Weak Solution Theory for Maxwell′s Equations in the Semistatic Limit Case

Milani,; Picard,

doi:10.1016/s0022-247x(85)71121-3

Cited by 12 publications

(6 citation statements)

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“…In Section 4 we prove under stronger assumptions the causality of the solution operator and its independence of the particular choice of the exponential weight (see Theorem 4.10 for the precise statement). Finally, we apply our results to a semistatic quasilinear variant of Maxwell's equations and thereby generalise the result of [12].…”

Section: Introductionmentioning

confidence: 61%

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Well-posedness for a general class of differential inclusions

Trostorff

2020

Journal of Differential Equations

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We consider an abstract class of differential inclusions, which covers differentialalgebraic and non-autonomous problems as well as problems with delay. Under weak assumptions on the operators involved, we prove the well-posedness of those differential inclusions in a pure Hilbert space setting. Moreover, we study the causality of the associated solution operator. The theory is illustrated by an application to a semistatic quasilinear variant of Maxwell's equations.

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Section: Introductionmentioning

confidence: 61%

“…(b) In case of Z being the inverse of a Lipschitz-continuous, c-monotone mapping, σ a positive constant number and κ = 0, the well-posedness result was already obtained in [12] employing a Galerkin approximation method.…”

Section: An Application To Semistatic Quasilinear Maxwell's Equationsmentioning

confidence: 98%

Well-posedness for a general class of differential inclusions

Trostorff

2020

Journal of Differential Equations

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“…BVPs of the type (92) have been studied by many authors, see the classical works [25,16,40] and the references therein. For the recent research works, see for instance [54,38,4,71,7,6], just name a few.…”

Section: Corollarymentioning

confidence: 99%

Variational and operator methods for Maxwell-Stokes system

Pan¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.

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“…In this paper we use frequently the results and techniques developed for Maxwell's equations and div-curl systems, in particular the div-curl-gradients inequalities, which can be found in various references including [DaL1,DaL3,Ce,GR,Sc,BW,W,KY,MMT,MP1,MP2,Pi,AuA,CD,Co,ABD,AS]. We also use frequently the results on exterior problems from [NW].…”

Section: Outlinesmentioning

confidence: 99%

Meissner states of type II superconductors

Pan

2018

J Elliptic Parabol Equ

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This paper concerns mathematical theory of Meissner states of a bulk superconductor of type II, which occupies a bounded domain Ω in R 3 and is subjected to an applied magnetic field below the critical field HS. A Meissner state is described by a solution (f, A) of a nonlinear partial differential system called Meissner system, where f is a positive function on Ω which is equal to the modulus of the order parameter, and A is the magnetic potential defined on the entire space such that the inner trace of the normal component on the domain boundary ∂Ω vanishes. Such a solution is called a Meissner solution. Various properties of the Meissner solutions are examined, including regularity, classification and asymptotic behavior for large value of the Ginzburg-Landau parameter κ. It is shown that the Meissner solution is smooth in Ω, however the regularity of the magnetic potential outside Ω can be rather poor. This observation leads to the ides of decomposition of the Meissner system into two problems, a boundary value problem in Ω and an exterior problem outside of Ω. We show that the solutions of the boundary value problem with fixed boundary data converges uniformly on Ω as κ tends to ∞, where the limit field of the magnetic potential is a solution of a nonlinear curl system. This indicates that, the magnetic potential part A of the solution (f, A) of the Meissner system, which has same tangential component of curl A on ∂Ω, converges to a solution of the curl system as κ increases to infinity, which verifies that the curl system is indeed the correct limit of the Meissner system in the case of three dimensions.Published in:

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Weak Solution Theory for Maxwell′s Equations in the Semistatic Limit Case

Cited by 12 publications

References 0 publications

Well-posedness for a general class of differential inclusions

Well-posedness for a general class of differential inclusions

Variational and operator methods for Maxwell-Stokes system

Meissner states of type II superconductors

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