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Lq-Helmholtz Decomposition and Lq-Spectral Theory for the Maxwell Operator on Periodic Domains

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Lq-Helmholtz Decomposition and Lq-Spectral Theory for the Maxwell Operator on Periodic Domains Lq-Helmholtz decomposition and Lq-spectral theory for the Maxwell operator on periodic domains Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN der Fakult¨at fur¨ Mathematik des Karlsruhe Instituts fur¨ Technologie (KIT) genehmigte DISSERTATION von Jens Babutzka aus Sindelfingen Tag der mundlichen¨ Prufung:¨ 27.01.2016 Referent: Priv.-Doz. Dr. Peer Christian Kunstmann Korreferent: Prof. Dr. Lutz Weis This document is licensed under the Creative Commons Attribution 3.0 DE License (CC BY 3.0 DE): http://creativecommons.org/licenses/by/3.0/de/ Abstract We prove the existence of the Helmholtz decomposition q Cd q q L (Ωp; ) = Lσ(Ωp) ⊕ G (Ωp); d d for periodic domains Ωp ⊆ R with respect to a lattice L ⊆ R , i.e. Ωp = Ωp + z for all 3+" z 2 L, and for all q 2 I = ( 2+" ; 3 + "), where " = "(Ωp) is given by 1 • " = 1 if Ωp 2 C , i.e. I = (1; 1). • " > 1 if Ωp is Lipschitz and d = 2. • " > 0 if Ωp is Lipschitz and d ≥ 3. The proof of the Helmholtz decomposition builds upon [Bar13], where periodic operators are extended continuously from L2 to operators defined on Lq by using Bloch theory and 2 Fourier multiplier theorems. Here, we prove that the Helmholtz projection P2 on L has an extension to a bounded linear operator on Lq. The same approach yields the existence of the Leray decomposition q Cd q q L (Ωp; ) = Lσ;Dir(Ωp) ⊕ GDir(Ωp); which is related to the weak Dirichlet problem, while the Helmholtz decomposition is related to the weak Neumann problem. Following the approach in [KU15a], we define the Maxwell operator on periodic domains by using the form method, and this allows us to extend the operator on Lq-spaces, where 1;1 q 2 [3=2; 3] if @Ωp is Lipschitz and q 2 [6=5; 6] if @Ωp 2 C , by using generalized Gaussian estimates. Furthermore, we show spectral independence of q for the Maxwell operator and prove a spectral multiplier theorem for a shifted version of the operator. Furthermore, we get that the Stokes operator with Dirichlet boundary conditions generates an analytic semigroup for periodic domains of C3-class, and are able to show several results for the incompressible Navier-Stokes equations which make use of [GHHS12] and [GK15]. Acknowledgements During my thesis I was member of the Research Training Group 1294 "Analysis, Simula- tion und Design nanotechnologischer Prozesse" at the Karlsruher Institute of Technology. I gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG). I am really thankful for getting this scholarship, which allowed me to focus totally on research. I want to thank all employees of the faculty for their engagement in teaching and research, especially those who were involved in the GRK and SFB (CRC 1173). Besides I want to thank: • My supervisor Priv-Doz. Dr. Peer Kunstmann for his kind support all the time. He not only suggested this interesting topic, but had always an open door for me and gave my helpful advices during our countless discussions. • Prof. Dr. Marlis Hochbruck and Prof. Dr. Lutz Weis for their involvement, which brought me into the GRK. Furthermore, I thank Prof. Dr. Lutz Weis for taking the co-examination. • All my dear colleagues in the GRK and SFB-team for the great atmosphere within all mathematical and non-mathematical events during this time. • My kind room mates Silvana Avramska-Lukarska and Leonid Chaichenets and all other members of our workgroup for the excellent, kind atmosphere and the regu- lar joining of my lunch announcements. Besides, I want to thank Prof. Dr. Dirk Hundertmark and Prof. Dr. Tobias Lamm for being the best chiefs imaginable. • Dr. Kaori Nagato-Plum for her (private) invitation to the Christmas party and her effort in doing all the organization stuff. • Markus Antoni, Johannes Eilinghoff and Sebastian Schwarz for their friendship and our regular games evenings. • My colleague and (chess) friend Carlos Hauser. • My long-term friends Bernhard, Patrick and Markus G. • My parents, my brothers and the rest of my family for their support throughout my whole life. • My special thanks go to Ute for everything, especially for her love and mental and moral support during the last years. i Contents Abstract Acknowledgements i 1 Introduction 1 2 Preliminaries 12 2.1 Basic Notations . 12 2.2 Semigroups . 19 2.3 Sesquilinear forms . 22 2.4 Differentiable and analytical maps . 25 2.5 R-boundedness . 26 2.6 The H1-calculus . 31 2.7 The Helmholtz decomposition . 