Oberseminar Partielle Differentialgleichungen Detail

Vortragende Person/Vortragende Personen:
Dr. Sebastien Boyaval (Ecole des Ponts ParisTech)

Many Partial Differential Equations (PDEs) have been proposed to model viscoelastic flows. Seminal hyperbolic PDEs have been proposed by Maxwell in 1867 for 2D elastic fluids with stress relaxation, to ensure propagation of 1D shear waves at finite-speed while capturing the viscosity of real fluid continua. But actual computations of multi-dimensional viscoelastic flows using Maxwell’s PDEs have remained limited, at least without additional diffusion that blurs the hyperbolic character of Maxwell’s PDEs. We propose a new system of PDEs to model 3D viscoelastic flows of Maxwell fluids. Our system, quasilinear and symmetric-hyperbolic, unequivocally models smooth flows on small times, while ensuring propagation of waves at finite-speed. It is inspired by the KBKZ integral reformulation of Maxwell PDEs (proposed independently by Kaye and Bernstein, Kearsley, Zapas), but it is more versatile because purely differential, using additional variables for the fluid material properties. Our system rigorously unifies fluid models with elastodynamics for compressible solids, and it can be manipulated for various applications of the viscoelastic flow concept in environmental hydraulics (shallow-water flows) or materials engineering (non-isothermal flows).