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Intro; Preface; Contents; On Effective PDEs of Quantum Physics; 1 Introduction; 2 Hartree and Gross-Pitaevski Equations; 2.1 Origin and Properties; 2.1.1 Properties of the Hartree and Gross-Pitaevski Equations; 2.2 Particles Coupled to the Electromagnetic Field; 3 The (Generalized) Hartree-Fock Equations; 3.1 Formulation and Properties; 3.1.1 Exchange Energy Term; 3.2 Static gHF Equations; 3.3 Coupling to the Electromagnetic Field; 3.4 Static gHFem Equations; 3.4.1 Free Energy; 3.4.2 Electrostatics; 4 Density Functional Theory; 4.1 Crystals; 4.2 Macroscopic Perturbations
5 Hartree-Fock-Bogoliubov Equations6 Bogoliubov-de Gennes Equations; 6.1 Formulation; 6.2 Symmetries; 6.3 Conservation Laws; 6.4 Stationary Bogoliubov-de Gennes Equations; 6.5 Free Energy; 6.6 Ground/Gibbs States; 6.7 Symmetry Breaking; 6.8 Stability; 6.8.1 Normal States; 6.8.2 Superconducting States; 6.8.3 Mixed States; 6.8.4 Magnetic Flux Quantization; 7 Existence of Periodic Solutions by the Variational Technique; References; Critical Exponents for Differential Inequalities with Riemann-Liouville and Caputo Fractional Derivatives; 1 Introduction; 1.1 Notation; 2 Global Weak Solutions
3 A Suitable Test Function4 Proof of Theorem 1; 5 Proof of Theorem 2; 6 Decay Estimates for the Fractional Subdiffusive Equation; 6.1 Proof of Lemma 3; 6.2 Decay Estimates; 6.3 Proof of Theorems 3 and 4; References; Weakly Coupled Systems of Semilinear Effectively Damped Waves with Different Time-Dependent Coefficients in the Dissipation Terms and Different Power Nonlinearities; 1 Introduction; 1.1 Notations; 2 Main Results; 2.1 Low Regular Data; 2.2 Data from Energy Space; 2.3 Data from Sobolev Spaces with Suitable Regularity; 2.4 Large Regular Data; 3 Philosophy of Our Approach
3.1 Proof of Theorem 2.13.2 Proof of Theorem 2.6; 3.3 Proof of Theorem 2.8; 3.4 Proof of Theorem 2.11; 4 Concluding Remarks; Appendix; References; Incompressible Limits for Generalisations to Symmetrisable Systems; 1 Introduction; 1.1 An Example: The Incompressible Limit for the Euler System; 1.2 An Example: The Quasineutral Limit for the Euler-Poisson System; 2 Assumptions and Main Results; 3 The Uniform Existence Interval; 4 The Incompressible Limit; 5 An Application; 6 Concluding Remarks; References; The Critical Exponent for Evolution Models with Power Non-linearity; 1 Introduction
5 Hartree-Fock-Bogoliubov Equations6 Bogoliubov-de Gennes Equations; 6.1 Formulation; 6.2 Symmetries; 6.3 Conservation Laws; 6.4 Stationary Bogoliubov-de Gennes Equations; 6.5 Free Energy; 6.6 Ground/Gibbs States; 6.7 Symmetry Breaking; 6.8 Stability; 6.8.1 Normal States; 6.8.2 Superconducting States; 6.8.3 Mixed States; 6.8.4 Magnetic Flux Quantization; 7 Existence of Periodic Solutions by the Variational Technique; References; Critical Exponents for Differential Inequalities with Riemann-Liouville and Caputo Fractional Derivatives; 1 Introduction; 1.1 Notation; 2 Global Weak Solutions
3 A Suitable Test Function4 Proof of Theorem 1; 5 Proof of Theorem 2; 6 Decay Estimates for the Fractional Subdiffusive Equation; 6.1 Proof of Lemma 3; 6.2 Decay Estimates; 6.3 Proof of Theorems 3 and 4; References; Weakly Coupled Systems of Semilinear Effectively Damped Waves with Different Time-Dependent Coefficients in the Dissipation Terms and Different Power Nonlinearities; 1 Introduction; 1.1 Notations; 2 Main Results; 2.1 Low Regular Data; 2.2 Data from Energy Space; 2.3 Data from Sobolev Spaces with Suitable Regularity; 2.4 Large Regular Data; 3 Philosophy of Our Approach
3.1 Proof of Theorem 2.13.2 Proof of Theorem 2.6; 3.3 Proof of Theorem 2.8; 3.4 Proof of Theorem 2.11; 4 Concluding Remarks; Appendix; References; Incompressible Limits for Generalisations to Symmetrisable Systems; 1 Introduction; 1.1 An Example: The Incompressible Limit for the Euler System; 1.2 An Example: The Quasineutral Limit for the Euler-Poisson System; 2 Assumptions and Main Results; 3 The Uniform Existence Interval; 4 The Incompressible Limit; 5 An Application; 6 Concluding Remarks; References; The Critical Exponent for Evolution Models with Power Non-linearity; 1 Introduction