32 3 The Helmholtz decomposition on periodic domains 42 3.1 Equality of two solenoidal vector field spaces on periodic domains . 42 3.2 The Bloch transform and multiplier theorems . 54 2 d 3.3 Fibre operators for the Helmholtz projection on L (Ω#; C )......... 59 q d 3.4 The fibre operators on L (Ω#; C )....................... 66 3.5 The weak Dirichlet problem on periodic domains . 73 4 Applications 83 2 4.1 Photonic crystals and the Maxwell operator on L (Ωp)............ 83 4.2 Gaussian estimates and a spectral multiplier theorem . 90 q 4.3 The Maxwell operator on L (Ωp)........................ 96 4.4 Navier-Stokes equations and the Stokes operator . 98 5 Physical Appendix 104 5.1 Maxwell's equations . 104 5.2 The Navier-Stokes equations . 107 Index 111 Bibliography 113 CHAPTER 1 Introduction In 1858 von Helmholtz [Hel58] introduced the vector field decomposition 1 Rd Cd u = r(div F ∗ u) − curl (curl F ∗ u); u 2 Cc ( ; ); ( 1 2π log jxj; d = 2; F (x) : = 1 2−d d(2−d)jB(0;1)j jxj ; d = 3; where F is the fundamental solution of the Laplace operator. Since div curl = 0, the decomposition consists of a gradient part and a solenoidal vector field. This ansatz can be used to prove the existence of the Helmholtz decomposition on Lq(Rd; Cd) for all q 2 (1; 1), i.e. a unique decomposition into a gradient part and a solenoidal vector field. From there on mathematicians analysed this decomposition and generalized it to arbitrary domains Ω ⊆ Rd. This decomposition, if it exists, is known as the Helmholtz decomposition and one common today's formulation for a domain Ω ⊆ Rd looks as follows. We say that the Helmholtz decomposition exists on Lq(Ω; Cd), if for all f 2 Lq(Ω; Cd) there q q exist unique functions rp 2 G (Ω), g 2 Lσ(Ω) so that f = g + rp; and jjgjjLq(Ω;Cd) + jjrpjjLq(Ω;Cd) ≤ CjjfjjLq(Ω;Cd); q q where C = C(q; Ω). Here, the spaces Lσ(Ω) and G (Ω) are defined as follows: ||·|| Lq(Ω;Cd) q 1 q 1 Cd Lσ(Ω) = Cc,σ(Ω) = fu 2 Cc (Ω; ) j div u = 0 in Ωg ; q q Cd 1;q q 1 G (Ω) = frp 2 L (Ω; ) j p 2 Wloc (Ω)g = frp 2 L (Ω) j p 2 Lloc(Ω)g: q d q d If the decomposition exists, there is a bounded projection operator Pq : L (Ω; C ) ! L (Ω; C ) q q satisfying kernel(Pq) = G (Ω) and image(Pq) = Lσ(Ω). Pq is called the Helmholtz projec- tion. The Helmholtz decomposition exists on Lq(Ω; Cd) if and only if the weak formulation of the classical Neumann problem ∆p = div u in Ω; @p = u · ν on @Ω: @ν 1 1 Introduction has for all u 2 Lq(Ω; Cd) a unique solution p 2 Gq(Ω), compare Theorem 2.43 and Remark 2.44. The Helmholtz decomposition can be used in physical models considering source-free vector fields, which is in fact a very common assumption. In this work we examine the Maxwell equations (note that the magnetic field is source-free) and the incompressible Navier-Stokes equations for fluids. We discuss below (p.4 ff.) how the Helmholtz projection enters in these systems. We shall come back to applications, but we first outline the existence theory of the Helmholtz decomposition. On half-spaces there is, as in the case of Ω = Rd, a fundamental solution for the Laplace operator with Neumann boundary condition, which yields the existence of Helmholtz de- composition for all q 2 (1; 1). From a mathematical point of view the Hilbert space case q = 2 is by far the easiest, and in fact one can easily prove the existence of the Helmholtz decomposition on any domain Ω ⊆ Rd in that case by using the Lax-Milgram lemma, com- pare Theorem 2.45. In the study of (nonlinear) differential equations it often does not suffice to consider the case q = 2. In the theory of the three dimensional Navier-Stokes equations the spaces Lq for q ≥ 3 play an important role. For example, an important tool in the study is the Fujita- Kato iteration scheme [FK64], which yields a unique mild solution in the case q > d = 3. Here, the iterations enlarges step by step the interval on which the mild solution is unique. The exponent q > d is needed to have the embedding of W 1;q into the space of continuous functions. Also the case q = d is of special interest, cf. for example [Gig86], because one can use the scaling invariance of the Navier-Stokes equations on Ld under the scaling u(x; t) = λu(λx, λ2t); p(x; t) = λ2p(λx, λ2t): Furthermore, in nonlinear Maxwell equations and heat equations nonlinear terms behaving like ujujp−1, p > 1 have to be studied. So here, one also is obviously reliant on using Lq-theory. At least since 1986 [Bou86] it became clear that the Helmholtz decomposition fails for some unbounded (even C1-) domains and also fails for some bounded Lipschitz domains for some range of q.
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