Local Wellposedness of Nonlinear Maxwell Equations

Spitz, Martin

doi:10.5445/ir/1000078030

Cited by 4 publications

(32 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We can also treat more general domains with a uniform C m+2 -boundary satisfying some extra properties. See [18,Chapter 5] for details. (3) In [19] we also show the finite speed of propagation of solutions of (1.5).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The proof again follows standard ideas, see [18,Lemma 2.21] for details. As indicated in the introduction, we will reduce (1.5) via localization to a halfspace problem with variable coefficients below.…”

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 95%

“…The proof relies on a term by term analysis of the derivatives of the products combined with an appropriate application of the Sobolev embedding theorem and Hölder's inequality, see [18,Lemma 2.22].…”

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 99%

“…Choosing the covering of ∂G fine enough, there exist numbers τ > 0 and k(i) ∈ {1, 2, 3} such that |µ i 3,k(i) | ≥ τ on R 3 + for all i ∈ N, see Lemma 5.1 in [18]. Here we denote by µ i 3,j the coefficients for which A i 3 = 3 j=1 A co j µ i 3,j .…”

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 99%

“…The results in Sections 3 and 4 are also true on the full space and follow by the same strategy, which is much easier in that case. See [18,Theorem 5.3] for a more detailed discussion. (2) The proof that Theorem 1.1 indeed follows from the corresponding result for problem (2.6) is very technical but follows from standard ideas, see [18, Theorem 5.6] once again.…”

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 99%

See 4 more Smart Citations

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Spitz

2019

Journal of Differential Equations

Self Cite

141

View full text Add to dashboard Cite

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)

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 95%

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 99%

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 99%

Section: Compatibility Conditions Function Spaces and Localizationmentioning

confidence: 99%

See 3 more Smart Citations

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Spitz

2019

Journal of Differential Equations

Self Cite

141

View full text Add to dashboard Cite

show abstract

On quasilinear Maxwell equations in two dimensions

Schippa

Schnaubelt

2022

Pure Appl. Analysis

View full text Add to dashboard Cite

New sharp Strichartz estimates for the Maxwell system in two dimensions with rough permittivity and non-trivial charges are proved. We use the FBI transform to carry out the analysis in phase space. For this purpose, the Maxwell equations are conjugated to a system of half-wave equations with rough coefficients, for which Strichartz estimates are similarly derived as in previous work by Tataru on scalar wave equations with rough coefficients. We use the estimates to improve the local well-posedness theory for quasilinear Maxwell equations in two dimensions.

show abstract

Local wellposedness of quasilinear Maxwell equations with conservative interface conditions

Schnaubelt¹,

Spitz²

2018

View full text Add to dashboard Cite

Local Wellposedness of Nonlinear Maxwell Equations

Cited by 4 publications

References 0 publications

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

On quasilinear Maxwell equations in two dimensions

Local wellposedness of quasilinear Maxwell equations with conservative interface conditions

Contact Info

Product

Resources

About