Short course #1 by Enrico Valdinoci (The University of Western Australia) on Nonlocal Theory of Surface Tension
Capillarity theory aims at understanding the displacement of a liquid droplet in a container. The modelization of surface tension is a rather delicate business, being the average outcome of the long-range attractive forces of molecules. We describe some recent results motivated by a nonlocal theory of capillarity, as related to the formation of droplets due to long-range interaction potentials, in connection with the theory of nonlocal minimal surfaces.
References.
[1] F. Maggi, E. Valdinoci, Capillarity problems with nonlocal surface tension energies. Comm. Partial Differential Equations 42 (2017), no. 9, 1403-1446.
[2] S. Dipierro, F. Maggi, E. Valdinoci, Asymptotic expansions of the contact angle in nonlocal capillarity problems. J. Nonlinear Sci. 27 (2017), no. 5, 1531-1550.
[3] S. Dipierro, F. Maggi, E. Valdinoci, Minimizing cones for fractional capillarity problems. Rev. Mat. Iberoam. 38 (2022), no. 2, 635-658.
[4] A. De Luca, S. Dipierro, E. Valdinoci, Nonlocal capillarity for anisotropic kernels. ArXiv:2202.03823
Short course #2: by Roberto Camassa (University of North Carolina - Chapel Hill) on Free surface problems in incompressible fluid dynamics
Lecture 1: Fundamentals of continuum mechanics, and illustration by simple experiments with fluids and mechanical systems: Continuum, and in particular fluid, mechanics can be given an axiomatic structure that relies on a minimal set of mathematical assumptions playing the role of postulates. From these, several subtle consequences can emerge, such as the distinction, from a kinematics viewpoint, between boundary and material surfaces and the implications on their time evolution. Thought experiments as well as actual ones with stratified fluids can serve to illustrate these and other aspects of the fundamental theory of continua. A good reference is R.E. Meyer, “Introduction to Mathematical Fluid Dynamics”, Dover, New York, 1971.
Lecture 2: Derivation of model equations governing the evolution of long waves at the surface of (shallow) ideal fluid layers: Layer averaging of conservation laws provides a natural, first-principles starting point for implementing asymptotic approximations based on scale separation of the expected dynamics, and it is especially efficient as it automatically encapsulates the boundary conditions at the free surface and at the bottom of the fluid layer. This technique, augmented by asymptotics based on the small ratio between vertical and horizontal scales, can in principle be carried out to all orders, but the first two are the most significant in isolating the essential elements, nonlinearity and dispersion, governing the time evolution of the system. Elements of this approach can be found in my paper with W. Choi, "Fully nonlinear internal waves in a two-fluid system," J. Fluid Mech. Vol. 396 (1999).
Lecture 3: Model applications and predictions: Free surface dynamics caused by flow over topography offers perhaps the most immediate example of model predictions that are qualitatively and quantitatively accurate, In particular, bifurcation diagrams stemming from equilibrium solutions can be analytically derived, and their predictions observed in laboratory experiments with a towed bottom bump, emulating the commonly occurring geophysical settings of ocean currents over submerged ridges or air current over mountain tops, see for instance the theory presented in D.D. Houghton and A. Kasahara, "Nonlinear shallow fluid flow over an isolated ridge," Comm. Pure Appl. Math. Vol. 21, 1-23 (1968), the theoretical, numerical and experimental work by S.J. Lee, G.T. Yates and T.Y. Wu, J. Fluid Mech. Vol. 199, 566-593 (1988), and, most recently, the numerical investigation of Zhao et al., J. Fluid Mech. Vol. 963, A32, doi:10.1017/jfm.2023.355 (2023). More work is in progress, but still at a very preliminary stage, in our group here at UNC.
Short curse #3 by Vlad Vicol (Courant Institute - New York University) on the Euler equations governing compressible fluids
These lectures are based on joint works with S. Shkoller (x8), T. Buckmaster (x5), and T. Drivas (x2).
Lecture 1. The first lecture introduced the compressible Euler equations. We have discussed the equivalent formulation in terms of velocity, sound speed, and specific entropy, and have shown that prior to the first singularity the systems are equivalent. We discussed several reductions arriving at the 1D isentropic system. For this system we have discussed Riemann variables and the three different wave speeds present in the system. We have discussed the fact that 1D Burgers naturally lives within the 1D isentropic Euler system. We have given three different proofs of finite time singularity for 1D Burgers and have discussed the corresponding blowup proofs for Euler, not just in 1D. We have discussed the various cusp singularities which may arise for 1D Burgers, either from Holder or from smooth initial data.
References.
[1] C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Vol. 3, Springer, 2005.
[2] P.D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, Journal of Mathematical Physics 5, no. 5, 611–613, 1964.
[3] Isaac Neal, Calum Rickard, Steve Shkoller, Vlad Vicol. A new type of stable shock formation in gas dynamics. arXiv:2303.16842 [math.AP], 2023.
Lecture 2. We discussed the fact that three different questions may be asked about the compressible Euler system: (i) shock formation from smooth initial datum; (ii) maximal hyperbolic development of smooth Cauchy data; (iii) the physical shock development problem. We have discussed problems (i)-(iii) in the context of 1D Burgers and the 1D Euler system, to contrast and compare. For the shock development problem we have discussed the Rankine-Hugoniot jump conditions and the equivalence of the physical entropy condition and the Lax geometric entropy conditions in the context of weak shocks. We have shown that regular shock solutions cannot be isentropic, and that generically these solutions have nonzero vorticity (except for radial symmetry or 1D solutions). We have discussed the weak singularities predicted by Landau and Lifshitz. We have discussed the full solution of shock formation and shock development in 1.5D - for 2D solutions with azimuthal symmetry. This includes a uniqueness statement and the development of weak singularities.
References.
[1] L.D. Landau and E.M. Lifshitz, Fluid mechanics, Pergamon Press, Oxford, 1987.
[2] M.P. Lebaud, Description de la formation d’un choc dans le p-systeme ` , J. Math. Pures Appl. (9) 73, no. 6, 523–565, 1994.
[3] H. Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math. J. 175, 125–164, 2004.
[4] T. Buckmaster, T.D. Drivas, S. Shkoller, V. Vicol. Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data. Annals of PDE 8, 26, 2022.
Lecture 3. We discussed the genuinely multi-dimensional Euler system. We have shown that for smooth initial data, the formation of the “very first gradient singularity”, the point-shock, can be attacked without the use of self-similar coordinates, by working with the Lagrangian coordinates associated with the fast acoustic characteristic. We have discussed the geometry of the maximal hyperbolic development of the Cauchy data in the context of 2D isentropic Euler. We have discussed Arbitrary-Lagrangian-Eulerian coordinates and the nonlinear evolution of the associated normal vector used to define the direction of the steepening shock front. This geometry is fully smooth, as opposed to the corresponding Eulerian geometry. In this language, shock formation, or better said, “pre-shock” formation, may be understood as the vanishing of the Jacobian of the ALE-map. We have discussed the three steps required to smoothly characterize the spacetime of maximal hyperbolic development.
References.
[1] T. Buckmaster, S. Shkoller, V. Vicol. Shock formation and vorticity creation for 3d Euler. Communications on Pure and Applied Mathematics, doi.org/10.1002/cpa.22067.
[2] S. Shkoller, V. Vicol. The geometry of maximal development for the Euler equations, Preprint, 2023.