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Turbulence and Diffusion : Scaling Versus Equations
This book is an introduction to the multidisciplinary field of anomalous diffusion in complex systems, with emphasis on the scaling approach as opposed to techniques based on the quantitative analysis of underlying transport equations. Typical examples of such systems are turbulent plasmas, convective rolls, zonal flow systems and stochastic magnetic fields. From the more methodological point of view, the approach relies on the general use of correlations estimates, quasilinear equations and continuous time random walk techniques. Yet, the mathematical descriptions are not meant to become a fixed set of recipes but rather develop and strengthen the reader's physical intuition and understanding on the underlying mechanisms involved.
Transport Equations in Biology
The book contains many ex- ples without mathematical analysis.In some other cases complete mathematical proofs are detailed, but the choice has been a compromise between technicality and ease of interpretation of the mathematical result. It is usual in the ?eld to see mathematics as a blackboxwhere to enter specifc models, often at the expense of simplifcations. Here, the idea is di?erent; the mathematical proof should be close to the ‘natural’ structure of the model and refect somehow its meaning in terms of applications. Dealing with frstorder PDEs,onecould think that the senotesarerelyingon the burden of using the method of characteristics and of defning weak solutions.
Transport Equations and Multi-D Hyperbolic Conservation Laws
The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements thanks to ideas and techniques related to the structure and fine properties of functions of bounded variation. This volume provides an up-to-date overview of the status and perspectives of two areas of research in PDEs, related to hyperbolic conservation laws. Geometric and measure theoretic tools play a key role to obtain some fundamental advances: the well-posedness theory of linear transport equations with irregular coefficients, and the study of the BV-like structure of bounded entropy solutions to multi-dimensional scalar conservation laws.
Stochastic Ordinary and Stochastic Partial Differential Equations : Transition from Microscopic to Macroscopic Equations
This book provides the first rigorous derivation of mesoscopic and macroscopic equations from a deterministic system of microscopic equations. The microscopic equations are cast in the form of a deterministic (Newtonian) system of coupled nonlinear oscillators for N large particles and infinitely many small particles. The mesoscopic equations are stochastic ordinary differential equations (SODEs) and stochastic partial differential equatuions (SPDEs), and the macroscopic limit is described by a parabolic partial differential equation. A detailed analysis of the SODEs and (quasi-linear) SPDEs is presented. Semi-linear (parabolic) SPDEs are represented as first order stochastic transport equations driven by Stratonovich differentials. The time evolution of correlated Brownian motions is shown to be consistent with the depletion phenomena experimentally observed in colloids.
Modelling the dispersion of radionuclides in the marine environment : An introduction
This book is a practical guide to the subject of numerical modelling of radioactivity dispersion in the marine environment. Thus, the techniques and numerical procedures required are explained in detail, with the aim of enabling the reader to build a real mathematical model. The book covers basic concepts and techniques, such as solving the advection-diffusion equation in a simple 1D form, as well as the most recent developments (full 3D models for non-conservative radionuclides including chemical reactions and speciation). A chapter is dedicated to the basic hydrodynamic modelling that is always required to simulate the dispersion of tracers in the sea; Eulerian and Lagrangian modelling techniques are also described. A chapter describes sensitivity and uncertainty analysis, the final stage in modelling works. A review on some published radionuclide dispersion models is also included.
Instability in Models Connected with Fluid Flows II
Instability in Models Connected with Fluid Flows II presents chapters from world renowned specialists. The stability of mathematical models simulating physical processes is discussed in topics on control theory, first order linear and nonlinear equations, water waves, free boundary problems, large time asymptotics of solutions, stochastic equations, Euler equations, Navier-Stokes equations, and other PDEs of fluid mechanics. Fields covered include: the free surface Euler (or water-wave) equations, the Cauchy problem for transport equations, irreducible Chapman--Enskog projections and Navier-Stokes approximations, randomly forced PDEs, stability of equilibrium figures of uniformly rotating viscous incompressible liquid, Navier-Stokes equations in cylindrical domains, Navier-Stokes-Poisson flows in a vacuum.
Exponentially Dichotomous Operators and Applications
In this monograph the natural evolution operators of autonomous first-order differential equations with exponential dichotomy on an arbitrary Banach space are studied in detail. Characterizations of these so-called exponentially dichotomous operators in terms of their resolvents and additive and multiplicative perturbation results are given. The general theory of the first three chapters is then followed by applications to Wiener-Hopf factorization and Riccati equations, transport equations, diffusion equations of indefinite Sturm-Liouville type, noncausal infinite-dimensional linear continuous-time systems, and functional differential equations of mixed type.
Macroscopic Transport Equations for Rarefied Gas Flows : Approximation Methods in Kinetic Theory
This book discusses classical and modern methods to derive macroscopic transport equations for rarefied gases from the Boltzmann equation, for small and moderate Knudsen numbers, i.e.as well as the new order of magnitude method, which avoids the short-comings of the classical methods, but retains their benefits.
Calculus of variations and nonlinear partial differential equations : With a historical overview by Elvira Mascolo : Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005
This volume provides the texts of lectures given by L. Ambrosio, L. Caffarelli, M. Crandall, L.C. Evans, N. Fusco at the Summer course held in Cetraro (Italy) in 2005. These are introductory reports on current research by world leaders in the fields of calculus of variations and partial differential equations. The topics discussed are transport equations for nonsmooth vector fields, homogenization, viscosity methods for the infinite Laplacian, weak KAM theory and geometrical aspects of symmetrization. A historical overview of all CIME courses on the calculus of variations and partial differential equations is contributed by Elvira Mascolo.
