Main website: **https://conferences.cirm-math.fr/3108.html**

**Sarka Necasova,**Institute of Mathematics, Czech Academy of Science, Czech Republic

**Raphael Danchin**, Universite Paris-Est Creteil Val-de-Marne, France

Our conference will be dedicated to recent research in theoretical mathematical fluid mechanics. Our aim is to bring together mathematicians from around the world working on four different aspects of mathematical fluid mechanics: possible blow-up, non-uniqueness results, flows in moving domains or fluid structure interaction, and the study of models for multi-component flows that have been derived recently.

**1. Blow-up or regularity? **

In his seminal paper of 1934, J. Leray established that for the incompressible Navier-Stokes equations (N-S), any finite energy divergence free velocity field gives rise to at least one global-intime weak solution satisfying the energy inequality. Whereas in dimension two, these solutions are unique and regular, these questions are still open in dimension three. In particular, the question of existence of a global regular solution for arbitrarily large regular data is one of the seven Millennium Prize Problems (Clay Mathematics Institute 2000).

Two main approaches have so far dominated the effort to solve this question. The first one consists in proving uniqueness and regularity of weak solutions constructed by Leray, and the second one is to show the local in time existence of a strong solution and then to prove that it can be possibly extended for all positive times. A key idea is the control of scaling invariant quantities, which are typically appropriate norms. Analogous regularity criteria have also been pointed out for more elaborated models like e.g. the incompressible Navier-Stokes system with variable density or the compressible Navier-Stokes system.

**2. Non-uniqueness **

It is easy to show formally that smooth solutions of the Euler equations (E) governing the evolution of incompressible inviscid fluids have constant energy. In dimension three, Onsager’s conjecture (1949) states that the minimal regularity of solutions ensuring this property is the spatial H¨older regularity C1/3(R3). On the one hand, in 1994, Constantin and Titi proved that, indeed, every solution of (E) with spatial regularity index greater than 1/3 conserves the kinetic energy. On the other hand, recently, Buckmaster, de Lellis-Szekelyhidi and Vicol showed that for every positive function E(t) on [0, T) and all 0 < α < 1/3, there exists a weak solution u ∈ Cα(T3 × [0, T]) of (E) with the kinetic energy E(t) at time t. This shows that the weak solutions generally do not preserve the kinetic energy.

It is a big challenging problem whether an analogous non-uniqueness result does hold for weak solutions of (N-S) in or outside the Leray-Hopf class L∞(0, T;L2) ∩ L2(0, T;H1). In this context, Albritton-Bru´e-Colombo proved very recently that there exist two distinct Leray-Hopf weak solutions with initial state zero and the same non-zero external force. The question of whether a similar result is also true if the external force is zero is currently a subject of an intensive research.

**3. Flow in moving domains and fluid-structure interaction **

The mathematical theory of incompressible flows in a moving domain goes back to the fundamental work of Fujita and Sauer 1970 (viscous case) and Kozono 1985 (inviscid case). Since then, many mathematicians have investigated the problem of existence, uniqueness and other properties of the mathematical model of a fluid flow around a translating rigid body. The problem of motion of a rigid body with a cavity, filled by a fluid, goes back to the famous Zhukovsky conjecture that the system itself stabilizes for time going to infinity regardless of the shape of the cavity or the type of fluid. The stability result, confirming Zhukovsky’s hypothesis, was recently proven by Galdi et al. in the incompressible case and, also in the compressible case.

Beginnings of mathematical analysis of systems that describe problems of fluid-structure interaction are of later date. The simplest case, when the structure is represented by one or more rigid bodies, is well developed for viscous incompressible fluids for t < t0, or possibly also t ≤ t0, where t0 is the first collision time. Nowadays there exist plenty of results in the case of an incompressible or compressible fluids and an elastic/rigid structure.

**4. Multi-components flows **

Several new rheological models were introduced during the last few years. One of the prototypes is the so-called compressible bi-fluid model which has drawn immense interest. The theory on global existence has been developed quite recently. For instance, Vasseur et al. established the global existence of weak solutions for a bi-fluid model when the densities are comparable.

Another model, which has currently been intensively investigated, was introduced in the seminal work by Fick, who summarized the diffusion of matter in liquids, and formulated a relation between the molar mass flux, the molar concentration of constituents and the phenomenological coefficients. Many mathematical results have also been proved in the direction of multi-component diffusion systems.

The themes of the proposed international conference yield an exciting challenge for all invited scientists, from young doctoral students and post-doctoral researchers to leading experts in the field. We plan to have 12 plenary speakers and around 25 speakers, who will give shorter talks, including a few post-doctoral students. We expect 70 participants. We also plan to organize a poster session for PhD students and early postdoctoral researchers in the vestibule of the lecture hall. This means that not less than 50 participants out of expected 70 will have an opportunity to present their recent research during the conference. This number is the maximum one that we can afford to avoid parallel sessions.

**Maria Colombo**(EPFL, Lausanne)**David Gerard-Varet**(Universit´e Paris Cite)**Alexander Kiselev**(Duke University, Durham, NC)**Irena Lasiecka**(The University of Memphis),**Xian Liao**(Technische Universit¨at Karlsruhe)**Vlad Vicol**(Courant Institute, NY)**Ewelina Zatorska**(University College, London)**Zhifei Zhang**(Peking University, Beijing)

In order to disseminate the contents of the talks worldwide, we plan to publish proceedings in an academic journal. If there is a possibility, we will prefer a publishing company, managed by a university or a learned society.

**Sarka Necasova** (Czech Academy of Science) is one of the two organizers of the conference and **Anne-Laure Dalibard** (ENS Paris) is a member of the Scientific Committee (which includes three people). One third of the keynote speakers are women: **Maria Colombo** (EPFL SB, Lausanne), **Irena Lasiecka** (University of Memphis), **Xian Liao** (KIT, Karlsruhe) and **Ewelina Zatorska** (Imperial College, London). The proportion is also one third in the list of prospective participants. This list includes many young researchers (post-doctoral or early career researchers) and will be completed if the project is accepted by PhD students.