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The present volume collects the contributions selected for publication in the Special Issue entitled "Microlocal and Time-Frequency Analysis" of the journal Mathematics, edited by Elena Cordero and S. Ivan Trapasso over 2020 and 2021.

Computing & information technology --- Computer science --- pseudodifferential operators --- Gevrey regularity --- sharp Gårding inequality --- p-evolution equations --- ultradifferentiable functions --- ultradistributions --- extended Gevrey regularity --- boundary values of analytic functions --- wave front sets --- Rio Papaloapan Bridge --- vibration signals --- damage identification --- wavelet energy accumulation method --- admissibility condition --- the continuous wavelet transform --- inversion formula --- semi-discrete wavelet transform --- tight frames --- covriance matrix --- polar duality --- uncertainty principle --- reconstruction problem --- BBM equation --- ill-posedness --- Fourier amalgam spaces --- Wiener amalgam spaces --- Fourier–Lebesgue spaces --- modulation spaces --- bounded measures --- convolution --- homogeneous Banach spaces --- integrated group representation --- Segal algebra --- Wiener amalgam space --- bounded uniform partition of unity --- locally compact groups

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Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if their spatial regularity was below 1/3-Hölder. In this book, Philip Isett uses the method of convex integration to achieve the best-known results regarding nonuniqueness of solutions and Onsager's conjecture. Focusing on the intuition behind the method, the ideas introduced now play a pivotal role in the ongoing study of weak solutions to fluid dynamics equations.The construction itself-an intricate algorithm with hidden symmetries-mixes together transport equations, algebra, the method of nonstationary phase, underdetermined partial differential equations (PDEs), and specially designed high-frequency waves built using nonlinear phase functions. The powerful "Main Lemma"-used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem-has been extended to a broad range of applications that are surveyed in the appendix. Appropriate for students and researchers studying nonlinear PDEs, this book aims to be as robust as possible and pinpoints the main difficulties that presently stand in the way of a full solution to Onsager's conjecture.

Fluid dynamics --- Mathematics. --- Beltrami flows. --- Einstein summation convention. --- Euler equations. --- Euler flow. --- Euler-Reynolds equations. --- Euler-Reynolds system. --- Galilean invariance. --- Galilean transformation. --- HighЈigh Interference term. --- HighЈigh term. --- HighЌow Interaction term. --- Hlder norm. --- Hlder regularity. --- Lars Onsager. --- Main Lemma. --- Main Theorem. --- Mollification term. --- Newton's law. --- Noether's theorem. --- Onsager's conjecture. --- Reynolds stres. --- Reynolds stress. --- Stress equation. --- Stress term. --- Transport equation. --- Transport term. --- Transport-Elliptic equation. --- abstract index notation. --- algebra. --- amplitude. --- coarse scale flow. --- coarse scale velocity. --- coefficient. --- commutator estimate. --- commutator term. --- commutator. --- conservation of momentum. --- continuous solution. --- contravariant tensor. --- convergence. --- convex integration. --- correction term. --- correction. --- covariant tensor. --- dimensional analysis. --- divergence equation. --- divergence free vector field. --- divergence operator. --- energy approximation. --- energy function. --- energy increment. --- energy regularity. --- energy variation. --- energy. --- error term. --- error. --- finite time interval. --- first material derivative. --- fluid dynamics. --- frequencies. --- frequency energy levels. --- h-principle. --- integral. --- lifespan parameter. --- lower indices. --- material derivative. --- mollification. --- mollifier. --- moment vanishing condition. --- momentum. --- multi-index. --- non-negative function. --- nonzero solution. --- optimal regularity. --- oscillatory factor. --- oscillatory term. --- parameters. --- parametrix expansion. --- parametrix. --- phase direction. --- phase function. --- phase gradient. --- pressure correction. --- pressure. --- regularity. --- relative acceleration. --- relative velocity. --- scaling symmetry. --- second material derivative. --- smooth function. --- smooth stress tensor. --- smooth vector field. --- spatial derivative. --- stress. --- tensor. --- theorem. --- time cutoff function. --- time derivative. --- transport derivative. --- transport equations. --- transport estimate. --- transport. --- upper indices. --- vector amplitude. --- velocity correction. --- velocity field. --- velocity. --- weak limit. --- weak solution.

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The present volume collects the contributions selected for publication in the Special Issue entitled "Microlocal and Time-Frequency Analysis" of the journal Mathematics, edited by Elena Cordero and S. Ivan Trapasso over 2020 and 2021.

