Well posedness for quasi-linear hyperbolic mixed problems with characteristic boundary

Ohkubo, Toshio

doi:10.14492/hokmj/1381517781

Cited by 13 publications

(6 citation statements)

References 8 publications

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“…In particular, no general theory for the corresponding linear problems is available and even a loss of regularity there is possible, see [26]. In [29] an additional structural assumption is proposed in order to prevent this loss of regularity and a quasilinear result is derived from it in [30]. However, the Maxwell system does not fulfill this structural assumption.…”

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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“…The second problem is the question of whether the solution of the problem in which Γ consists only of Γ o has full regularity or not (cf. [7,10,14]).…”

Section: Introductionmentioning

confidence: 99%

The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition

Yanagisawa

Matsumura

1991

Commun.Math. Phys.

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“…Making use of this fact, and the data (/,F) fixed in the space H l (K) x ^V. (0,r;R^), we first prove that the sequence of solutions to (11) remains bounded in A^([0,T];R£) as r} tends to 0. Next, by a sort of weak compactness method we find a solution to (1) …”

Section: «"(*?)■ «-(*!)■mentioning

confidence: 99%

“…Some characteristic equations enjoy the same property thanks to their special structure ( [7], [10], [11]). This is not always true of all the character istic problems, as illustrated by several equations including the one of ideal magneto-hydrodynamics ( [10], [13], [26]).…”

Section: Introductionmentioning

confidence: 99%

Regularity of Solutions of Initial Boundary Value Problems for Symmetric Hyperbolic Systems With Boundary Characteristic of Constant Multiplicity

Yamamoto¹

1998

Series on Advances in Mathematics for Applied Sciences

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We discuss the higher order regularity of solutions to initial boundary value prob lems for linear symmetric hyperbolic systems with boundary characteristic of con stant multiplicity. By means of the standard energy method it is shown that the solutions and their derivatives with respect to the time variable lie in certain weighted Sobolev spaces under a suitable compatibility condition between the data.

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Well posedness for quasi-linear hyperbolic mixed problems with characteristic boundary

Cited by 13 publications

References 8 publications

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition

Regularity of Solutions of Initial Boundary Value Problems for Symmetric Hyperbolic Systems With Boundary Characteristic of Constant Multiplicity

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