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Abstract Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in$$\mathbb R^3$$ . The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the$$x_1$$ -direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting$$H^3({\mathbb {R}}^3)$$ . In addition, explicit decay rates in$$H^2({\mathbb {R}}^3)$$ are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in$$\mathbb R^3$$ remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid.
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Higher-order error estimates for physics-informed neural networks approximating the primitive equations
Abstract Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates area priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the$$H^s$$ norm during the training.
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- PAR ID:
- 10433266
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Partial Differential Equations and Applications
- Volume:
- 4
- Issue:
- 4
- ISSN:
- 2662-2963
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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