Computing & information technology --- Computer science --- pseudodifferential operators --- Gevrey regularity --- sharp Gårding inequality --- p-evolution equations --- ultradifferentiable functions --- ultradistributions --- extended Gevrey regularity --- boundary values of analytic functions --- wave front sets --- Rio Papaloapan Bridge --- vibration signals --- damage identification --- wavelet energy accumulation method --- admissibility condition --- the continuous wavelet transform --- inversion formula --- semi-discrete wavelet transform --- tight frames --- covriance matrix --- polar duality --- uncertainty principle --- reconstruction problem --- BBM equation --- ill-posedness --- Fourier amalgam spaces --- Wiener amalgam spaces --- Fourier–Lebesgue spaces --- modulation spaces --- bounded measures --- convolution --- homogeneous Banach spaces --- integrated group representation --- Segal algebra --- Wiener amalgam space --- bounded uniform partition of unity --- locally compact groups --- pseudodifferential operators --- Gevrey regularity --- sharp Gårding inequality --- p-evolution equations --- ultradifferentiable functions --- ultradistributions --- extended Gevrey regularity --- boundary values of analytic functions --- wave front sets --- Rio Papaloapan Bridge --- vibration signals --- damage identification --- wavelet energy accumulation method --- admissibility condition --- the continuous wavelet transform --- inversion formula --- semi-discrete wavelet transform --- tight frames --- covriance matrix --- polar duality --- uncertainty principle --- reconstruction problem --- BBM equation --- ill-posedness --- Fourier amalgam spaces --- Wiener amalgam spaces --- Fourier–Lebesgue spaces --- modulation spaces --- bounded measures --- convolution --- homogeneous Banach spaces --- integrated group representation --- Segal algebra --- Wiener amalgam space --- bounded uniform partition of unity --- locally compact groups

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This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high co-dimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Elliptic operators. --- Markov processes. --- Population biology --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Differential operators, Elliptic --- Operators, Elliptic --- Partial differential operators --- Mathematical models. --- 1-dimensional integral. --- Euclidean model problem. --- Euclidean space. --- Hlder space. --- Hopf boundary point. --- Kimura diffusion equation. --- Kimura diffusion operator. --- Laplace transform. --- Schauder estimate. --- WrightІisher geometry. --- adjoint operator. --- backward Kolmogorov equation. --- boundary behavior. --- degenerate elliptic operator. --- doubling. --- elliptic Kimura operator. --- elliptic equation. --- forward Kolmogorov equation. --- function space. --- general model problem. --- generalized Kimura diffusion. --- heat equation. --- heat kernel. --- higher dimensional corner. --- higher regularity. --- holomorphic semi-group. --- homogeneous Cauchy problem. --- hybrid space. --- hypersurface boundary. --- induction hypothesis. --- induction. --- inhomogeneous problem. --- irregular solution. --- long time asymptotics. --- long-time behavior. --- manifold with corners. --- martingale problem. --- mathematical finance. --- model problem. --- normal form. --- normal vector. --- null-space. --- off-diagonal behavior. --- open orthant. --- parabolic equation. --- perturbation theory. --- polyhedron. --- population genetics. --- probability theory. --- regularity. --- resolvent operator. --- semi-group. --- solution operator. --- uniqueness.

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The topic of this book is the mathematical and numerical analysis of some recent frameworks based on differential equations and their application in the mathematical modeling of complex systems, especially of living matter. First, the recent new mathematical frameworks based on generalized kinetic theory, fractional calculus, inverse theory, Schrödinger equation, and Cahn–Hilliard systems are presented and mathematically analyzed. Specifically, the well-posedness of the related Cauchy problems is investigated, stability analysis is also performed (including the possibility to have Hopf bifurcations), and some optimal control problems are presented. Second, this book is concerned with the derivation of specific models within the previous mentioned frameworks and for complex systems in biology, epidemics, and engineering. This book is addressed to graduate students and applied mathematics researchers involved in the mathematical modeling of complex systems.

boundedness --- delay --- Hopf bifurcation --- Lyapunov functional --- stability --- SEIQRS-V model --- kinetic theory --- integro-differential equations --- complex systems --- evolution equations --- thermostat --- nonequilibrium stationary states --- discrete Fourier transform --- discrete kinetic theory --- nonlinearity --- fractional operators --- Cahn–Hilliard systems --- well-posedness --- regularity --- optimal control --- necessary optimality conditions --- Schrödinger equation --- Davydov’s model --- partial differential equations --- exact solutions --- fractional derivative --- abstract Cauchy problem --- C0−semigroup --- inverse problem --- active particles --- autoimmune disease --- degenerate equations --- real activity variable --- Cauchy problem --- electric circuit equations --- wardoski contraction --- almost (s, q)—Jaggi-type --- b—metric-like spaces --- second-order differential equations --- dynamical systems --- compartment model --- epidemics --- basic reproduction number