Sylabus vyberovky M610 Malek-Rokyta, LS 1995/96

Sylabus Vybranych kapitol z nelinearnich PDR (vyberovka M610 Malek-Rokyta)

Letni semestr 1995/96

Metody slabe konvergence v parcialnich diferencialnich rovnicich

1. Skalarni zakon zachovani

Motivace, slaba-* konvergence a prechod v nelinearite, poznamka o metode kompaktnosti (klasicke).

2. Slaba a silna konvergence v prostorech L^p

Kompaktnost omezenych mnozin v prostorech konecne dimenze (silna kompaktnost), reflexivnich (slaba kompaktnost), dualech k separabilnim prostorum (slaba-* kompaktnost).
Slaba kompaktnost v L^p, vnoreni L¹ do mer; konvergence norem; uniformne konvexni prostory a Kadec-Kleeova vlastnost.
Slabe konvergentni posloupnosti, ktere nekonverguji silne, konvergence lokalnich prumeru, oscilace a koncentrace.

3. Youngovy miry

Omezene posloupnosti v L^inf, veta o dualu k L¹(Q,C₀(R)), veta o existenci Youngovych mer.
Charakterizace rozdilu mezi silnou a slabou konvergenci pomoci Youngovych mer.

4. Div-curl lemma, Muratovo lemma

Div-curl lemma ve dvou dimenzich, v n dimenzich (ve strukture L²).
Kompaktnost mer a Muratovo lemma.

5. Aplikace na skalarni zakon zachovani

Parabolicka perturbace a stejnomerny L^inf odhad; odhad gradientu v L² zavisly na 1/epsilon; entropie a entropicka nerovnost.
Representujici Youngova mira; uziti div-curl a Muratova lemmatu k odvozeni Murat-Tartarovy rovnice; kompenzovana kompaktnost.
Dukaz dirakovskosti Youngovy miry pro skalarni rovnici v 1 dimenzi, Tartaruv a Vecchiho dukaz.

6. Zobecneni div-curl lemmatu

Charakteristicka mnozina prirazena linearnim diferencialnim formam; Tartarovo zobecneni div-curl na kvadraticke formy; dukaz pro dvojkovou strukturu pomoci Fourierovy transformace.

7. Hyperbolicke systemy a kompenzovana kompaktnost

System rovnic v jedne dimenzi, hyperbolicita; entropie a tok, entropicka podminka; parabolicka perturbace.
Odvozeni Murat-Tartarovy rovnice za predpokladu existence stjnomernych odhadu.
Systemy 2x2. Striktni hyperbolicita a ryzi (genuine) nelinearita. Riemannovy invarianty.
Laxova trida entropii s exponencialnim rustem; DiPernova veta o redukci nosice miry pro systemy 2x2.
p-system; prepis do Riemannovych invariantu; naznak dukazu o L^inf odhadu (nedokonceno).

Literatura

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