This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

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Hitchhiker's guide to the fractional Sobolev spaces

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

/pdf/a-user-s-guide-to-pde-models-for-chemotaxis-4zxd8lr4pw.pdf

A user’s guide to PDE models for chemotaxis

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Existence of local and global in time solutions for parabolic-elliptic and parabolic systems describing the aggregation of the cellular slime mould by the chemical attraction is studied. In particular, conditions for global in time existence, as well as nite time blow-up are given. Self-similar and radial solutions are also considered.

Local and global solvability of some parabolic systems modelling chemotaxis

The Debye system: existence and large time behavior of solutions

where ν denotes the outward unit normal vector to ∂Ω. For the potential φ we assume either (4.1) φ = 0 on ∂Ω, or instead of the Dirichlet boundary condition above (4.2) φ = En ∗ u with En the fundamental solution of the n-dimensional Laplacian. For the initial-boundary problem we add the condition (5) u(x, 0) = u0(x) ≥ 0. For the interpretation of (1)–(5) we refer the reader to the introduction of [9]. In the one-dimensional case solutions are global in time (see the reasoning in [7, Sec. 3] based on an idea from [14], and [18]). The first proof of nonexistence of solutions to (1)–(4) defined globally in time has been discovered for the radial problem in an n-dimensional ball, n ≥ 2, in [7, Theorem 3]. The critical value of mass of the initial

/pdf/existence-and-nonexistence-of-solutions-for-a-model-of-4kssoukmud.pdf

Existence and nonexistence of solutions for a model of gravitational interaction of particles, I

Fractal Burgers Equations

We study the convergence rates of solutions to drift-diffusion systems (arising from plasma, semiconductors and electrolytes theories) to their self-similar or steady states. This analysis involves entropy-type Lyapunov functionals and logarithmic Sobolev inequalities.

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Piotr Biler

Papers

Local and global solvability of some parabolic systems modelling chemotaxis

The Debye system: existence and large time behavior of solutions

Existence and nonexistence of solutions for a model of gravitational interaction of particles, I

Fractal Burgers Equations

Long time behavior of solutions to Nernst-Planck and Debye-Hückel drift-diffusion systems