The geometry of maximal development and shock formation for the Euler equations in multiple space dimensions. (English) Zbl 07895063
Summary: We construct a fundamental piece of the boundary of the maximal globally hyperbolic development (MGHD) of Cauchy data for the multi-dimensional compressible Euler equations, which is necessary for the local shock development problem. For an open set of compressive and generic \(H^7\) initial data, we construct unique \(H^7\) solutions to the Euler equations in the maximal spacetime region below a given time-slice, beyond the time of the first singularity; at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, containing the union of three sets: first, a co-dimension-2 surface of “first singularities” called the pre-shock; second, a downstream hypersurface called the singular set emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. We develop a new geometric framework for the description of the acoustic characteristic surfaces which is based on the Arbitrary Lagrangian Eulerian (ALE) framework, and combine this with a new type of differentiated Riemann variables which are linear combinations of gradients of velocity, sound speed, and the curvature of the fast acoustic characteristic surfaces. With these new variables, we establish uniform \(H^7\) Sobolev bounds for solutions to the Euler equations without derivative loss and with optimal regularity.
References:
| [1] | Abbrescia, L., Speck, J.: The emergence of the singular boundary from the crease in \({3D}\) compressible Euler flow. arXiv preprint. (2022). arXiv:2207.07107 [math.AP] |
| [2] | Abbrescia, L., Speck, J.: The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in \({1D}\) and multi-dimensions. arXiv preprint. (2023). arXiv:2308.07289 [math.AP] · Zbl 07783487 |
| [3] | Bianchini, S.; Bressan, A., Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math. (2), 161, 1, 223-342, 2005 · Zbl 1082.35095 |
| [4] | Buckmaster, T.; Drivas, T. D.; Shkoller, S.; Vicol, V., Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data, Ann. PDE, 8, 2, 1-199, 2022 · Zbl 1504.35192 |
| [5] | Buckmaster, T.; Shkoller, S.; Vicol, V., Formation of shocks for 2D isentropic compressible Euler, Commun. Pure Appl. Math., 75, 9, 2069-2120, 2022 · Zbl 1498.35405 |
| [6] | Buckmaster, T., Cao-Labora, G., Gómez-Serrano, J.: Smooth self-similar imploding profiles to 3D compressible Euler. arXiv preprint (2023). arXiv:2301.10101 [math.AP] · Zbl 1522.35379 |
| [7] | Buckmaster, T.; Shkoller, S.; Vicol, V., Formation of point shocks for 3D compressible Euler, Commun. Pure Appl. Math., 76, 9, 2073-2191, 2023 · Zbl 1527.35230 |
| [8] | Buckmaster, T.; Shkoller, S.; Vicol, V., Shock formation and vorticity creation for 3D Euler, Commun. Pure Appl. Math., 76, 9, 1965-2072, 2023 · Zbl 1527.35229 |
| [9] | Cao-Labora, G., Gómez-Serrano, J., Shi, J., Staffilani, G.: Non-radial implosion for compressible Euler and Navier-Stokes in \(\mathbb{T}^3\) and \(\mathbb{R}^3 \). arXiv preprint. (2023). arXiv:2310.05325 [math.AP] |
| [10] | Chen, G.-Q. G., Remarks on R. J. DiPerna’s paper: “Convergence of the viscosity method for isentropic gas dynamics” [Comm. Math. Phys. 91 (1983), no. 1, 1-30; MR0719807 (85i:35118)], Proc. Am. Math. Soc., 125, 10, 2981-2986, 1997 · Zbl 0888.35066 |
| [11] | Chen, R. M.; Vasseur, A. F.; Yu, C., Global ill-posedness for a dense set of initial data to the isentropic system of gas dynamics, Adv. Math., 393, 2021 · Zbl 1496.35293 |
| [12] | Chen, J., Cialdea, G., Shkoller, S., Vicol, V.: Vorticity blowup in 2D compressible Euler. Preprint (2024) |
| [13] | Chiodaroli, E.; De Lellis, C.; Kreml, O., Global ill-posedness of the isentropic system of gas dynamics, Commun. Pure Appl. Math., 68, 7, 1157-1190, 2015 · Zbl 1323.35137 |
| [14] | Choquet-Bruhat, Y.; Geroch, R., Global aspects of the Cauchy problem in general relativity, Commun. Math. Phys., 14, 329-335, 1969 · Zbl 0182.59901 |
| [15] | Christodoulou, D., The Formation of Shocks in 3-Dimensional Fluids, 2007, Zürich: European Mathematical Society (EMS), Zürich · Zbl 1138.35060 |
| [16] | Christodoulou, D., The Shock Development Problem, 2019, Zürich: European Mathematical Society (EMS), Zürich · Zbl 1445.35003 |
| [17] | Christodoulou, D.; Lisibach, A., Shock development in spherical symmetry, Ann. PDE, 2, 1, 2016 · Zbl 1402.35172 |
| [18] | Christodoulou, D.; Miao, S., Compressible Flow and Euler’s Equations, 2014, Somerville: International Press, Somerville · Zbl 1329.76002 |
| [19] | Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves, 1948, New York: Interscience, New York · Zbl 0041.11302 |
| [20] | Courant, R.; Hilbert, D., Methods of Mathematical Physics. Vol. II. Partial Differential Equations, 1989, New York: Wiley, New York · Zbl 0729.35001 |
