Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum (Q6638195)

The authors prove the existence, uniqueness, and continuous dependence on the data of local-in-time strong solutions to the Navier-Stokes equations describing non-Newtonian compressible fluids in the \(d\)-dimensional torus \(\Omega =\mathbb{T}^{d}\): \(\partial_{t}\rho +\operatorname{div}(\rho u)=0\), \(\partial_{t}(\rho u)+\operatorname{div}(\rho u\otimes u)-\operatorname{div}Su+\nabla p=\rho f\), in \(\Omega \times (0,T)\), where \(T\in (0,\infty ]\), \(u:\Omega \times (0,T)\rightarrow \mathbb{R}^{d}\) is the velocity field of the fluid, \(\rho: \Omega \times (0,T)\rightarrow \mathbb{R}\) its density, \(p=p(\rho)\) its pressure, \(p:[0,\infty)\rightarrow \mathbb{R}^{+}\) being a \(C^{2}\)-function of the density \(\rho\), \(Su=(S_{ij}u)_{1\leq i,j\leq d}\) the stress tensor and \(f\in C([0,T];L^{2}(\Omega))\cap L^{2}(0,T;L^{q}(\Omega))\) the external body force satisfying \(f_{t}\in L^{2}(0,T;H^{-1}(\Omega))\). The initial conditions: \(\rho (0)=\rho_{0}\in W^{1,q}(\Omega)\) and \(u(0)=u_{0}\in H^{2}(\Omega)\) are imposed in \(\Omega\), the initial density \(\rho_{0}\geq 0\) being allowed to vanish on some subset and \(u_{0}\) satisfying \(\int_{\Omega}u_{0}dx=0\). The stress tensor is given through the constitutive law: \(Su=2\mu (\left\vert D(u)\right\vert^{2})D(u)+\lambda (\operatorname{div}u)\operatorname{div}uI\), where \(D(u)\) is the symmetric part of the gradient, \(I\) denotes the identity in \(\mathbb{R}^{d}\), \(\mu \in C^{1}([0,\infty),\mathbb{ R})\) and \(\lambda \in C^{1}(\mathbb{R},\mathbb{R})\) satisfying ellipticity conditions.\ A strong solution \((\rho, u)\) to the above problem in \((0,T^{\ast})\times \Omega\) is a weak solution that satisfies the above system almost everywhere in \((0,T^{\ast})\times \Omega\) and the properties: \(\rho \in C([0,T^{\ast}];W^{1,q_{0}}(\Omega))\), \(u\in C([0,T^{\ast}];H^{2}(\Omega))\cap L^{2}(0,T^{\ast};W^{2,q_{0}}(\Omega))\), \(\rho_{t}\in C([0,T^{\ast}];L^{q_{0}}(\Omega))\), \(u_{t}\in L^{2}(0,T^{\ast};H^{1}(\Omega))\) and \(\sqrt{\rho}u_{t}\in L^{\infty}(0,T^{\ast};L^{2}(\Omega))\). \N\NThe main result proves that if \(d=3\), \(q\in (3,\infty)\), \(q_{0}=min\{6,q\}\), \(S\) satisfies \(-\operatorname{div}Su_{0}+\nabla p(\rho_{0})=\rho_{0}^{1/2}g\), for some \(g\in L^{2}(\Omega)\), and for \(p\in (1,\infty)\), the non-linear elliptic problem \(-\operatorname{div}Su=f\) has for each \(f\in L_{0}^{p}(\Omega)^{d}\) a unique solution \(u\in W^{2,p}(\Omega)^{d}\) with \(\int_{\Omega}udx=0\), and there exists a positive constant \(C\) such that \(\left\Vert u\right\Vert_{W^{2,p}(\Omega)^{d}}\leq C\left\Vert f\right\Vert_{L^{p}(\Omega)}\), there exist a time \(T^{\ast}\in (0,T]\) and a unique strong solution \((\rho, u)\) to the above non-linear problem. For the proof, the authors introduce for \(\delta >0\) \(\rho_{0}^{\delta}=\rho_{0}+\delta\) and \(u_{0}^{\delta}\) the solution to the non-linear elliptic problem: \(-\operatorname{div}Su_{0}^{\delta}=(\rho_{0}^{\delta})^{1/2}g-\nabla p_{0}^{\delta}\), where \(p_{0}^{\delta}=p(\rho_{0}^{\delta})\). They construct iteratively approximate solutions to the Navier-Stokes system, starting with \(u^{0}=0\) and for \(k\geq 1\), taking \((\rho^{k},u^{k})\) as the unique smooth solution to the following quasi-linear problem: \(\rho_{t}^{k}+u^{k-1}\cdot \nabla \rho^{k}+\rho^{k}\operatorname{div}u^{k-1}=0\), \(\rho^{k}u_{t}^{k}+\rho^{k}u^{k-1}\cdot \nabla u^{k}-\operatorname{div}Su^{k}+\nabla p^{k}=\rho^{k}f\), \((\rho^{k},u^{k})\mid_{t=0}=(\rho_{0}^{\delta},u_{0}^{\delta})\). They prove a priori estimates in higher norms on the approximate solutions and they let \(k\rightarrow \infty\), then \(\delta \rightarrow 0\), using Banach-Alaoglu theorem, compact embeddings, and Aubin-Lions lemma, among different tools. Using the momentum and continuity equations in the above system, they prove the uniqueness of the strong solution and a continuous dependence on the data. The paper ends with a blow-up result. If \(T^{\ast}\) is the maximal existence time of the strong solution and \(T^{\ast}<T\), then \(limsup_{t\rightarrow T^{\ast}}(\left\Vert \rho \right\Vert W^{1,q_{0}}+\left\Vert u(t)\right\Vert_{H^{1}})=\infty\). The authors essentially use Grönwall's inequality.