Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour (Q6061973)

The authors consider the problem written in Eulerian coordinates as: \(\rho _{t}+\operatorname{div}(v\rho )=0\), \(\rho v_{t}+\rho v\cdot \nabla v-\Delta v=-\rho \nabla K\rho \), posed in \([0,\infty )\times \mathbb{T}^{d}\), where \(\mathbb{T }^{d}=\mathbb{R}^{d}/(2\pi \mathbb{Z)}^{d}\) is the \(d\)-dimensional torus, \( \rho \) the density, \(v\) the gas flow, \(K\) the operator \((-\Delta )^{-1}\) restricted to the periodic functions with zero mean, namely \(\Psi =K(\rho -\{\rho \})\) satisfies \(-\Delta \Psi =\rho -\{\rho \})\), where \(\{\rho \}=\int_{\mathbb{T}^{d}}\rho dx\). The initial conditions \(\rho \mid _{t=0}=\rho _{0}\), \(v\mid _{t=0}=v_{0}\) are added. The main result of the paper proves that for \(d\geq 3\) and \(p\in (\min(d/2,2),d)\), there exists \( \epsilon >0\) such that for every \(\rho _{0}-1\in B_{p,1}^{d/p}(\mathbb{T} ^{d})\) and \(v_{0}\in B_{p,1}^{d/p-1}(\mathbb{T}^{d})\) satisfying \(\int_{ \mathbb{T}^{d}}\rho _{0}v_{0}=0\) and \(\left\Vert v_{0}\right\Vert _{B_{p,1}^{d/p-1}(\mathbb{T}^{d})}+\left\Vert \rho _{0}-1\right\Vert _{B_{p,1}^{d/p}(\mathbb{T}^{d})}\leq \epsilon \), there exists a unique global-in-time solution \((\rho ,v)\) to the above problem such that \(\rho -1\in C_{b}([0,\infty );B_{p,1}^{d/p}(\mathbb{T}^{d}))\), \(v_{t},\nabla ^{2}v\in L^{1}((0,\infty );B_{p,1}^{d/p-1}(\mathbb{T}^{d}))\), with \( \left\Vert \rho -1\right\Vert _{L^{\infty }((0,\infty );B_{p,1}^{d/p}( \mathbb{T}^{d}))}+\left\Vert v_{t}\right\Vert _{L^{1}((0,\infty );B_{p,1}^{d/p-1}(\mathbb{T}^{d}))}+\left\Vert v\right\Vert _{L^{1}((0,\infty );B_{p,1}^{d/p+1}(\mathbb{T}^{d}))}\leq C\epsilon \), where \(C=C(d,p)>1\) is a constant. Here \(B_{p,q}^{s}(\mathbb{T}^{d})\) are Besov spaces. For the proof, the authors write the above problem in Lagrangian coordinates as: \(\eta _{t}+\eta \operatorname{div}_{u}u=0\), \(\eta u_{t}-\Delta _{u}u=-\eta \nabla u(-\Delta u)^{-1}a\), in \([0,\infty )\times \mathbb{T}^{d}\), where \( a=\eta -1\), with initial conditions \(\eta \mid _{t=0}=\rho _{0}\), \(u\mid _{t=0}=v_{0}\). They also introduce the compressible Stokes system: \( a_{t}+\operatorname{div}u=h\), \(u_{t}-v \Delta u+\nabla (Ka)=g\), in \([0,\infty )\times \mathbb{T}^{d}\), with the initial conditions \(a\mid _{t=0}=a_{0}\), \(u\mid _{t=0}=u_{0}\) in \(\mathbb{T}^{d}\), which is a linearization of the preceding problem. Assuming \(s\in \mathbb{R}\), \(p\in \lbrack 1,\infty ]\), \(a_{0}\in B_{p,1}^{s+1}\), \(u_{0}\in B_{p,1}^{s}\), \(g\in L^{1}B_{p,1}^{s}\), \(h\in L^{1}B_{p,1}^{s+1}\), they prove that this linearized system has a unique solution which belongs to appropriate Besov spaces, together with \(a_{t}\) and \(u_{t}\). They then prove an existence for the problem written in Lagrangian coordinates, using Banach contraction theorem, among other tools. They conclude proving the equivalence of the Eulerian and Lagrangian formulations.