The speaker: Professor Pavel Plotnikov (Lavrentyev Institute of Hydrodynamics of Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia)
Title of the talk: “Concentrations and singularities of solutions to the Navier-Stokes equations of compressible isentropic flows”.
Time: Thursday, 8 of April 2021, 15:00 - 16:00.
Abstract. The talk is devoted to the theory of weak solutions to compressible Navier-Stokes equation with critical and subcritical adiabatic constants. In 2001, Feireisl, Novotny, and Petzeltova proved the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic flows in the three space dimension on condition that the adiabatic constant is greater than critical value 3/2. The critical and subcritical cases are still poor investigated. The main difficulty lies in the fact that in the critical and subcritical cases, the finite energy can be concentrated on sets of arbitrarily small measure. This leads to the so-called concentration problem. In the present work, we prove the absence of concentrations of the kinetic energy tensor in the critical case. We also give the derivation of estimates of non-stationary potentials of the pressure function. These estimates allow us to estimate from below the Hausdorff dimension of the support of the concentrations-defect measure. The case of rotationally symmetric solutions with adiabaticconstant equals 1 is studied in details. In this case we prove that the concentrations-defect measure of the kinetic energy tensor is a matrix-valued measure, which is concentrated on the symmetry axis and depends only on the time variable. In particular, the divergence of the concentrations-defect measure equals zero.
ZOOM Id: 836 7313 8944
We are pleased to inform you that the information on previous talks is available on the seminar web page https://researchseminars.org/seminar/Cafe_Analysis_and_Fluid see Past Talks section.
Organizing Committee: Professors Zhen Lei, Mikhail Korobkov, Adele Ferone, Remigio Russo and Konstantin Pileckas.
This seminar is organized by Fudan University (Shanghai, China) with the collaboration of Universita degli studi della Campania "Luigi Vanvitelli" (Caserta, Italy) and Vilnius University (Vilnius, Lithuania). This seminar is meant to gather renowned experts working on Geometric and Real Analysis with applications to PDEs and Mathematical Physics and represents an opportunity for presenting high-impact research, as well as facilitating the exchange of ideas between researchers.