Degenerate Diffusions
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Degenerate Diffusions
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Author : Panagiota Daskalopoulos
language : en
Publisher: European Mathematical Society
Release Date : 2007
Degenerate Diffusions written by Panagiota Daskalopoulos and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.
The book deals with the existence, uniqueness, regularity, and asymptotic behavior of solutions to the initial value problem (Cauchy problem) and the initial-Dirichlet problem for a class of degenerate diffusions modeled on the porous medium type equation $u_t = \Delta u^m$, $m \geq 0$, $u \geq 0$. Such models arise in plasma physics, diffusion through porous media, thin liquid film dynamics, as well as in geometric flows such as the Ricci flow on surfaces and the Yamabe flow. The approach presented to these problems uses local regularity estimates and Harnack type inequalities, which yield compactness for families of solutions. The theory is quite complete in the slow diffusion case ($ m>1$) and in the supercritical fast diffusion case ($m_c
Degenerate Diffusions
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Author : Wei-Ming Ni
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Degenerate Diffusions written by Wei-Ming Ni and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This IMA Volume in Mathematics and its Applications DEGENERATE DIFFUSIONS is based on the proceedings of a workshop which was an integral part of the 1990- 91 IMA program on "Phase Transitions and Free Boundaries". The aim of this workshop was to provide some focus in the study of degenerate diffusion equations, and by involving scientists and engineers as well as mathematicians, to keep this focus firmly linked to concrete problems. We thank Wei-Ming Ni, L.A. Peletier and J.L. Vazquez for organizing the meet ing. We especially thank Wei-Ming Ni for editing the proceedings. We also take this opportunity to thank those agencies whose financial support made the workshop possible: the Army Research Office, the National Science Foun dation, and the Office of Naval Research. A vner Friedman Willard Miller, Jr. PREFACE This volume is the proceedings of the IMA workshop "Degenerate Diffusions" held at the University of Minnesota from May 13 to May 18, 1991.
Degenerate Diffusions With Advection
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Author : Yuming Zhang
language : en
Publisher:
Release Date : 2019
Degenerate Diffusions With Advection written by Yuming Zhang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with categories.
Flow of an ideal gas through a homogeneous porous medium can be described by the well-known Porous Medium Equation $(PME)$. The key feature is that the pressure is proportional to some powers of the density, which corresponds to the anti-congestion effect given by the degenerate diffusion. This effect is widely seen in fluids, biological aggregation and population dynamics. If adding an advection, the equation can be naturally contextualized as a population moving with preferences or fluids in a porous medium moving with wind. Furthermore we may consider drifts that depend on the solution itself by a non-local convolution, which describe the interaction between particles in a swarm model or a model for chemotaxis. In this dissertation, we study those PDEs. In the first two chapters, we consider local advection transportation driven by a known vector field. Chapter 1 is devoted to investigate the H\"{o}lder regularity of solutions in terms of bounds of the vector field in the space $L_x^{p}$. By a scaling argument, we find that $p=d$ is critical (where $d$ is the space dimension). Along with a De Giorgi-Nash-Moser type arguments, we prove H\"{o}lder regularity of solutions after time $0$ in the subcritical regime $p>d$. And we give examples showing the loss of uniform H\"{o}lder continuity of solutions in the critical regime even for divergence-free drifts. In Chapter 2, we are interested in the geometric properties of the free boundary for the solution ($u$): $\partial\{u>0\}$. First it is shown that, if the initial data has super-quadratic growth at the free boundary, then the support strictly expands relative to the streamline. We then proceed to show the nondegeneracy and $C^{1,\alpha}$ regularity of the free boundary, when the solution is directionally monotone in space variable in a local neighborhood. The main challenge lies in establishing a local non-degeneracy estimate, which appears new even for the zero drift case. In Chapter 3 and 4, we consider more general drifts which depends on the solution itself by a non-local convolution. If considering a swarm model or a model for chemotaxis, the non-local drift describes the interaction effect between particles as swarms of locusts or cells. Chapter 3 discusses the vanishing viscosity limit of the equation in a bounded and convex domain. The limit agrees with the first-order system with a projection operator on the boundary proposed by Carrillo, Slepcev and Wu. Thus our result gives another justification of their equation. We apply the gradient flow method and we explore bounded approximations of singular measures in the generalized Wasserstein distance, which I believe, is independently interesting and might be useful in other contexts. Chapter 4 considers singular kernels of the form $(-\Delta)^{-s} u$ with $s\in (0,\frac{d}{2})$. With $s=1$ we recover the well-known Patlak-Keller-Seger equation which is an macroscopic description of the chemotaxis phenomenon. The competition between non-local attractive interactions and the diffusion is one of the core of subject of diffusion-aggregation equations. We study well-posedness, boundedness and H\"{o}lder regularity of solutions in most of the subcritical regime. Several open questions will be discussed.
