Quantitative analysis of passive intermodulation and surface roughness (Q6590510)

The present paper deals with the wellposedness of the Maxwell-Fourier system in a bounded Lipschitz domain \(\Omega\) of \(\mathbb{R}^3 \), under the electrical conductivity \(\sigma (u) = \sigma_0 /(1+\alpha_{NL}(u)u)\), where \(\sigma_0\) is a positive constant and \(u\alpha_{NL}(u)\in C^1(\mathbb{R}) \) is such that \(\alpha_1 \leq 1+u\alpha_{NL}(u) \leq\alpha_2\) and \(| \frac{d}{du} (u\alpha_{NL}(u))|\leq M (1+u\alpha_{NL}(u))^2 \), for some constants \(\alpha_1 , \alpha_2, M >0\). The unknown functions are the magnetic and electric fields, \( \mathbf{H}\) and \(\mathbf{E} \), respectively, and the temperature \(u \).\N\NFirst, the authors study the existence of weak solution \((\mathbf{E}, u) \in \mathbf{H}(\mathrm{curl}, \Omega)\times L^q(0,T; W^{1,q}(\Omega))\), for some \(1<q<5/4\), to the high-frequency heating problem\N\begin{align*}\N\nabla\times(\eta\nabla \times\mathbf{E})+ \xi\mathbf{E} =\mathbf{0};\\\N\partial_t u -\nabla\cdot(k \nabla u)= \sigma |\mathbf{E}|^2,\N\end{align*}\Nin \(Q_T=\Omega\times ]0,T[\), accomplished by the boundary conditions \( \mathbf{n}\times\mathbf{E}=\mathbf{m}\) on \(\partial\Omega\), and \(k \nabla u \cdot \mathbf{n} =h\) on \(\partial\Omega\times [0,T]\). Here, the thermal conductivity \(k=k(x,t,u)\) is a real valued function, while the coefficients \(\eta=\eta(x,u)\) are \(\xi=\xi(x,u)\) are complex valued functions that include dissipative effects. The existence result is stated, under suitable assumptions on the initial data \(u_0\) and the boundary data \(h\) and \( \mathbf{m}\), and some Lipschitz continuity and boundedness assumptions on the remaining parameters, for \(\Omega\) assumed to be a polyhedron, which is simply connected and has a connected boundary \(\partial\Omega\). The uniqueness of solutions and continuous dependence on data are proved by the energy method, under additional regularity assumptions \( \nabla u, \mathbf{E},\nabla\times\mathbf{E}\in L^{2/s}(0,T; L^{3/(1-s)}(\Omega))\), for \(0<s<\delta\), with \(0<\delta<1\) being characterized in terms of boundary trace [\textit{C. Amrouche} et al., Math. Methods Appl. Sci. 21, No. 9, 823--864 (1998, Zbl 0914.35094)].\N\NSecondly, the authors study the existence of weak solution \((\mathbf{E}, \mathbf{H}, u) \in [ L^\infty (0,T; L^2(\Omega))]^2 \times L^q(0,T; W^{1,q}(\Omega))\), for some \(1<q<5/4\), to the main problem\N\begin{align*}\N\partial_t(\varepsilon\mathbf{E})+ \sigma\mathbf{E} = \nabla \times\mathbf{H};\\\N\partial_t \mathbf{H} +\nabla\times\mathbf{E}=\mathbf{0};\\\N\partial_t u -\nabla\cdot(k \nabla u)= \sigma |\mathbf{E}|^2,\N\end{align*}\Nwhere \(\varepsilon =\varepsilon(x,t)\) is bounded, measurable, \(C^1\) in time, and satisfies \(\partial_t\varepsilon\) is bounded and nonnegative, and \(k\in C^1(Q_T)\) is lower bounded by a positive constant and Lipschitz on \(u\), under suitable assumptions on the initial and boundary data.\N\NBoth existence results are proved via the Schauder fixed point theorem, which may be applied by using the unique SOLA (solution obtained as a limit approximations) from \(L^1-\)theory on PDE due to the existence of the Joule effect.\N\NFinally, the authors discuss the wellposedness of the magneto-quasistatic problem\N\begin{align*}\N\partial_t (\mu\mathbf{H})+ \nabla\times (\sigma^{-1}\nabla\times\mathbf{H})=\mathbf{0};\\\N\partial_t u -\nabla\cdot(k \nabla u)= \sigma^{-1} |\nabla\times\mathbf{H}|^2.\N\end{align*}