=0\), and satisfies \(\frac{1}{p}\left\Vert \rho (t)\right\Vert_{L^{p}}^{p}+\frac{1}{p^{\prime}}\left\Vert u(t)\right\Vert_{L^{p^{\prime}}}^{p^{\prime}}+M\left\Vert R(t)\right\Vert_{L^{1}}<e(t)\) for all \(t\in \lbrack 0,T]\). The space \(\mathcal{X}_{0}\) of all subsolutions is equipped with the metric \(d_{\mathcal{X}}\) defined through: \(d_{\mathcal{X}}((\rho ,u),(\rho ^{\prime},u^{\prime}))=sup_{t\in \lbrack 0,T]}(\left\Vert \rho (t)-\rho ^{\prime}(t)\right\Vert_{L^{1}}+\left\Vert u(t)-u^{\prime}(t)\right\Vert_{W^{1,\widetilde{p}}})\). The completion \(\mathcal{X}\subset C([0,T],L^{1}\times W^{1,\widetilde{p}})\) of \(\mathcal{X}_{0}\) in the topology induced by \(d_{\mathcal{X}}\) is a complete metric space and thus a Baire space and it has infinite cardinality. The authors define the functionals \(E(\rho ,u)(t)=\frac{1}{p}\left\Vert \rho (t)\right\Vert_{L^{p}}^{p}+\frac{1}{p^{\prime}} \left\Vert u(t)\right\Vert_{L^{p^{\prime}}}^{p^{\prime}}\), \(i(\rho ,u)(t)=e(t)-E(\rho ,u)(t)\), and \(I(\rho ,u)=max_{t\in \lbrack 0,T]}i(\rho ,u)(t)\). The first main result of the paper proves that the set of functions \((\rho ,u)\in \mathcal{X}\) which are strongly continuous in time \((\rho ,u)\in C([0,T];L^{p}\times L^{p^{\prime}})\), solve the transport equation in the sense of distributions, and have energy profile \(e\), that is \(E(\rho ,u)(t)=e(t)\) for all \(t\in \lbrack 0,T]\), is residual in \(\mathcal{X}\). The proof is decomposed in three steps. The functional \(I\) is first proved to be a Baire-1-map on \(\mathcal{X}\). The authors prove that \(I\) is upper semicontinuous, proceeding by contradiction and using Banach-Alaoglu theorem. In the second step, the authors prove that if \((\rho ,u)\in \mathcal{X}\) is a point of continuity of \(I\), then \(I(\rho ,u)=0\). They again proceed by contradiction. Finally, they prove that if \((\rho ,u)\in \mathcal{X}\) is such that \(I(\rho ,u)=0\), then \((\rho ,u)\) is strongly continuous in \(L^{p}\times L^{p^{\prime}}\) and a solution to the above problem. The authors here use mainly convexity and dominated convergence arguments. The second main result proves the existence of a constant \(M\geq 1 \), such that if \(p\in (1,\infty)\) and \(\widetilde{p}\geq 1\) are such that \(\frac{1}{p}+\frac{1}{\widetilde{p}}>1+\frac{1}{d}\), for any \(\delta >0\) and any smooth solution \((\rho_{0},u_{0},R_{0})\) of the continuity defect equation, there is another smooth solution \((\rho_{1},u_{1},R_{1})\) with estimates \(\frac{1}{p}\left\Vert (\rho_{1}-\rho_{0})(t)\right\Vert_{L^{p}}^{p}+\frac{1}{p^{\prime}}\left\Vert (u_{1}-u_{0})(t)\right\Vert_{L^{p^{\prime}}}^{p^{\prime}}\leq M\left\Vert R_{0}(t)\right\Vert_{L^{1}} \), \(\left\Vert (u_{1}-u_{0})(t)\right\Vert_{W^{1,\widetilde{p}}}+\left\Vert R_{1}(t)\right\Vert_{L^{1}}\leq \delta \), \(\left\Vert (\rho_{1}-\rho_{0})(t)\right\Vert_{L^{1}}+\left\Vert (u_{1}-u_{0})(t)\right\Vert_{L^{1}}\leq \delta \), \(\left\Vert (\rho_{1}-\rho_{0})(u_{1}-u_{0})(t)\right\Vert_{L^{1}}\geq \left\Vert R_{0}(t)\right\Vert_{L^{1}}-\delta \), for all \(t\in \lbrack 0,T]\). For the proof, the authors follow that from a previous paper by \textit{S. Modena} and \textit{G. Sattig} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 37, No. 5, 1075--1108 (2020; Zbl 1458.35363)]. They finally adapt this methodology to prove the genericity of distributional solutions to the Navier-Stokes equations \(\partial_{t}v+v\cdot \nabla v+\nabla p=\Delta v\), \(divv=0\), with Sobolev regularity.