Admissible singular solutions to gas dynamics systems with non-positive pressure

Abstract

<p>Karakteristika hiperboličnih sistema zakona odrržanja je da čak i u slučaju glatkog po-četnog uslova re&scaron;enja uglavnom razvijaju prekide u konačnom vremena. Zbog toga se posmatraju slaba re&scaron;enja koja dati sistem zadovoljavaju u distributivnom smislu i mogu biti čak i neograničena &scaron;to se ispoljava kroz pojavu Dirakove delta funkcije u re&scaron;enju. U ovoj disertaciji se akcenat stavlja na analizu protoka sti&scaron;ljivog neviskoznog fluida koji ne menja pravac prilikom kretanja. Protok je opisan Ojlerovim sistemom iz gasne dinamike koji se sastoji iz zakona održanja mase, količine kretanja i energije, dok su karakteristike fluida određene konstitutivnim relacijama. U slučaju izentropskog ili izotermnog protoka sistem se svodi na zakone održanja mase i količine kretanja. Glatka re&scaron;enja takvog sistema automatski zadovoljavaju zakon održanja energije, dok prelaskom na slabu formulaciju dolazi do gubitka energije. Za predstavnike sistema gasne dinamike sa nepozitivnim pritiskom su uzeti sistem gasne dinamike bez pritiska i model za&nbsp; Čapliginov gas i njegova uop&scaron;tenja. Data su re&scaron;enja Rimanovih problema za te sisteme koja se mogu predstaviti kao kombinacija klasičnih elementarnih talasa i senka talasa koji aproksimiraju re&scaron;enja u obliku delta udarnih talasa i koji omogućavaju re&scaron;avanje početnog problema koji u početnom uslovu sadrži delta funkciju. Na primeru modela za uop&scaron;ten Čapliginov gas dokazano je da uslov prekompresivnosti nije jači od entropijskog uslova, &scaron;to je prvi takav rezultat u literaturi. Dalje su kori&scaron;ćena re&scaron;enja Rimanovih problema, kao i problema singularne interakcije i dat je algoritam za konstrukciju globalnog dopustivog približnog re&scaron;enja početnog problema za sistem gasne dinamike bez pritiska. Algoritam je univerzalan i ideja se može pro&scaron;iriti na veliki broj sistema zakona održanja i veliki broj početnih uslova. Diskutovane su promene energije u približnom re&scaron;enju i posle interakcija. Dobijeno približno re&scaron;enje slabo konvergira u prostoru Radonovih mera sa predznakom.</p>

<p>&nbsp;</p><p class="MsoNormal">A solutions to hyperbolic conservation laws systems starting out as smooth often develop singularities in a finite time. As a consequence, we are forced to look for weak solutions that satisfy the system in distributional sense. Those solutions are often unbounded, which is expressed through the appearance of Dirac delta function. In this theses we study a one-dimensional, compressible and inviscid flow of a fluid. The process is described by compressible Euler gas dynamics system which consists of conservation laws of mass, linear momentum and energy, while the characteristics of the fluid are described using constitutive relations. In the case of isentropic or isothermal flow the system reduces to conservation laws of mass and linear momentum. The energy is conserved for smooth solutions to such systems, but while passing to the weak formulation the energy is being dissipated. As representatives, we&nbsp; consider pressureless gas dynamics system, as well as Chaplygin gas model and its generalizations. We give the solutions to Riemann problems which can be represented as a combinations of classical elementary waves and shadow waves that approximate the solutions in the form of delta shock and allow as to solve the problems with initial data containing delta function. We use generalized Chaplygin gas model as demonstration of the fact that overcompressibility condition is not stronger that entropy condition, which is the first result of that kind in the literature. Further, we use solutions&nbsp; to the Riemann problems, as well as singular interaction problems to give the algorithm for construction of global admissible approximate solution to the pressureless gas dynamics initial value problem. The algorithm is universal and idea can be applied to large number of conservation laws systems and large number of initial data. We discuss&nbsp; energy changes in approximate solution and after the interactions. 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