Degenerate Diffusion Operators Arising In Population Biology Am 185
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Author : Charles L. Epstein
language : en
Publisher: Princeton University Press
Release Date : 2013-04-04
Degenerate Diffusion Operators Arising In Population Biology Am 185 written by Charles L. Epstein and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-04 with Mathematics categories.
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 codimensional 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.
Degenerate Nonlinear Diffusion Equations
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Author : Angelo Favini
language : en
Publisher: Springer
Release Date : 2012-05-08
Degenerate Nonlinear Diffusion Equations written by Angelo Favini and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-08 with Mathematics categories.
The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asymptotic behaviour, discretization schemes, coefficient identification, and to introduce relevant solving methods for each of them.
On Large Deviations Of Invariant Measures For Degenerate Diffusions
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Author : Lukasz Stettner
language : en
Publisher:
Release Date : 1987
On Large Deviations Of Invariant Measures For Degenerate Diffusions written by Lukasz Stettner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Differentiable dynamical systems categories.
The Regularity Of General Parabolic Systems With Degenerate Diffusion
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Author : Verena Bögelein
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-01-28
The Regularity Of General Parabolic Systems With Degenerate Diffusion written by Verena Bögelein and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-28 with Mathematics categories.
The aim of the paper is twofold. On one hand the authors want to present a new technique called $p$-caloric approximation, which is a proper generalization of the classical compactness methods first developed by DeGiorgi with his Harmonic Approximation Lemma. This last result, initially introduced in the setting of Geometric Measure Theory to prove the regularity of minimal surfaces, is nowadays a classical tool to prove linearization and regularity results for vectorial problems. Here the authors develop a very far reaching version of this general principle devised to linearize general degenerate parabolic systems. The use of this result in turn allows the authors to achieve the subsequent and main aim of the paper, that is, the implementation of a partial regularity theory for parabolic systems with degenerate diffusion of the type $\partial_t u - \mathrm{div} a(Du)=0$, without necessarily assuming a quasi-diagonal structure, i.e. a structure prescribing that the gradient non-linearities depend only on the the explicit scalar quantity.
Recurrence And Invariant Measures For Degenerate Diffusions
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Author : Wolfgang Kliemann
language : de
Publisher:
Release Date : 1982
Recurrence And Invariant Measures For Degenerate Diffusions written by Wolfgang Kliemann and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982 with categories.
Stability And Functional Central Limit Theorems For Degenerate Diffusions
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Author : Gopal Krishna Basak
language : en
Publisher:
Release Date : 1989
Stability And Functional Central Limit Theorems For Degenerate Diffusions written by Gopal Krishna Basak and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with Central limit theorem categories.
Highly Degenerate Diffusions For Sampling Molecular Systems
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Author : Emad Noorizadeh
language : en
Publisher:
Release Date : 2010
Highly Degenerate Diffusions For Sampling Molecular Systems written by Emad Noorizadeh and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with categories.
This work is concerned with sampling and computation of rare events in molecular systems. In particular, we present new methods for sampling the canonical ensemble corresponding to the Boltzmann-Gibbs probability measure. We combine an equation for controlling the kinetic energy of the system with a random noise to derive a highly degenerate diffusion (i.e. a diffusion equation where diffusion happens only along one or few degrees of freedom of the system). Next the concept of hypoellipticity is used to show that the corresponding Fokker-Planck equation of the highly degenerate diffusion is well-posed, hence we prove that the solution of the highly degenerate diffusion is ergodic with respect to the Boltzmann-Gibbs measure. We find that the new method is more efficient for computation of dynamical averages such as autocorrelation functions than the commonly used Langevin dynamics, especially in systems with many degrees of freedom. Finally we study the computation of free energy using an adaptive method which is based on the adaptive biasing force technique.