Members:
- Francesca Crispo
- Angelica Pia Di Feola
- Paolo Maremonti
- Giulio Starita
- Alfonsina Tartaglione
Keywords: - Fluid Dynamics
- Navier-Stokes equations: existence, uniqueness, suitable weak solutions, regularity, energy equality,steady flows, boundary data;
- Non-Newtonian power-law models: global regularity, high regularity, extinction properties;
- Elasticity theory: elastic solids; viscoelastic solids; stress relaxation; creep ;
- Diffusion phenomena: singular p-Laplacian system, p(t,x)-Laplacian system, global regularity, high regularity, Burgers equations.
Research Profile:
The Mathematical Physics group at the Department of Mathematics and Physics of University of Campania “Luigi Vanvitelli” has an extensive experience in the analytic theory of Newtonian and non-Newtonian fluids, in Elasticity theory, and Mathematical Models for Continuum Mechanics which has led to a wide literature published in international journals over the years.
- Fluid Dynamics
Navier-Stokes equations
The Navier-Stokes equations are a model for the dynamics of an incompressible and viscous fluid. As it is well known this model is one of the most interesting to describe fluid phenomena, and hence one of the most employed in the applications.
In spite of this, the 3D mathematical theory of the Navier-Stokes equations is still an open problem, investigated by many specialists of PDEs with great interest in the last decades, also sponsored by the millennium prize.
The chief mathematical question is to understand if the Hadamard principles of well posedness hold. It is known that for all initial data with finite kinetic energy there exists a weak solution to the Navier-Stokes equations, which is defined for all time instant. But the weak formulation is not satisfactory, as a regularity result (as for 2D) is not achieved. That is, we do not know if a weak solution is unique and regular in such a way to be consistent with the laws of continuous mechanics leading to the model.
Hence from a mathematical-physical point view the importance to investigate on the regularity of a weak solution is evident.
The mathematical issue consists in achieving a regularity criterium for weak solutions.
There are several attempts in this sense, all based on scaling invariant metrics. Actually, they work in the case of small data. But among these regularity criteria we can single out the one due to Caffarelli-Kohn-Nirenberg, that works locally in space-time variables connected by the parabolic scaling. Further developments are possible in the wake of this criterium.
Members of the group also analyze issues like existence, uniqueness and regularity for steady solutions to the various boundary value problems (with adherence or slip boundary conditions) governed by Navier-Stokes equations, as well as their Stokes and Oseen linearizations, under different regularity hypotheses on the boundary and on the boundary data.
Non-Newtonian power-law models
There are plenty of experimental studies highlighting for many incompressible fluids, like blood, the viscosity may decrease or increase with increasing shear rates in a suitable range of shear rates.
Hence the Navier-Stokes model is not anymore satisfactory, and these fluids are power-law fluids, characterized by a generalized viscosity that is a power of the share-rate. An increasing or decreasing generalized viscosity describes, respectively, shear thickening fluids (like batter) or shear thinning fluids (like blood, latex paint, lubricants). The industrial, medical, biological and engineering applications of such fluids motivate the growing attention devoted to their mathematical study and brings the importance of mathematical analysis in these applications areas.
Despite their applicative importance and considerable achievements from experimental point of view, the mathematical analysis of the models for these fluids is still far from being complete, and, on the other hand, it is crucial for the validation of the models. While the existence of weak solutions is well understood, the uniqueness and global regularity still have partial contributions.
Members of the group plan to continue the mathematical analysis of these models, in the sense of improvements of uniqueness and regularity results of solutions. Our interest in regularity is directed towards continuity results of the velocity gradient, and high regularity, in the sense of high integrability of the second derivatives, for solutions of boundary and initial boundary value problems. For the nonlinear character of the operator this appears to be a remarkable regularity property. - Elasticity theory
Members of the research group are interested in the study of elastic and viscoelastic bodies.
The interest for elastic solids regards the analysis of the properties of the solutions to the equation of motion by evaluating their extension to unbounded body configurations.
The equilibrium problem for elastic bodies is also analyzed to try and get existence and uniqueness results to the boundary value problems (with displacement, traction and other boundary conditions), besides asymptotic (in space) properties of the solutions to the equilibrium equations.
The viscoelastic solids are also examined: nonlinear models are developed governing the stress relaxation and creep behavior of bodies with material properties observed in biological tissues. - Diffusion phenomena
Members of the group are interested in the study of the p-Laplacian system, both in the elliptic and in the parabolic case. As the principal non-linear operator is close to the one for power-law fluids,we mainly study high regularity properties of solutions with the goal of trying to next extend them to the fluid dynamics setting, wondering whether the addition of a new dynamic variable, the pressure, and a new equation arising from the incompressibility of the fluid, allow us to preserve the same properties obtained for solutions to the “corresponding” elliptic or parabolic systems.
We are also involved in studying properties of solutions of p(x) and p(t,x)-Laplacian systems. These issues naturally fit into the framework of the regularity of minimizers of functionals with non-standard growth. We also mention that similar kind of systems govern the steady or unsteady motion of electrorheological fluids.
Further, we consider the Burgers equation with principal operator given by the p-Laplacian. Classical Burgers equations (with the Laplacian as principal operator) are considered both as a model for various applications and as an approximation of the Navier-Stokes system.
The aim is to test whether in the nonlinear singular case some fundamental results known for the classical Burgers equations, such as the maximum principle, can be reproduced.
Recent Publications
