Global regularity for the one-dimensional stochastic Quantum-Navier-Stokes equations
A talk in the SPDEvent series by
Lorenzo Pescatore from University of L'Aquila
| Abstract: | In this talk I will present some new results concerning the analysis of the stochastically forced 1D Quantum-Navier-Stokes equations. In particular for $x\in \mathbb{T}$, the 1-dimensional flat torus, and $t\in [0,T],$
the system under studying is the following: $\text{d}\rho+\partial_x(\rho u)\text{d}t=0 $ $\text{d}(\rho u)+[ \partial_x(\rho u^2+p(\rho))]\text{d}t=[\partial_x(\mu(\rho)\partial_x u)+ \rho \partial_x \bigg( \dfrac{\partial_{xx} \sqrt{\rho}}{\sqrt{\rho}}\bigg) ]\text{d}t +\mathbb{G}(\rho,\rho u)\text{d}W. $ The unknowns $\rho>0$ and $u\in\mathbb{R}$ denote the density and the velocity of the fluid, while $p(\rho)=\rho^{\gamma} , \quad \gamma > 1 , \quad \mu(\rho)= \rho^\alpha, \quad \alpha \geq 0,$ represent the isoentropic pressure and the viscosity coefficient. The stochastic forcing term $\mathbb{G}(\rho, u)\text{d}W$ is a multiplicative noise defined on a stochastic basis with a complete right-continuous filtration $(\Omega, \mathfrak{F},(\mathfrak{F}_t)_{t \ge 0},\mathbb{P})$ together with a cylindrical $(\mathfrak{F}_t)$-Wiener process $W(t).$ The related stochastic integral is understood in the It$\hat{\text{o}}$ sense. Our analysis is focused on the existence of solutions to \eqref{stoc quantum} which are strong both in PDEs and probability sense. In particular, we prove the local well-posedness of the problem up to a maximal stopping time $\tau$ which depends on the $W^{2,\infty}$ norm of the solution $(\rho,u)$ and we derive some a priori estimates in the case of the viscosity exponent $\alpha \in [0, \frac{1}{2}],$ which allow us to extend the local strong solution to a global one by controlling the arising of vacuum states of the density. The analysis is performed for a wide class of density dependent viscosity coefficients and as a byproduct of our results we also get the global well-posedness for the deterministic case. |