Download Entropy Methods for Diffusive Partial Differential Equations by Ansgar Jüngel PDF

By Ansgar Jüngel

ISBN-10: 3319342185

ISBN-13: 9783319342184

ISBN-10: 3319342193

ISBN-13: 9783319342191

This booklet provides a variety of entropy equipment for diffusive PDEs devised via many researchers during the previous few a long time, which permit us to appreciate the qualitative habit of strategies to diffusive equations (and Markov diffusion processes). functions contain the large-time asymptotics of suggestions, the derivation of convex Sobolev inequalities, the lifestyles and distinctiveness of susceptible ideas, and the research of discrete and geometric constructions of the PDEs. the aim of the ebook is to supply readers an creation to chose entropy equipment that may be present in the study literature. with a purpose to spotlight the center strategies, the consequences will not be acknowledged within the widest generality and many of the arguments are just formal (in the feel that the useful environment isn't designated or enough regularity is supposed). The textual content can be compatible for complicated grasp and PhD scholars and will function a textbook for exact classes and seminars.

Show description

Read or Download Entropy Methods for Diffusive Partial Differential Equations PDF

Similar functional analysis books

Nonlinear Functional Analysis

This graduate-level textual content bargains a survey of the most rules, strategies, and techniques that represent nonlinear practical research. It positive aspects vast statement, many examples, and engaging, hard workouts. subject matters contain measure mappings for countless dimensional areas, the inverse functionality concept, the implicit functionality thought, Newton's tools, and lots of different matters.

A Basis Theory Primer: Expanded Edition

The classical topic of bases in Banach areas has taken on a brand new existence within the smooth improvement of utilized harmonic research. This textbook is a self-contained creation to the summary conception of bases and redundant body expansions and its use in either utilized and classical harmonic research. The 4 elements of the textual content take the reader from classical practical research and foundation idea to trendy time-frequency and wavelet conception.


This ebook offers the most effects and strategies on inverse spectral difficulties for Sturm-Liouville differential operators and their functions. Inverse difficulties of spectral research consist in getting better operators from their spectral features. Such difficulties frequently seem in arithmetic, mechanics, physics, electronics, geophysics, meteorology and different branches of ordinary sciences.

Additional resources for Entropy Methods for Diffusive Partial Differential Equations

Sample text

Proof The symmetry of L sym follows from the symmetry of D(x) since Rd L sym (u)vu −1 ∞ dx = − Rd u∞∇ u u∞ D∇ v dx = u∞ Rd u L sym (v)u −1 ∞ dx for suitable functions u, v. To prove that L as is anti-symmetric, we first integrate by parts in Rd L as (u)vu −1 ∞ dx = − Rd −1 u D F · u −1 ∞ ∇v + v∇u ∞ dx. 34), we have −1 −1 −1 0 = div(D Fu ∞ )uvu −2 ∞ = (div D) · Fuvu ∞ − uv(D F) · ∇u ∞ + D : ∇ Fuvu ∞ , where D : ∇ F = Rd i, j Di j ∂xi F j . 35), we obtain L as (u)vu −1 ∞ dx = − =− Rd Rd u D F · ∇v + (div D) · Fuv + D : ∇ Fuv u −1 ∞ dx u div(D Fv)u −1 ∞ dx = − Rd u L as (v)u −1 ∞ dx, which shows the lemma.

Solution algorithms for quantifier elimination problems have been implemented, for instance, in Mathematica. There are also specialized tools like QEPCAD (Quantifier Elimination by Partial Cylindrical Algebraic Decomposition); see [7]. The advantage of these algorithms is that the solution is complete and exact; the algorithm delivers a full proof. However, because of the complexity of the algorithms, their time and memory consumption is very high. An alternative is given by the sum-of-squares (SOS) method [26, 27].

With the polynomial variables ξ1 , ξ2 , etc. Setting ξ = (ξ1 , ξ2 , ξ3 , ξ4 ) ∈ R4 , Pα corresponds to S0 (ξ ) = −ξ1 ξ3 , I1 corresponds to T1 (ξ ) = (α + β − 3)ξ14 + 3ξ12 ξ2 , I2 corresponds to T2 (ξ ) = (α + β − 2)ξ12 ξ2 + ξ22 + ξ1 ξ3 , I3 corresponds to T3 (ξ ) = (α + β − 1)ξ1 ξ3 + ξ4 . 9) We call Ti shift polynomial since they allow us to “shift” partial derivatives from one term to another one. e. if ∃c1 , c2 , c3 ∈ R : ∀ξ ∈ R4 : (S0 + c1 T1 + c2 T2 + c3 T3 )(ξ ) ≥ 0. 10) Such problems are called polynomial decision problems; they are well known in real algebraic geometry.

Download PDF sample

Rated 4.42 of 5 – based on 15 votes