CONFERENCE
Nonlinear PDEs in Fluid Dynamics (2576)
EDP non linéaires en dynamique des fluides
Dates: 9-13 May 2022
Place: CIRM (Marseille Luminy, France)
DESCRIPTION
 Aims
The aim of the conference is to be an internationally viewed important forum in the year 2022, which
•  brings together many leading experts in PDE and fluid dynamics for one week to Luminy,
•  discusses the most actual state, approaches and methods for the above topics,
•  intensifies the interdisciplinary cooperation of 'pure' mathematicians working in harmonic analysis, spectral theory, evolution equations, convex integration with colleagues working on viscous and inviscid fluid models, geophysical flows, complex fluids and stochastic fluid dynamics.

State of Research
The modern theory of partial differential equations plays an important role in various areas of mathematics, natural sciences and in particular in fluid dynamics. Of special interest within fluid dynamics are:
•  global well-posedness of the classical Euler- and Navier-Stokes equations,
•  geophysical flows as primitive and SQG equations,
•  one or two-phase free boundary value problems with or without phase transition,
•  liquid crystals and more generally complex fluids,
•  viscoeleastic fluids,
•  additive and multiplicative white noise perturbations and stochastic boundary conditions
•  flows on manifolds,
•  stability and asymptotic properties,
•  nonhomogeneous fluids,
•  analysis in critical spaces.

This field has seen tremendous scientific activity in the last years, which are due to the facts that the underlying governing equations have a very complex structure and on the other hand that these equations form a fundamental building block for other disciplines of mathematics and science.
The mathematical investigations of these topics often follow their traditional lines but essential progress on certain problems or equations was, however, often based on techniques which were well-known only in other areas or disciplines. For example, the recent results on the regularity of weak solutions of the Euler and the Navier-Stokes equations by De Lellis and Szekelyhidi [7], [3] and Buckmaster and Vicol [4], respectively, are based on the method of convex integration; well-posedness results for the Ericksen-Leslie model for liquid crystals [12] are based on the theory of quasilinear parabolic problems and thermo- dynamical consistency; and recent progress on two-phase problems [19] rest on properties on functional calculi [13]. This holds also for quasigeostrophic equations [6] on domains and manifolds of the primitive equations [11], [10] or for SPDEs in critical spaces.

This is where the international conference comes into its own: besides the investigations and representations of the most actual results on the mathematical structures of the underlying equations and phenomena, special emphasis will be put to foster the interaction between more fundamental methods within pure mathematics, see e.g., [9],[13], such as convex integration, evolution equations, functional calculus, spectral theory, maximal regularity, Fourier multipliers, Littlewood-Paley theory, interpolation theory, stochastic integrals and tools for critical spaces with current questions arising in the theory of the Navier-Stokes and the Euler equations, with deterministic or stochastic models of complex fluids such as liquid crystals, phase transitions or free boundary value problems.

One point will be to discuss the non-uniqueness properties of certain classes of weak solutions to the Euler, Navier-Stokes and transport equations, see [3], [7], [18]. We aim to initiate a more intense exchange of ideas and methods of experts working in this field with other members of the community in order to find ideas and approaches for difficult and open problems in related areas, such as:
•  recent non-uniqueness results for the DiPerna-Lions theory on the transport equation [18].
Other areas of these discussions concern e.g.,
•  geophysical equations such as the inviscid primitive equations,
•  Prandtl’s equation or related equations for boundary layers [17],
•  fluid equations on Lipschitz domains [15],[20] or
•  coupled parabolic-hyperbolic systems arsing in fluid-elastic interaction or viscoelastic fluids.
Another focal point will be the analysis of fluid models in critical spaces, see e.g., [8]. They arise, for example, in the study of global solutions to one or two-phase free boundary value problems when one introduces Lagragian coodinates. Aiming to control the L1-norm of the solutions globally asks again for the interaction of researchers with different backgrounds. Also nonlocal models ask for new methods concerning stability and asymptotics.
As written above, many of the above more applied questions would not been treatable without certain methods developed in pure analysis. A further inner development of these areas in pure mathematics is hence very important. 
SCIENTIFIC COMMITTEE
ORGANIZING COMMITTEE
SPEAKERS (list tba)
References
[1] K. Abe, Y. Giga, M. Hieber, Stokes resolvent estimates in spaces of bounded functions. Ann. Sci. Ec. Norm. Super., 48:521-543, 2015.
[2] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn. Ann. of Math. 156 (2002), 633-654.
[3] T. Buckmaster, C. De Lellis, P. Isett, L. Szekelyhidi, Anamolous dissipation for 1/5-Hölder Euer flows. Ann. of Math. 182, (2015), 127-172.
[4] T. Buckmaster, V. Vicol. Nonuniqueness of weak solutions to the Navier-Stokes equations. Ann. of Math., 189:101-144, 2019.
[5] C. Cao, E. Titi. Global wellposedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. of Math. 166 (2007), 245–267.
[6] P. Constantin, H. Nguyen. Global weak solutions for SQG in bounded domains. Comm. Pure Appl. Math.71 (2018), 2323-2333.
[7]  C. De Lellis, L. Szekelyhidi, On turbulence and geometry:  from Nash to Onsager. Notices Amer. Math. Soc.66, (2019), 677-685.
[8]  R.  Danchin,  P.  Mucha.    Critical  functional  framework  and  maximal  regularity  in  action  on  systems  of incompressible flows.  Mem. Soc. Math. France, 143, 2015.
[9] R. Denk, M. Hieber, J. Prüss.  R-Boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc., vol.166, 2003.
[10] Y. Giga, M. Gries, M. Hieber, A. Hussein, T. Kashiwabara.  The hydrostatic Stokes operator and well- posedness of the primitive equations on spaces of bounded functions. J. Funct. Anal., to appear.
[11] M. Hieber, T. Kashiwabara,   Strong well-posedness of the three dimensional primitive equations in the Lp-setting. Archive Ration. Mech. Anal., 221 (2016), 1077–1115.
[12]  M. Hieber, J. Prüss.  Dynamics of Ericksen-Leslie equations with general Leslie stress II: The compressible isotropic case.  Arch. Rational Mech. Anal., 233:1441-1468, 2019.
[13] T. Hytönen, J. Van Neerven, M. Veraar, L. Weis, Analysis in Banach spaces, Vol. II, Springer, 2016.   [14] H. Koch, D. Tataru, Wellposedness for the Navier-Stokes equations. Advances Math.  157, (2001), 22-35.
[15]  P. Kunstmann, L. Weis, New criteria for the H∞-calculus ad the Stohes operator on bounded Lipschitz domains. J. Evol. Eq 17, (2017), 287-409.
[16] J. Li, E. Titi. Recent advances concerning certain classes of geophysical flows. In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 933-972, Springer, 2016.
[17]  Y. Maekawa, On the inviscid limit problem of the vorticity equation for viscous incompressible flows in the half-plane. Comm. Pure Appl. Math, 67, (2014), 1045-1128.
[18]  S. Modena, L. Szekelyhidi.  Non-uniqueness for the transport equation with Soboled vector fields.  Annalso of PDE, 4, 2018.
[19] J. Prüss, G. Simonett. Moving Interfaces and Quasilinear Parabolic Evolution Equations. Birkhaüser Monographs in Math., 2016.
[20] P. Tolksdorf.  On the Lp-theory of the Navier-Stokes equations on three dimensional Lipschitz domains. Math. Ann., 371:445-460, 2018.
SPONSORS OF THIS SEMESTER (to be completed)