The project CRISIS is conceived to give rigorous mathematical grounds to the description of heterogeneity effects, singular behaviours and multi-scale processes which occur and interplay in several real world phenomena related to fluid mechanics, and improve their theoretical understanding.
We have in mind situations which range from the context of geophysical flows to the one of collective dynamics.
Heterogeneities are ubiquitous in nature:
they may appear in complex phenomena, as a result of the interaction of different agents and components (e.g. density or temperature variations, presence of a magnetic field),
or at the boundary of the physical domain, where interactions with the external world occur.
The key observation is that non-homogeneities are often responsible for the emergence of singular structures. The word "singular" refers here to some lack of regularity, typically due to high concentrations and/or fast oscillations, which may involve, for instance, the vorticity of the fluid
(think e.g. to typhoons), the fluid density (like in mixtures, in multi-phase flows or in crowd dynamics), or defects in complex fluids (like liquid crystals).
In all those cases, some quantity is strongly localised in some region of space, and experiences a jump across the interface delimiting that region.
In many situations the interface is simply advected by the flow, but in some cases of interest (arising e.g. in saturation phenomena) it presents a free boundary, meaning that
some mass exchanges may happen between the two regions separated by the interface.
For applications, it is of upmost importance to give a qualitative description of the dynamics of the interface; yet, the mathematical difficulties for doing that are deep and intricate,
since this requires to go beyond the classical well-posedness theory for non-linear PDEs.
Another fundamental observation, which stands at the core the project CRISIS, is that heterogeneity may appear also in multi-scale processes, as a result of
the interplay between several factors, which contribute to the mean motion by acting at different scales (in time and/or in space).
Such a complexity makes the derivation of reduced models necessary, both for theoretical and application purposes.
From the physical viewpoint, the importance of the various factors is assessed by dimensional analysis, by introducing some adimensional parameters,
whose values may be very large or very small, depending on the specific considered regime. The usual way for obtaining reduced models consists in performing asymptotic expansions
with respect to those parameters, and retaining only the leading order terms in the equations; yet, this approach is merely formal.
On the contrary, the rigorous justification of those approximations generates several mathematical difficulties: first of all, one is led to facing singular perturbation problems;
in addition, new analytical tools are needed to capture, at the level of the limit equations, the multiple scales which interact in the system.
In the context outlined above, the primary goals of the project CRISIS are:
- (G1) to give a better understanding of the effects due to heterogeneities on fluid motion;
- (G2) to improve the description of the dynamics of interfaces;
- (G3) to make progress in the understanding of the role of the boundary;
- (G4) to rigorously derive reduced models for fluid flows in presence of multiple scales.
The proposed research plan focuses on several aspects of the mathematical theory of non-homogeneous fluids, and consists of three main working programmes:
- (WP1) the study of systems of non-homogeneous fluids in a low/critical regularity framework;
- (WP2) the analysis of certain free-boundary problems;
- (WP3) the study of singular limits in the context of geophysical flows.
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