IV. Partial differential equations
My research in the field of PDEs is focused (amongst others) on problems related to the parabolic p-Laplacian
or more general problems as
with vector fields
(Carathéodory functions) satisfying the monotonicity condition
and growth condition
where
Moreover, the inhomogeneities satsify
My research interest beside problems based on parabolic equations is focused on obstacle problems and systems, especially on existence, regularity, stability theory and nonstandard growth problems.
Motivation of parabolic problems and nonstandard growth problems with p(x)- or p(x,t)-growth exponent:
The study of the parabolic p-Laplacian is motivated from fluid mechanics. More precisely, the parabolic p-Laplacian is a mathematical model of the behaviour of compressible fluid in a homogeneous isotropic rigid porous medium. Starting with a continuity equation and using a nonlinear Darcy law one can derive the parabolic p-Laplacian, cf.
Wu, Zhao, Yin and Li. Nonlinear diffusion equations. World Scientific, 2001.
In general, parabolic problems are often motivated by physical aspects. In particular, evolutionary equations and systems can be used to model physical processes, e.g. heat conduction or diffusion processes. For example the basic equation of fluid mechanics is the Navier-Stokes equation. Some properties of solutions to the system of a modified Navier-Stokes equation describing electro-rheological fluids are studied in [3]. Such fluids are of high technological interest, because of their ability to change the mechanical properties under the influence of exterior electro-magnetic field, see [8]. Many electro-rheological fluids are suspensions consisting of solid particles and a carrier oil. These suspensions change their material properties dramatically if they are exposed to an electric field, see [9]. The stationary model with p(x)-growth is studied, e.g. in [2]. Furthermore, for the restoration in image processing one uses also diffusion models with nonstandard growth condition, please see [1,5-7]. In the context of parabolic problems with p(x,t)-growth applications are, e.g. the models for flows in porous media [4] or nonlinear parabolic obstacle problems, please see my publication list.
R. Aboulaich, D. Meskine and A. Souissi. New diffusion models in image processing. Computers & Mathematics with Applications, 56(2008), no.4, p.874-882.
E. Acerbi and G. Mingione. Regularity results for stationary electro-rheological fluids. Arch. Rational Mech. Anal., 164(2002), no.3, p.213-259.
E. Acerbi, G. Mingione and G.A. Seregin. Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21(2004), no.1, p.25-60.
S. Antontsev and S. Shmarev. A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions. Nonlinear Anal., 60(2005), p.515-545.
Y. Chen, S. Levine and M. Rao. Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics, 66(2006), no.4, p.1383-1406.
F. Li, Z. Li, L. Pi and Ling, Variable exponent functionals in image restoration. Appl. Math. Comput. 216(2010), no.3, p.870-882.
P. Harjulehto, P. Hästö, V. Latvala and O. Toivanen. Critical variable exponent functionals in image restoration. Applied Mathematics Letters, 26(2013), no.1, p.56-60.
M. Růžička. Electrorheological fluids: modeling and mathematical theory. Springer-Verlag, Heidelberg, 2000.
M. Růžička. Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math., 49(2004), no.6, p.565-609.
For an overview on nonstandard growth problems, please see the books:
Stanislav Antontsev and Sergey Shmarev. Evolution PDEs with Nonstandard Growth Conditions. Atlantis Studies in 258 Differential Equations, Atlantis Press, 2015.
Lars Diening, Petteri Harjulehto, Peter Hästö and Michael Růžička. Lebesgue and Sobolev spaces with variable exponents. Springer-Verlag, 2011.
Selected presentations: