Geometric structure of mass concentration sets for pressureless Euler alignment systems. (English) Zbl 1490.35330
Summary: We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel \(\varphi\) and unidirectional velocity \(\mathbf{u} = (u, 0, \dots, 0)\). We demonstrate a striking correspondence between the entropy function \(e_0 = \partial_1 u_0 + \phi \ast \rho_0\) and the limiting ‘concentration set’, i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of \(C^1\) hypersurfaces: the image of the zero set of \(e_0\) under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of \(e_0\). In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35B35 | Stability in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
Euler alignment; flocking; concentration of mass; emergent dynamics; Cucker-Smale; collective behaviorReferences:
| [1] | Brenier, Y.; Grenier, E., Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35, 6, 2317-2328 (1998) · Zbl 0924.35080 |
| [2] | Carrillo, J. A.; Choi, Y.-P.; Tadmor, E.; Tan, C., Critical thresholds in 1D Euler equations with non-local forces, Math. Models Methods Appl. Sci., 26, 1, 185-206 (2016) · Zbl 1356.35174 |
| [3] | Chen, G.-Q.; Liu, H., Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Phys. D, Nonlinear Phenom., 189, 1-2, 141-165 (2004) · Zbl 1098.76603 |
| [4] | Cucker, F.; Smale, S., Emergent behavior in flocks, IEEE Trans. Autom. Control, 52, 5, 852-862 (2007) · Zbl 1366.91116 |
| [5] | Cucker, F.; Smale, S., On the mathematics of emergence, Jpn. J. Math., 2, 1, 197-227 (2007) · Zbl 1166.92323 |
| [6] | Ha, S.-Y.; Liu, J.-G., A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7, 2, 297-325 (2009) · Zbl 1177.92003 |
| [7] | Ha, S.-Y.; Huang, F.; Wang, Y., A global unique solvability of entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation, J. Differ. Equ., 257, 5, 1333-1371 (2014) · Zbl 1315.35151 |
| [8] | Ha, S.-Y.; Kim, J.; Zhang, X., Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11, 5, 1157-1181 (2018) · Zbl 1409.34046 |
| [9] | Ha, S.-Y.; Park, J.; Zhang, X., A first-order reduction of the Cucker-Smale model on the real line and its clustering dynamics, Commun. Math. Sci., 16, 7, 1907-1931 (2018) · Zbl 1459.82178 |
| [10] | Ha, S.-Y.; Kim, J.; Park, J.; Zhang, X., Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231, 1, 319-365 (2019) · Zbl 1409.34045 |
| [11] | He, S.; Tadmor, E., Global regularity of two-dimensional flocking hydrodynamics, C. R. Math. Acad. Sci. Paris, 355, 7, 795-805 (2017) · Zbl 1375.35563 |
| [12] | Lear, D.; Shvydkoy, R., Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems, Anal. PDE (2022), in press |
| [13] | Leslie, T. M., On the Lagrangian trajectories for the one-dimensional Euler Alignment model without vacuum velocity, C. R. Math., 358, 4, 421-433 (2020) · Zbl 1447.35270 |
| [14] | Leslie, T. M.; Shvydkoy, R., On the structure of limiting flocks in hydrodynamic Euler Alignment models, Math. Models Methods Appl. Sci., 29, 13, 2419-2431 (2019) · Zbl 1428.92130 |
| [15] | Leslie, T. M.; Tan, C., Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-Alignment system · Zbl 1515.35197 |
| [16] | Liu, J.-G.; Pego, R. L.; Slepčev, D., Least action principles for incompressible flows and geodesics between shapes, Calc. Var. Partial Differ. Equ., 58, 5, Article 179 pp. (2019) · Zbl 1428.35376 |
| [17] | Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44 (1995), Cambridge University Press: Cambridge University Press Cambridge, Fractals and rectifiability · Zbl 0819.28004 |
| [18] | Nguyen, T.; Tudorascu, A., Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. Math. Anal., 40, 2, 754-775 (2008) · Zbl 1171.35432 |
| [19] | Nilsson, B.; Shelkovich, V. M., Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks, Appl. Anal., 90, 11, 1677-1689 (2011) · Zbl 1237.35111 |
| [20] | Shelkovich, V. M., Transport of mass, momentum and energy in zero-pressure gas dynamics, (Hyperbolic Problems: Theory, Numerics and Applications. Hyperbolic Problems: Theory, Numerics and Applications, Proc. Sympos. Appl. Math., vol. 67, Part 2 (2009), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 929-938 · Zbl 1190.35155 |
| [21] | Shvydkoy, R.; Tadmor, E., Eulerian dynamics with a commutator forcing II: flocking, Discrete Contin. Dyn. Syst., 37, 11, 5503-5520 (2017) · Zbl 1372.35227 |
| [22] | Shvydkoy, R.; Tadmor, E., Topologically-based fractional diffusion and emergent dynamics with short-range interactions, SIAM J. Math. Anal., 52, 6, 5792-5839 (2020) · Zbl 1453.92371 |
| [23] | Tadmor, E.; Tan, C., Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc., Math. Phys. Eng. Sci., 372, Article 20130401 pp. (2014) · Zbl 1353.76064 |
| [24] | Tadmor, E.; Wei, D., A variational representation of weak solutions for the pressureless Euler-Poisson equations (2011), arXiv preprint |
| [25] | Tan, C., On the Euler-Alignment system with weakly singular communication weights, Nonlinearity, 33, 4 (2020) · Zbl 1442.35479 |
| [26] | Weinan, E.; Rykov, Y. G.; Sinai, Y. G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys., 177, 2, 349-380 (1996) · Zbl 0852.35097 |
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