## Partial differential equations: from theory towards applications

The fundamental laws upon which the study of earth science and fluid mechanics is based are generally expressed by partial differential equations, often nonlinear and highly complex: their study requires the application of various methods of advanced mathematics and is a research field of high theoretical and practical relevance.

The qualitative and quantitative study of such models requires the development and the application of sophisticated mathematical tools, and it represents a relevant and topical research field from the mathematical point of view.

A list of issues considered by the research groups in mathematics within this PhD programme follow.

**Capillary equations and related topics **

In fluid mechanics capillarity is the ability of a liquid to flow in narrow spaces without the assistance of, or in opposition to, external forces. It occurs because of inter-molecular attractive forces between the liquid and the solid surrounding surfaces. More in general, capillarity is the set of phenomena due to interactions between the molecules of a fluid and of another material, which can be a solid, a liquid or a gas, on their separation surface (interface), said capillarity surface. With the advent of miniaturized technology, also systems combining the effects of electrostatic and capillary forces are receiving more and more attention, in order to fully understand how these electro-capillary systems operate at small scales. In very recent years some new models have been introduced with the aim of taking into account of the full effects of capillarity.

From the mathematical point of view these phenomena are governed by highly nonlinear partial differential equations, which exhibit various peculiarities: the possible lack of existence or uniqueness of solutions, the general lack of regularity, the possible development of singularities either in the interior or at the boundary. The study of these equations, which involve the mean curvature operator, is a classical subject, but still of great actuality, due to the large number of open problems, having a great interest even from the point of view of the applications.

Their analysis requires the use of sophisticated tools of advanced mathematics: geometric measure theory, variational methods in spaces of functions of bounded variation, non-smooth analysis techniques, combined with other typical techniques of nonlinear analysis, such as critical points theory, topological degree, bifurcation techniques. Proposed research projects include, but are not limited to, the study of:

a) existence, non-existence, multiplicity, and qualitative properties (such as regularity and stability) of solutions of the capillarity equations, on bounded or unbounded domains, which appear as sub-critical points (not necessarily minimizers) of the functional representing the mechanical energy of the system, with special attention to superlinear/sublinear indefinite problems.

b) micro-electro-capillary systems, which lead to a class of partial differential equations involving the mean curvature operator with a singular perturbation, whose study has been, for the moment, confined to low dimensional problems, possibly with symmetries. A challenging problem is to build a mathematical theory (existence, multiplicity and qualitative properties) of these anisotropic mean curvature equations governing electro-capillary systems in the general N-dimensional case.

**Euler and Navier-Stokes equations**

Important mathematical models for the motion of a fluid are the Euler and the Navier-Stokes equations. Concerning Euler equations, very deep results have been obtained recently with the use of the so called convex integration. This technique has been utilized in proving the non-uniqueness of weak solutions. This mathematical property suggests that new ways of selection of solutions having a physical meaning are necessary. At the same time a new perspective on the proposed model would be of great interest. Proposed researches concern non-uniqueness for weak solutions to the Euler equation in different situations, e.g., in domains with boundaries, in infinite domains, with non-isotropic viscosity, in presence of the action of the Coriolis force.

Concerning Navier-Stokes equations, starting from the pioneering work of J. Leray in the 30's of the last century, the study of this topic has become another central subject in fluid mechanics. Some important tools in the study of Navier-Stokes equation are related to Fourier analysis and more generally to the theory of pseudo and para-differential equations, to harmonic analysis, to non-linear profile decompositions and to backward uniqueness for parabolic equations. In this setting, there have been recent interesting results utilizing notions of generalized solutions (weak, mild, strong, etc.) belonging to more refined functional spaces (Sobolev, Besov, BMO, etc.).

Proposed researches concern existence and uniqueness for mild solutions to the Navier-Stokes equation in various different situations e.g. in domains with boundaries, in infinite domains, with non-isotropic viscosity, in presence of the action of the Coriolis force, when initial data are given in Besov spaces of negative index, or in BMO spaces, or which are fast oscillating.

**Inverse problems for PDEs towards applications **

One leading aspect of nowadays research in Partial Differential Equations is the one of Inverse Problems. The notion of inverse problem arises when, starting with a given problem, called the direct problem, the role among part of the data and part of the unknowns is exchanged. The unknowns, instead of being the solution of the equations modelling the direct problem, are, depending on the cases under examination, coefficients, inhomogeneous terms, nonlinearities of the same equations, or else parameters of other sorts, which for instance, characterize the geometry of the domain to which the equations apply.

