Backus problem in geophysics: a resolution near the dipole in fractional Sobolev spaces
Backus problem in geophysics: a resolution near the dipole in fractional Sobolev spaces
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地球物理学中的巴科斯问题：分数索博列夫空间中偶极子附近的分辨率
DOI：
10.1007/s00030-022-00749-4
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Onodera Michiaki
中科院分区：
文献类型：
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作者：
Kan Toru;Magnanini Rolando;Onodera Michiaki
我们考虑地球物理学中的巴克斯问题。这包括当在地球表面测量相关磁场的强度时，重建地球外部的谐和电势。因此，边界条件是(严重)非线性的。万有引力的情况是很容易理解的。它存在于单极子附近的局域分辨率，即由点质量产生的势。在本文中，我们考虑地磁的情况。这包括将所谓的偶极子附近的磁场强度线性化，偶极子是模拟磁体螺线管势的调和函数。这个问题是相当困难的，因为与线性化问题相关的分解算子通常是无界的。事实上，在这个框架下，巴克斯问题的存在结果在文献中是不存在的。在这项工作中，我们在轴对称的情况下，局部地求解了Backus问题的地磁版本。在数学方面，我们证明了球体外部的调和函数的存在，其中给定的(边界)场的强度足够接近偶极子的强度，并且具有与偶极子相同的轴对称性。我们还证明了可以通过规定球体赤道圆上位势的平均值来选择唯一的解。我们将这些解表示为球谐函数的级数。泛函框架需要使用球面上的分数次Soblev Hilbert空间，并赋予其谱范数。一个关键因素是合适的子空间的代数结构。
We consider Backus’s problem in geophysics. This consists in reconstructing a harmonic potential outside the Earth when the intensity of the related field is measured on the Earth’s surface. Thus, the boundary condition is (severely) nonlinear. The gravitational case is quite understood. It consists in the local resolution near a monopole, i.e. the potential generated by a point mass. In this paper, we consider the geomagnetic case. This consists in linearizing the field’s intensity near the so-called dipole, a harmonic function which models the solenoidal potential of a magnet. The problem is quite difficult, because the resolving operator related to the linearized problem is generally unbounded. Indeed, existence results for Backus’s problem in this framework are not present in the literature. In this work, we locally solve the geomagnetic version of Backus’s problem in the axially symmetric case. In mathematical terms, we show the existence of harmonic functions in the exterior of a sphere, with given (boundary) field’s intensity sufficiently close to that of a dipole and which have the same axial symmetry of a dipole. We also show that unique solutions can be selected by prescribing the average of the potential on the equatorial circle of the sphere. We obtain those solutions as series of spherical harmonics. The functional framework entails the use of fractional Sobolev Hilbert spaces on the sphere, endowed with a spectral norm. A crucial ingredient is the algebra structure of suitable subspaces.