By Ansgar Jüngel
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.
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Additional resources for Entropy Methods for Diffusive Partial Differential Equations
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 . 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.