Research on Inverse Problems is driven by applications. In fact, whenever indirect means of measurements, data collection or imaging are used, from the point of view of mathematical modelling, one gets into an inverse problem. Such occurrences are encountered in daily practice, in the most diverse fields, for instance, image reconstruction in microscopy as well as in astronomy, in seismic data processing and innumerable industrial applications such as non-destructive testing of materials by mechanical, thermal, electrical or electromagnetic measurements. The future of scientific and technological development is in many ways linked with the development of Inverse Problems.

Usually, Inverse Problems do not satisfy the Hadamard postulates of well-posedness, and, often, they are extremely nonlinear. In most cases, in order to overcome such obstacles, it is impossible to invoke all-purpose, readymade, theoretical procedures. Instead, it is necessary for each problem, or group of problems, to single out the suitable approach and trade-off the intrinsic ill-posedness with original ideas and a deep use of mathematical methods from various areas.

The mathematical treatment of Inverse Problems involves several steps from theoretical aspects of uniqueness and conditional stability, to more concrete computational aspects of devising algorithms apt to handle incomplete and noisy measurements. Depending on the advancement of knowledge on the various topics treated, one needs to face all such kind of steps, towards the complete resolution of Inverse Problems.

Major objects of investigation will arise from medical imaging, structural building testing and design and geophysical exploration. More specifically, in this last area, research shall focus on: theoretical analysis of stability, convergence of iterative algorithms, construction of ad-hoc cost functionals for inverse boundary value problems arising in earth sciences, such as the so-called Direct Current method for geo-electrical prospection and for Full Waveform inversion from acoustic or electromagnetic measurements.

**Nonlinear elliptic equations and systems**

Nonlinear elliptic equations and systems appears naturally in different context of pure and applied science. We study a common and fundamental aspect of this problems, namely existence, non-existence and qualitative aspects of the equilibria of related evolution problems. A fundamental tool for the general understanding of the stability of an evolution phenomena is the precise knowledge of the stability of the so-called steady states (equilibria).

Specific research projects include:

a) quasi-linear elliptic systems: qualitative properties, a priori estimates, asymptotic behavior, Liouville theorems in non-Euclidean structures

b) higher order semilinear problems: coercive equations and inequalities, a priori estimates and representation of solutions for problems in R^n; polisuper-harmonicity properties.

c) representation formulas for solutions to linear sub-elliptic problems: degenerate and singular problems.

**Nonlinear Schroedinger equation**

The nonlinear Schroedinger equation appears naturally in the context of the theory of propagation of light in nonlinear optical fibers, in the theory of Bose-Einstein condensation, in the study of lasers. While generally local existence and well posedness is known, much has yet to be understood about the long time evolution of its solutions. The nonlinear Schroedinger equation exibits a diverse set of interesting patterns (solitons, kinks, vortices, breathers etc.). The most famous open problem is the soliton resolution conjecture, which is not know for any nonlinear Schroedinger equation with solitons, and which states that any finite energy solution for generic equations breaks up in a finite number of solitons diverging the one from the other.

Here we are focused on problems of stability of solitons. There is a rather complicated interaction between dispersing radiation with some discrete oscillators which are damped because they lose energy through nonlinear interaction with radiation. Dispersion of radiation is a linear phenomenon, but when the nonlinearity is strong, it can be difficult to prove.

**Numerical-analytical problems for integro-differential equations arising in biophysics**

Some aspects of the macromolecules of the life, like DNA, can be understood by considering their behaviour as charged molecules interacting with ions. They are modelized as two like-charged surfaces immersed in a solution composed of rod-like ions.

The presence of counter-ions (ions of the opposite charge with respect to the surfaces) in the solution may give rise to a non-intuitive attraction between the equally charged macromolecules.

Normally, the multivalent ions in the electrolyte solutions are not point-like, but they posses internal structures, i.e. the individual charges of the ion are located at separated positions within the ion.

In case of rod-like ions, a Poisson-Boltzmann theory, a combination of electrostatic theory and statistical mechanics, provides a mathematical model given by a second order integro-differential equation equipped with Neumann boundary condition.

Models with partial differential equations arise by considering the macromolecules as cylinders rather than surfaces.

Interesting analytical questions, like existence and uniqueness of the solution, and numerical questions, like convergence of the numerical schemes, are involved in such equations.