Session 24. Nonlinear PDEs with applications in materials science and biology

Applications of Orlicz spaces in the theory of nonlinear PDE: non-Newtonian fluids and abstract problems

Aneta Wróblewska-Kamińska, Institute of Mathematics, Polish Academy of Sciences, Poland
We are interested in the existence of solutions to strongly nonlinear partial differential equations. We concentrate mainly on problems which come from dynamics of non-Newtonian fluids of a nonstandard rheology and abstract theory of elliptic and parabolic equations. In considered problems the nonlinear highest order term is monotone and its behaviour -- coercivity/growth condition -- is given with help of some general convex function. In our research we would like to cover both cases: sub- and super-linear growth of nonlinearity. Such a formulation requires a general framework for the function space setting, therefore we work with non-reflexive and non-separable Orlicz and Musielak-Orlicz spaces. Within the presentation we would like to emphasise problems we have met during our studies, their reasons and methods which allow us to achieve existence results.
References
  1. A. Wróblewska-Kamińska, Unsteady flows of non-Newtoniana fluids in generalized Orlicz spaces , Discrete and Continuous Dynamical Systems - A, 33 (2013), no 6, 2565-2592.
  2. E. Emmrich, A. Wróblewska-Kamińska, Convergence of a full discretization of quasilinear parabolic equations in isotropic and anisotropic Orlicz spaces . SIAM Journal on Numerical Analysis, 2013.
  3. A. Wróblewska-Kamińska, Existence result for the motion of several rigid bodies in an incompressible non-Newtonian fluid with growth conditions in Orlicz spaces . Nonlinearity 27 (2014) 685-716.
  4. \newblock P. Gwiazda, P. Wittbold, A. Wróblewska, A. Zimmermann. \newblock {\em Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces . Journal of Differential Equations, 253 (2012) 635--666.
  5. P. Gwiazda, P. Minakowski, A. Wróblewska-Kamińska. Elliptic problems in generalized Orlicz-Musielak spaces, Central European Journal or Mathematics, 10, no. 6 (2012), 2019-2032.
  6. P. Gwiazda, A. \'Swierczewska-Gwiazda, A. Wróblewska, Generalized Stokes system in Orlicz spaces, Discrete and Continuous Dynamical Systems A, 32 (2012), Issue 6, 2125-2146.
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