Transport equation and compressible fluid flows 1000-1M18RTP
Firstly we will introduce the transport equation, examine its basic properties, and check when it conserves some basic quantities (e.g. the norm in the space L^p).
The next part of the lecture will be devoted to the theory of renormalizations for the transport equation (DiPerna-Lions theory) showing that the regularity of the transport coefficients in the appropriate Sobolev space ensures that the solutions have the so-called renormalization property.
After learning the necessary mathematical tools, we will examine the systems which include the transport equation. Among the considered equations there will appear: inhomogeneous Navier-Stokes equations, or compressible Navier-Stokes equations. We will also study biological models based on the transport equation.
Type of course
Learning outcomes
The student knows the transport equation and knows when the Cauchy problem is well-posed, knows the concept of the renormalized solutions. In addition, she/he is familiar with models of fluid mechanics, where the transport equation is a fundamental ingredient and knows how to apply the new tools to the analysis of these systems of equations.
Assessment criteria
Oral exam.
Bibliography
1. DiPerna, R. J.;Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), no. 3, 511–547
2. Lions, Pierre-Louis, Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 1996
3. Perthame, Benoît, Transport equations in biology. Frontiers in Mathematics.Birkhäuser Verlag, Basel, 2007
Additional information
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