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Abstract We consider thed-dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain,$$d = 2,3$$ with homogeneous boundary conditions, and subject to external sources assumed to cause instability. The initial conditions for both fluid and magnetic equations are taken of low regularity. We then seek to uniformly stabilize such MHD system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, static, feedback controls, which are localized on an arbitrarily small interior subdomain. In additional, they will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space$$\displaystyle \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega )\times \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega ), \, 1< p < \frac{2q}{2q-1}, \, q > d,$$ of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to$$\textbf{L}^3(\Omega )$$ for$$d = 3$$ ). Showing maximal$$L^p$$ -regularity up to$$T = \infty $$ for the feedback stabilized linear system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem.
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On the Mass Transfer in the 3D Pitaevskii Model
Abstract We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in$${\mathbb {T}}^3$$ for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.
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- Award ID(s):
- 2306910
- PAR ID:
- 10507091
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Journal of Mathematical Fluid Mechanics
- Volume:
- 26
- Issue:
- 3
- ISSN:
- 1422-6928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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