Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Arch Rational Mech Anal

2017

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Given a Hilbert space H, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on H. We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of "very weak solutions" to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on R n , uniformly elliptic operators of different orders on domains, Hörmander's sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

Section: Landau Hamiltonian In 2dsupporting

confidence: 78%

“…The setting of the present paper is different (since we assume that L has a discrete spectrum), and in [54] the authors proved the existence, uniqueness and consistency for the case when L is the Landau Hamiltonian on R n (see the example in Sect. 3.1).…”

Section: U(t) + A(t)lu(t) = F (T) T ∈ [0 T ]mentioning

confidence: 99%

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

Arch Rational Mech Anal

2017

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“…Note that in[RT17b,RT18,RT19a] the wave equation for the Landau Hamiltonian with a singular magnetic field was studied.The reader is referred to [MRT19a, MRT19b, RT17c, RT18c, RT19b] for more examples of operators L and its applications.• The restricted fractional Laplacian.…”

mentioning

confidence: 99%

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Journal of Inverse and Ill-Posed Problems

Torebek

2019

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A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous leftinvariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.

“…We note that this construction goes in the opposite direction to the investigations devoted to the development of the global theory of pseudo-differential operators associated to a fixed operator, as in the papers [12,13,[25][26][27], where one is given an operator L acting in H with the system of eigenfunctions U and eigenvalues . In this case we could 'control' only one parameter, i.e.…”

Section: Definition 41 We Associate To the Pair (U ) A Linear Operatormentioning

confidence: 99%

“…In the case of the eigenfunctions having no zeros the corresponding global theory of pseudo-differential operators has been recently developed in [25]. The assumption on eigenfunctions having no zeros has been subsequently removed in [26], and some applications of such analysis to the wave equation for the Landau Hamiltonian were carried out in [27,29], as well as for general operators with discrete spectrum in [28], and for nonlinear PDE in [30]. The analysis in these papers relied on the spectral properties of a fixed operator acting in H = L 2 (M) for a smooth manifold M with or without boundary.…”

Section: Introductionmentioning

confidence: 99%

Convolution, Fourier analysis, and distributions generated by Riesz bases

2018

Monatsh Math

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