Strong well-posedness of the Q-tensor model for liquid crystals: the case of arbitrary ratio of tumbling and aligning effects \(\xi\) (Q6546530)

The authors consider a bounded domain \(\Omega \subset \mathbb{R}^{n}\) with smooth boundary in dimensions \(n=2,3\) and they analyze the Beris-Edwards model which describes biaxial liquid crystals and which is written as: \( \partial _{t}u+(u\cdot \nabla )u-\nu u+\nabla p=\operatorname{div}(\tau (Q,H)+\sigma (Q,H))\), \(\operatorname{div}u=0\), \(\partial _{t}Q+(u\cdot \nabla )Q-S(\nabla u,Q)=\Gamma H\), in \( \Omega \times (0,T)\), \((u,\partial _{\nu}Q)=(0,0)\), in \(\partial \Omega \times (0,T)\), \((u,Q)\mid _{t=0}=(v_{0},Q_{0})\), in \(\Omega \). Here \( S(\nabla u,Q)=(\xi D(u)+W(u))(Q+\mathbb{I}/n)+(Q+\mathbb{I}/n)(\xi D(u)-W(u))-2\xi (Q+\mathbb{I}/n)tr(Q\nabla u)\), \(H(Q)=\lambda Q-aQ+b(Q^{2}-tr(Q^{2})\mathbb{I}/n)-ctr(Q^{2})Q\), \(\tau (Q,H)=-\lambda \nabla Q\odot \nabla Q-\xi (Q+\mathbb{I}/n)H-\xi H(Q+\mathbb{I}/n)+2\xi (Q+ \mathbb{I}/n)tr(QH)\), \(\sigma (Q,H)=QH-HQ=\lambda (Q\Delta Q-\Delta QQ)=\lambda \sigma (Q,\Delta Q)\), where \(u\) is the velocity, \(p\) the pressure, and \(Q\) the molecular orientation of the liquid crystal, \(\Gamma \) , \(\lambda \), \(\nu \) and \(a\) are positive constants, \(\xi \in \mathbb{R}\) an arbitrary constant, and \(b\) and \(c\) constants, the authors assuming \(\nu =\Gamma =\lambda =a=b=c=1\). The first main result proves the existence of a unique strong solution to this problem.\ Let \(n=2,3\), \(p>\frac{4}{4-n}\), \( \xi \in \mathbb{R}\) be arbitrary and assume that \(v_{0}=(u_{0},Q_{0})\in \{u\in B_{2,p}^{2-2/p}(\Omega )\cap L^{2\sigma}(\Omega )\), \(u=0\) on \( \partial \Omega \}\times \{Q\in B_{2,p}^{3-2/p}(\Omega )\), \(\partial _{\nu}Q=0\) on \(\partial \Omega \}\). Then there exists \(T=T(v_{0})>0\) such that there exists a unique strong solution \(v=(u,Q)\) to the above problem lying in the regularity class \(v\in H^{1,p}(0,T;L^{2\sigma}(\Omega )\times H^{1}(\Omega ;\mathbb{S}_{0}^{n}))\cap L^{p}(0,T;H^{2}(\Omega )\times H^{3}(\Omega ;\mathbb{S}_{0}^{n}))\), where \(\mathbb{S}_{0}^{n}\) is the space of complex, symmetric, and traceless \(n\times n\)-matrices. The second main result proves that if \(p>\frac{4}{4-n}\), \(\xi \in \mathbb{R}\) , the equilibrium \(v^{\ast}=0\) of the problem is stable in some space \( X_{\gamma}\), i.e., there exists \(\delta >0\) such that the strong solution \( v(t)\) to the problem with the initial value \(v_{0}\in X_{\gamma}\) and \( \left\Vert v_{0}\right\Vert _{X_{\gamma}}\leq \delta \) exists globally and converges exponentially to \(0\) in \(X_{\gamma}\) as \(t\rightarrow \infty \). For the proofs, the authors analyze the properties of the \(Q\)-tensor, they state well-posedness results for quasilinear evolution equations, they analyze the linearization of the quasilinear formulation of the problem and its maximal regularity properties, here using methods from Schur complements and \(\mathcal{J}\)-symmetry. They finally estimate the nonlinear terms of the problem.