Regularity of weak solution to Maxwell's equations and applications to microwave heating (original) (raw)
Related papers
On Maxwell's Equations with a Temperature Effect, II
Communications in Mathematical Physics, 1998
In this paper we study Maxwell's equations with a thermal effect. This system models an induction heating process where the electric conductivity σ strongly depends on the temperature u. We focus on a special one-dimensional case where the electromagnetic wave is assumed to be parallel to the y-axis. It is shown that the resulting hyperbolic-parabolic system has a global smooth solution if the electrical conductivity σ(u) grows like u q with 0 ≤ q < 8 + 4 √ 3. A fundamental element in this paper is the establishment of a maximum principle for wave equations with damping. This maximum principle provides an a priori bound for the first derivative with respect to both x and t of the solution without the imposition of any differentiability assumptions nor bounds on the coefficient of the damping term. The use of a nonlinear multiplier then permits (via a bootstrap procedure) the estimation of successively higher L p-norms of the temperature function u.
On Maxwell's Equations with a Temperature Effect, II
Communications in Mathematical Physics, 1998
In this paper we study Maxwell's equations with a thermal effect. This system models an induction heating process where the electric conductivity σ strongly depends on the temperature u. We focus on a special one-dimensional case where the electromagnetic wave is assumed to be parallel to the y-axis. It is shown that the resulting hyperbolic-parabolic system has a global smooth solution if the electrical conductivity σ(u) grows like u q with 0 ≤ q < 8 + 4 √ 3. A fundamental element in this paper is the establishment of a maximum principle for wave equations with damping. This maximum principle provides an a priori bound for the first derivative with respect to both x and t of the solution without the imposition of any differentiability assumptions nor bounds on the coefficient of the damping term. The use of a nonlinear multiplier then permits (via a bootstrap procedure) the estimation of successively higher L p-norms of the temperature function u.
Existence and regularity of weak solutions for a thermoelectric model
Nonlinearity, 2019
This paper concerns a time-independent thermoelectric model with two different boundary conditions. The model is a nonlinear coupled system of the Maxwell equations and an elliptic equation. By analyzing carefully the nonlinear structure of the equations, and with the help of the De Giorgi-Nash estimate for elliptic equations, we obtain existence of weak solutions on Lipschitz domains for general boundary data. Using Campanato's method, we establish regularity results of the weak solutions.
Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer
We study the behavior of the electromagnetic field in a biological cell modelized by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain our asymptotic conditions in a simple way. This paper extends to time-harmonic Maxwell equations the previous works presented in . Asymptotic transmission conditions at any order are given in Appendix 1. Conditions de transmission approchées pour leś equations de Maxwell en régime harmonique dans un milieuà couche mince Résumé : Nousétudions le comportement asymptotique du champélectromagnétique dans une cellule biologique plongée dans un milieu ambiant. La cellule est composée d'un cytoplasme entouré d'une fine membrane. Nous obtenons des conditions de transmission sur le bord du cytoplasmeéquivalentesà la couche mince. Notre approche est essentiellement géométrique et basée sur un changement de variables adéquat dans la couche mince. Quelques notions de calcul différentiel sont rappelées afin d'obtenir directement notre développement asymptotique. Par ailleurs des estimations d'erreur sont démontrées. En appendice, nous présentons le développement asymptotiqueà tout ordre. 2 Using the notations of the electrical engineeering community, q = ω 2`ǫ − i σ ω´, where ω is the frequency, ǫ the permittivity and σ the conductivity of the domain [3]. INRIA inria-00347971, version 3 -14 Sep 2009 Péron& Poignard where
Electromagnetic waves in an inhomogeneous medium
Journal of Mathematical Analysis and Applications, 1992
In this paper we consider the electromagnetic wave problem in an inhomogeneous medium. We tirst prove uniqueness of the solution using Rellich and Cauchy-Kowalewska theorems. Then we explicitly compute the Dirichlet-Neumann operator on the sphere, we reduce the equations to a problem on a truncated domain, and we give a variational formulation. This formulation reads as a compact perturbation of a coercive operator, which leads to the existence of the solution' according to Fredholm's alternative. r,
IEEE Transactions on Plasma Science, 1999
Microwave heating processes involve electromagnetic and thermal effects coupled together through the local temperature dependence of the material dielectric properties. This paper presents a one-dimensional model for the coupled electromagnetic-thermal process and demonstrates its solutions for typical problems. The local temperature dependence of the lossy dielectric medium is taken into account in two different time scales. One is the heat-generation time scale due the microwave radiation, and the other is the temperature diffusion time scale. The two timescale approach minimizes the computation time and provides an efficient simulation tool for the analysis of various phenomena. The two-scale model presented in this paper is benchmarked by a comparison of its numerical results with other models published in the literature. Several examples of microwave heating processes in various materials are simulated. Effects of heatwave propagation in matter are predicted by the model. The results show the temporal and spatial evolution of the temperature and power-dissipation profiles. Variations in the (microwave) impedance profile in the medium due to the heating are computed. A further development of this model, including more complicated geometries and various loss mechanisms, may yield useful numerical tools for the synthesis and design of microwave heaters in which the heated material acts as a nonlinear load in the microwave circuit.
On a transmission problem for the time–harmonic Maxwell equations
Riassunto: In questo lavoro si considera il problema di trasmissione per le equazioni di Maxwell armoniche nel tempo, per un diffusore a infiniti strati omogenei, costituiti da materiali diversi. Si dimostra l'esistenza e l'unicità della soluzione. Inoltre si costruisce una rappresentazione integrale del campo esterno totale e si esamina il comportamento asintotico dell'onda diffusa nella regione di radiazione.
The conductive problem for Maxwell's equations at low frequencies
Applied Mathematics Letters, 1997
We study the scattering problem in the case where a low frequency plane electromagnetic wave is incident upon a body covered by a thin layer of very high conductivity. We develop convergent series solutions in powers of the wavenumber for the electric field, and the scattering amplitude. Moreover, we evaluate the leading term of the scattering amplitude and of the scattering croes-section.
Mathematical Models and Methods in Applied Sciences, 2000
We analyze the solution of the time-harmonic Maxwell equations with vanishing electric permittivity in bounded domains and subject to absorbing boundary conditions. The problem arises naturally in magnetotellurics when considering the propagation of electromagnetic waves within the earth's interior. Existence and uniqueness are shown under the assumption that the source functions are square integrable. In this case, the electric and magnetic fields belong to H(curl; Ω). If, in addition, the divergences of the source functions are square integrable and the coefficients are Lipschitz-continuous, a stronger regularity result is obtained. A decomposition of the space of square integrable vector functions and a new compact imbedding result are exploited. *
On the Uniqueness of a Solution to Anisotropic Maxwell’s Equations
Topics in Operator Theory, 2010
In the present paper we consider Maxwell's equations in an anisotropic media, when the dielectric permittivity ε and the magnetic permeability µ are 3 × 3 matrices. We formulate relevant boundary value problems, investigate a fundamental solution and find a Silver-Müller type radiation condition at infinity which ensures the uniqueness of solutions when permittivity and permeability matrices are real valued, symmetric, positive definite and proportional ε = κµ, κ > 0.