slideshow 3

Seminar on Partial Differential Equations

usually takes place each Tuesday at 09:00 in IM, rear building, ground floor
Chair: Šárka Nečasová, Milan Pokorný

Gradient Polyconvexity in the Framework of Rate-Independent Processes

Petr Pelech
Institute of Information Theory and Automation, Czech Academy of Sciences
Tuesday, 10. April 2018 - 9:00 to 10:00
The talk treats mathematical aspects of evolutionary material models for shape-memory alloys at finite-strains. The difficulty of related mathematical analysis consists in the non-linear and non-convex dependence of the energy on the deformation gradient. One possible way, how maintain the analysis tractable, is to suppose that the energy depends also on the second deformation gradient and is convex in it. We relax this assumption by using the recently proposed concept of gradient-polyconvexity. Namely, we consider energies which are convex only in gradients of non-linear minors (i.e. cofactor and determinant in three dimension) of the deformation gradient. As a result, the whole second deformation gradient needs not to be integrable. Yet, at the same time, the obtained compactness is sufficient and, moreover, additional physically desirable properties(e.g. local invertibility) can be shown. We extend the previous result for hyperelastic materials by incorporating a rate-independent... more

Nonlinear ellitpic and parabolic equations beyond the natural duality pairing

Miroslav Bulíček
Charles University
Tuesday, 27. March 2018 - 9:00 to 10:00
Many real-world problems are described by nonlinear partial differential equations. A promiment example of such equations is nonlinear (quasilinear) elliptic system with given right hand side in divergence form div f data. In case data are good enough (i.e., belong to L^2), one can solve such a problem by using the monotone operator therory, however in case data are worse no existence theory was available except the case when the operator is linear, e.g. the Laplace operator. For this particular case one can however establish the existence of a solution whose gradient belongs to L^q whenever f belongs to L^q as well. From this point of view it would be nice to have such a theory also for general operators. However, it cannot be the case as indicated by many counterexamples. Nevertheless, we show that such a theory can be built for operators having asymptotically the radial structrure, which is a natural class of operators in the theory of PDE. As a by product we develop new... more

Non-uniqueness of entropy solutions to the 2-d Riemann problem for the Euler equations

Simon Markfelder
Julius-Maximilians-Universität Würzburg
Tuesday, 20. March 2018 - 9:00 to 10:00
In this talk we consider the compressible (full) Euler equations in two space dimensions together with Riemann initial data. The issue of the talk is the question on uniqueness of weak entropy solutions to this problem. This issue has been studied for the isentropic Euler equations by E. Chiodaroli, C. De Lellis and O. Kreml (among others) and the aim is now to extend the results to full (i.e. non-isentropic) Euler. We consider a special class of Riemann data, namely those for which the 1-d self-similar solution consists of two shocks and possibly a contact discontinuity. We show that for this class there exist infinitely many weak entropy solutions, which are generated by convex integration. This is joint work with H. Al Baba, C. Klingenberg, O. Kreml and V. Mácha.

Measure-valued solutions and Navier-Stokes-Fourier system

Jan Březina
Tokyo Institute of Technology
Tuesday, 13. March 2018 - 9:00 to 10:00
Encouraged by the ideas and results obtained when studying measure-valued solutions for the Complete Euler system we introduce measure-valued solutions to the Navier–Stokes–Fourier system and show weak-strong uniqueness. Namely, we identify a large class of objects that we call dissipative measure–valued (DMV) solutions, in which the strong solutions are stable. That is, a (DMV) solution coincides with the strong solution emanating from the same initial data as  long as the latter exists.