Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary

Ohkubo, Toshio

doi:10.14492/hokmj/1381758116

Cited by 19 publications

(26 citation statements)

References 11 publications

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“…Even if A(ν) is singular (the characteristic case), one can recover normal from tangential regularity under certain structural conditions on the problem, see e.g. [12,13]. However, these conditions fail for the Maxwell system (1.5) (which is characteristic as the boundary matrix 3 j=1 A co j ν j is singular) with perfectly conducting boundary conditions, cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Spitz

2019

Journal of Differential Equations

141

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In this article we develop the local wellposedness theory for quasilinear Maxwell equations in H m for all m ≥ 3 on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in H 3 is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in H m with m ≥ 3 of the corresponding initial boundary value problem if the material laws and the data are accordingly regular and compatible. We further characterize finite time blowup in terms of the Lipschitz norm and we show that the solutions depend continuously on their data. Finally, we establish the finite propagation speed of the solutions.2010 Mathematics Subject Classification. 35L50, 35L60, 35Q61. Key words and phrases. nonlinear Maxwell equations, perfectly conducting boundary conditions, quasilinear initial boundary value problem, hyperbolic system, local wellposedness, blow-up criterion, continuous dependance. 1 2 MARTIN SPITZThe macroscopic Maxwell equations in a domain G readfor all t ≥ t 0 . In (1.1) we consider the Maxwell system with the boundary conditions of a perfect conductor, where ν denotes the outer normal unit vector of G. System (1.1) has to be complemented by constitutive relations between the electric fields and the magnetic fields, the so called material laws. We choose E and H as state variables and express D and B in terms of E and H. The actual form of these material laws is a question of modelling and different kinds have been considered in the literature. The so called retarded material laws assume that the fields D and B depend also on the past of E and H, see [2] and [33] for instance. In dynamical material laws the material response is modelled by additional evolution equations for the polarization or magnetization, see e.g.[1], [11], [16], or [17]. In this work we concentrate on the instantaneous material laws, see [6] and [14]. Here the fields D and B are given as local functions of E and H, i.e., we assume that there are functions θ 1 , θ 2 : G × R 6 → R 3 such that D(t, x) = θ 1 (x, E(t, x), H(t, x)) and B(t, x) = θ 2 (x, E(t, x), H(t, x)). The most prominent example is the so called Kerr nonlinearity, wherewith ϑ : G → R 3×3 and the vacuum permittivity and permeability set equal to 1 for convenience. We further assume that the current density decomposes as J = J 0 + σ 1 (E, H)E, where J 0 is an external current density and σ 1 denotes the conductivity. If we insert these material laws into (1.1) and formally differentiate, we obtainfor the evolutionary part of (1.1). The resulting equation is a first order quasilinear hyperbolic system. In order to write (1.1) in the standard form of first order systems, we introduce the matrices J 1 =   0 0 0 0 0 −1 0 1 0   , J 2 =   0 0 1 0 0 0 −1 0 0   , J 3 =   0 −1 0 1 0 0 0 0 0   and A co j = 0 −J j J j 0 (1.3)

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Spitz

2019

Journal of Differential Equations

141

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“…(these definitions are based on those used in [12]). The spaces X N2 ( $ ), with the norms # ) # N2 , are Banach spaces.…”

Section: Construction Of the Solutionmentioning

confidence: 99%

“…Choose an arbitrary ¹'¹(e). Regarding problems (P $ ) as mixed hyperbolic boundary value problems, all the conditions of Theorems 1 and 2 in paper [12] are satisfied by these problems (the assumption that e,0 for t)0 ensures that the compatibility conditions at t"0 in [12] are satisfied). Theorem 2 of [12] now shows that problems (P $ ) have unique solutions (…”

Section: Construction Of the Solutionmentioning

confidence: 99%

“…The construction of the solution (J, )3X N ( ) in the proof of Theorem 2.1 used Ohkubo's results in [12]. However, the exponential (in time) weight in the estimates in [12] prevents us from proving (3.1) using these results. To obtain (3.1) we turn to a semi-group construction of the solution in the time interval [¹, R) (to simplify the notation we now write ¹"¹(e)).…”

Section: Behaviour Of Solutions As Tprmentioning

confidence: 99%

“…Problems (P $ ) are solved by using the Laplace transform to transform the hyperbolic problems into elliptic problems. In this paper we will also construct (J, ) from solutions of (P $ ), but we will obtain the solutions of these problems directly using the theory of uniformly characteristic hyperbolic systems, see [12]. This approach yields better regularity results for (J, ) than those obtained in the papers [2,5,13] (we say more about this in Remark 2.4 below), and also enables us to discuss the asymptotic behaviour of (J, ) as time tPR, which is not considered in the papers cited above.…”

Section: Introductionmentioning

confidence: 99%

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1999

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On the L 2-Well Posedness of an Initial Boundary Value Problem for the Linear Elasticity in Two and Three Space Dimensions

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Serre²

2008

Hyperbolic Problems: Theory, Numerics, Applications

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Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary

Cited by 19 publications

References 11 publications

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

The well-posedness of the electric field integral equation for transient scattering from a perfectly conducting body

On the L 2-Well Posedness of an Initial Boundary Value Problem for the Linear Elasticity in Two and Three Space Dimensions

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