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.

Some issues considered by the research group in applied mathematics within this PhD programme are the following:

*Inverse problems in geophysics and fluid dynamics*: theory and computation of inverse gravimetry problem; detection of immersed obstacles by transient fluid measurements; identification of blood flow parameters by inverse methods; theoretical and computational analysis of Krylov methods for solving linear inverse problems.

*Navier-Stokes equation*: study of N-S equation 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 are fast oscillating.

*Capillary surfaces*: study of the existence, the multiplicity, and the qualitative properties (such as regularity and stability) of solutions of anisotropic mean curvature equations governing capillarity phenomena, including the description of micro-electro-capillary systems.

*Nonlinear Schroedinger equation*: study of dispersion of radiation and nonlinear damping of discrete modes of Hamiltonian systems surrounded by continuous modes, in the frame of the discussion of the stability of solitons.

*Modeling the behaviour of the molecules of the life*: study of like-charge attraction of macromolecules immersed in electrolyte solutions.

**Inverse problems in geophysics and fluid dynamics**

The inverse gravimetric problem consists in detecting the earth's mass density, or its inhomogeneities, from measurements of its gravitational field. From a pure mathematical point of view, it is nothing else than the inverse problem for Newtonian potential, which is a well-known underdetermined ill-posed problem. For the purpose of this application, a relevant issue is to take into account mechanical constraints and additional geophysical information (known for the earth's structure) in order to reduce, or possibly eliminate, the underdetermination of the inverse problem.

Recent results of stability in the determination of immersed bodies in stationary fluids are available. Such techniques can be extended to non-stationary models for the determination of immersed bodies or unaccessible river beds, from measurements taken on the fluid's surface.

The dynamics of blood's circulation has been extensively developed, from the modeling and the computational points of view. Various questions arise in the identification of parameters, not directly accessible to measurements, for which the inverse problems methods may provide a solution.

In the framework of linear ill-posed problems, Krylov projection methods represent an essential tool since their development, which dates back to the early 50's. In recent years, the use of these methods in a hybrid fashion or to solve regularized problems has received a great attention especially for large-scale problems arising in imaging and more generally for problems arising from the discretization of equations involving compact operators.

For these kind of problems many Krylov type methods are generally very fast, since they exhibit a convergence rate which is quite close to the decay rate of the singular values of the operator. Moreover, the projective nature of these methods makes them attractive for solving regularized problems, since the projected operators inherits the spectral properties of the underlying one. As consequence, many existing parameter selection strategies can be efficiently employed, without additional computational effort. Important open questions concerns the theoretical properties of Krylov methods when coupled with a Tikhonov-like regularization, even if they are already used to this purpose. Moreover, the use of these methods as a tool for solving nonlinear inverse problems still needs a systematic theoretical and computational analysis.

**Capillary surfaces 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, their general lack of regularity, their possible discontinuous dependence on boundary data, the possible breaking of symmetry. 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 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 tools of nonlinear analysis (critical points theory, topological degree, bifurcation methods).

Proposed researches concern the study of the existence, the multiplicity, and the 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. In particular, micro-electro-capillary systems 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. Challenging problem is to build a mathematical theory (existence, multiplicity and qualitative properties) of these electro-capillary systems in the general N-dimensional case.

**Navier-Stokes equation**

One important mathematical model for the motion of a viscous fluid is the so called Navier-Stokes equation. Starting from the pioneering work of J. Leray in the 30's of the last century, the study of the N.-S. equation has become one of the central subjects in the theory of partial differential equations and it has been attacked with the more sophisticated tools of the functional analysis. Nevertheless the theory of N.-S. equation is far to be completely understood.

Some important tools in the study of N.-S. equation are related to Fourier analysis and more generally to the theory of pseudo and para-differential equations, and to harmonic analysis. In this setting, making use of the concept of generalized solutions (weak, mild, strong, etc.) which are element of more and more precise functional spaces (Sobolev, Besov, BMO etc.), many interesting results has been recently obtained.

Proposed researches concern existence and uniqueness for mild solutions to the N.-S. 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.

**Nonlinear Schroedinger equation**

The nonlinear Schroedinger equation (NLS) 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 NLS 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 NLS 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.

**Analytical and numerical 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.