Entropijska rješenja kvazilinearnih jednadžbi prvog reda

master's thesis

• Title (english): Entropic solutions of first order quasilinear equations
• Author: Ivana Lončar
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj rad obrađuje pojam entropijskih rješenja kvazilinearnih jednadžbi prvog reda, te pokazuje njihovo postojanje i jedinstvenost. Kvazilinearne parcijalne diferencijalne jednadžbe prvog reda često se javljaju u matematičkim modelima jer zakoni sačuvanja pripadaju u tu skupinu jednadžbi. Kako općenito takve jednadžbe nisu dovoljno glatke, promatraju se slaba rješenja, odnosno pripadna potklasa entropijskih rješenja. Prvo poglavlje daje pregled teorije za rješavanje Hamilton-Jacobijeve jednadžbe i pripadnog inicijalnog problema te navodi neke tehničke tvrdnje koje koristimo kasnije. Drugo poglavlje se bavi skalarnim zakonima sačuvanja kao najjednostavnijem slučaju kvazilinearne jednadžbe. Ovdje je uveden Laxov entropijski uvjet pomoću kojeg je definiran pojam entropijskog rješenja. Zatim je korištenjem Lax-Oleinik formule pokazano postojanje i jedinstvenost takvog rješenja. Konačno, razvijena teorija je primijenjena na Riemannov problem. Treće poglavlje ilustrira ekvivalentnu formulaciju entropijskog rješenja. Pojam entropijskog rješenja obrađen je kroz koncept entropije i toka entropije. Uvodi se entropijska nejednakost kao način prepoznavanja fizikalnih rješenja medu svim slabim rješenjima. Na kraju smo Riemannov problem ponovno riješili koristeći ovaj pristup te očekivano dobili isto entropijsko rješenje.

Abstract (english)

In this thesis we have studied the notion of entropy solutions of first-order quasilinear partial differential equations (PDEs), and have shown their existence and uniqueness. Firstorder quasilinear PDEs often occur in mathematical models because the conservation laws belong to this group of equations. As in general such equations are not smooth enough, weak solutions, ie. the corresponding subclass of entropy solutions are observed. The first chapter provides an overview of the theory for solving the Hamilton-Jacobi equation and the corresponding initial problem, and lists some technical statements that we use later. The second chapter deals with scalar conservation laws as the simplest case of a quasilinear equation. An entropy condition is introduced here by which the notion of an entropy solution is defined. Then, the existence and uniqueness of such a solution was demonstrated using the Lax-Oleinik formula. Finally, the developed theory was applied to the Riemann problem. The third chapter illustrates the equivalent formulation of the entropy solution. The notion of entropy solution is studied through the concept of entropy and entropy flux. Entropy inequality is introduced as a way of recognizing physical solutions among all weak solutions. In the end, we solved the Riemann problem again using this approach and, as expected, obtained the same entropy solution.

• Keywords: kvazilinearne jednadžbe prvog reda; Hamilton-Jacobijeve jednadžba; Lax-Oleinik formula; Riemannov problem
• Keywords (english): quasilinear partial differential equations (PDEs); Hamilton-Jacobi equation; Lax-Oleinik formula; Riemann problem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:329734
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-02-26 13:25:59