Stochastic forcing in hydrodynamic models with non-local interactions (Q2100019)

The authors start from the Navier-Stokes system with extra terms representing the non-local interactions in a domain \((0,T)\times \mathbb{T}\) where \(\mathbb{T}\) denotes the 3D torus \(([-1,1]|\times \{-1,1\})^{3}\) and written as \(\partial _{t}\rho +\operatorname{div}(u)=0\), \(\partial _{t}(\rho u)+\operatorname{div}(\rho u\otimes u)+\nabla p(\rho )-\operatorname{div}S(Du)=-\rho \int_{\mathbb{T}}\nabla _{x}K(x-y)\rho (y)dy+\rho \int_{\mathbb{T}}\psi (x-y)\rho (y)(u(y)-u(x))dy\), where \(u\) is the velocity, \(\rho \) is the density and \(p\) the pressure. They assume that \(p(\rho )\) is a given pressure satisfying \(p\in C([0,\infty ))\cap C^{2}(0,\infty )\), \(p(0)=0\), \(p^{\prime }(\rho )>0\) whenever \(\rho >0\) and \(\lim_{\rightarrow \infty }p^{\prime }(\rho )/\rho ^{\gamma -1}=p_{\infty }>0\), for some \(\gamma >3/2\). They introduce the potential \(P\) as \(P(\rho )=\rho \int_{1}^{\rho }\frac{p(s)}{s^{2}}ds\). In the previous Navier-Stokes equations, \(S\) denotes a dissipative term defined as \(S(Du)=\mu (Du)+(\lambda +\mu )\operatorname{div}uI\).\ The kernel \(K=K(x)\) includes the repulsive-attractive interaction force among individuals and satisfies \(K\in C^{2}(\mathbb{T})\), \(K(-x)=K(x)\). \(\psi =\psi (x)\) gives the local averaging measuring the consensus in orientation and satisfies \(\psi \geq 0\) and \(\psi (-x)=\psi (x)\). The authors add a stochastic forcing term and finally consider the stochastic system \(d\rho +\operatorname{div}(u)dt=0\), \(d\rho u+\operatorname{div}(u\otimes u)dt+\nabla p(\rho )dt-\operatorname{div}S(Du)dt=-\rho \int_{\mathbb{T}}\nabla _{x}K(x-y)\rho (y)dydt+\rho \int_{\mathbb{T}}\psi (x-y)\rho (y)(u(y)-u(x))dydt+\mathbb{G}(\rho ,\rho u)dW\) in \((0,T)\times \mathbb{T}\). The forcing term \(\mathbb{G}(\rho ,\rho u)\) is defined as \(\mathbb{G}(\rho ,u)dW=\sum_{k=1}^{\infty }G_{k}(x,\rho ,\rho u)dW_{k}\), where \( W=(W_{k})_{k\in \mathbb{N}}\) is a cylindrical (\(\mathfrak{F}_{t}\))-Wiener process in a separable Hilbert space \(\mathfrak{U}\) given by a formal sum \( W(t)=\sum_{k=1}^{\infty }e_{k}W_{k}(t)\), \((e_{k})_{k\in \mathbb{N}}\) being a complete orthonormal system in \(\mathfrak{U}\), \((W_{k})_{k\in \mathbb{N}}\) a sequence of mutually independent real-valued (\(\mathfrak{F}_{t}\))-Wiener processes, and \(\mathbb{G}(\rho ,\rho u):\mathfrak{U}\rightarrow L^{1}( \mathbb{T})\). The coefficients \(G_{k}\) are continuously differentiable and satisfy \(\left\vert G_{k}(x,\rho ,q)\right\vert \leq g_{k}(\rho +\left\vert q\right\vert )\), \(\left\vert \nabla _{\rho ,q}Gk(x,\rho ,q)\right\vert \leq g_{k}\), for a sequence.\(\{g_{k}\}_{k=1}^{\infty }\subset \lbrack 0,\infty )\) such that \(\sum_{k=1}^{\infty }g_{k}^{2}<\infty \). The main result proves the existence of a dissipative martingale solution to the stochastic problem if \(\int_{L^{1}\times L^{1}}\left\vert \int_{\mathbb{T}}\left[ \frac{1}{2} \frac{q^{2}}{\rho }+P(\rho )\right] dx\right\vert ^{r}d\Lambda \leq c<\infty \) for a Borel probability measure \(\Lambda \) defined on the space \(L^{1}( \mathbb{T})\times (L^{1}(\mathbb{T}))^{3}\) such that \(\Lambda \{\rho \geq 0\}=1\), \(\Lambda \{0<\underline{\rho }\leq \int_{\mathbb{T}}\rho dx\leq \overline{\rho }<\infty \}=1\) for some \(\underline{\rho }\) and \(\overline{ \rho }\) and for some \(r\geq 4\). For the proof, the authors build an approximate system, adding the artificial viscosity term \(\varepsilon \Delta \rho \) in the continuity equation and the term \(\varepsilon \Delta (\rho u)\) in the momentum equation to keep the energy inequality valid. They also consider the modified pressure \(p_{\delta }(\rho )=p(\rho )+\delta (\rho ^{2}+\rho ^{6})\) and they use the method of the Galerkin approximations.