| [21] | Coutand, D.; Shkoller, S., Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Am. Math. Soc., 20, 3, 829-930, 2007 · Zbl 1123.35038 |
| [22] | Coutand, D.; Shkoller, S., Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206, 2, 515-616, 2012 · Zbl 1257.35147 |
| [23] | Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics, 2016, Berlin: Springer, Berlin · Zbl 1364.35003 |
| [24] | DeTurck, D. M., Deforming metrics in the direction of their Ricci tensors, J. Differ. Geom., 18, 1, 157-162, 1983 · Zbl 0517.53044 |
| [25] | DiPerna, R. J., Convergence of the viscosity method for isentropic gas dynamics, Commun. Math. Phys., 91, 1, 1-30, 1983 · Zbl 0533.76071 |
| [26] | Eperon, F. C.; Reall, H. S.; Sbierski, J. J., Predictability of subluminal and superluminal wave equations, Commun. Math. Phys., 368, 585-626, 2019 · Zbl 1416.83016 |
| [27] | Fournier, J.-D.; Frisch, U., L’équation de Burgers déterministe et statistique, J. Méc. Théor. Appl., 2, 5, 699-750, 1983 · Zbl 0573.76058 |
| [28] | Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18, 697-715, 1965 · Zbl 0141.28902 |
| [29] | Glimm, J.; Lax, P. D., Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, 1970, Providence: Am. Math. Soc., Providence · Zbl 0204.11304 |
| [30] | Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17, 2, 255-306, 1982 · Zbl 0504.53034 |
| [31] | Holzegel, G.; Klainerman, S.; Speck, J.; Wong, W. W.-Y., Small-data shock formation in solutions to 3D quasilinear wave equations: an overview, J. Hyperbolic Differ. Equ., 13, 1, 1-105, 2016 · Zbl 1346.35005 |
| [32] | Landau, L. D.; Lifshitz, E. M., Course of Theoretical Physics. Vol. 6. Fluid Mechanics, 1987, Oxford: Pergamon Press, Oxford · Zbl 0655.76001 |
| [33] | Lax, P. D., Hyperbolic systems of conservation laws. II, Commun. Pure Appl. Math., 10, 537-566, 1957 · Zbl 0081.08803 |
| [34] | Lax, P. D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5, 611-613, 1964 · Zbl 0135.15101 |
| [35] | Lax, P. D., The formation and decay of shock waves, Am. Math. Mon., 79, 227-241, 1972 · Zbl 0228.35019 |
| [36] | Lebaud, M.-P., Description de la formation d’un choc dans le \(p\)-système, J. Math. Pures Appl. (9), 73, 6, 523-565, 1994 · Zbl 0832.35092 |
| [37] | Liu, T. P., Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations, J. Differ. Equ., 33, 1, 92-111, 1979 · Zbl 0379.35048 |
| [38] | Luk, J.; Speck, J., Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math., 214, 1, 1-169, 2018 · Zbl 1409.35142 |
| [39] | Luk, J.; Speck, J., The stability of simple plane-symmetric shock formation for 3D compressible Euler flow with vorticity and entropy, Anal. PDE, 17, 3, 831-941, 2024 · Zbl 07846447 |
| [40] | Luo, T.-W., Yu, P.: On the stability of multi-dimensional rarefaction waves I: the energy estimates. arXiv preprint (2023). arXiv:2302.09714 |
| [41] | Luo, T.-W., Yu, P.: On the stability of multi-dimensional rarefaction waves II: existence of solutions and applications to Riemann problem. arXiv preprint (2023). arXiv:2305.06308 |
| [42] | Majda, A. J., The existence of multidimensional shock fronts, Mem. Am. Math. Soc., 43, 281, 1983 · Zbl 0517.76068 |
| [43] | Majda, A. J., The stability of multidimensional shock fronts, Mem. Am. Math. Soc., 41, 275, 1983 · Zbl 0506.76075 |
| [44] | Majda, A. J., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, 1984, New York: Springer, New York · Zbl 0537.76001 |
| [45] | Merle, F.; Raphaël, P.; Rodnianski, I.; Szeftel, J., On the implosion of a compressible fluid I: smooth self-similar inviscid profiles, Ann. Math. (2), 196, 2, 567-778, 2022 · Zbl 1497.35384 |
| [46] | Merle, F.; Raphaël, P.; Rodnianski, I.; Szeftel, J., On the implosion of a compressible fluid II: singularity formation, Ann. Math. (2), 196, 2, 779-889, 2022 · Zbl 1497.35385 |
| [47] | Miao, S.; Yu, P., On the formation of shocks for quasilinear wave equations, Invent. Math., 207, 2, 697-831, 2017 · Zbl 1362.35248 |
| [48] | Neal, I., Shkoller, S., Vicol, V.: A characteristics approach to shock formation in 2d Euler with azimuthal symmetry and entropy. arXiv preprint (2022). arXiv:2302.01289 [math.AP] |
| [49] | Neal, I.; Rickard, C.; Shkoller, S.; Vicol, V., A new type of stable shock formation in gas dynamics, Commun. Pure Appl. Anal., 2023 · doi:10.3934/cpaa.2023118 |
| [50] | Rauch, J., BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one, Commun. Math. Phys., 106, 3, 481-484, 1986 · Zbl 0619.35073 |
| [51] | Riemann, B., Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abh. K. Ges. Wiss. Göttingen, 8, 43-66, 1860 |
| [52] | Sideris, T. C., Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys., 101, 4, 475-485, 1985 · Zbl 0606.76088 |
| [53] | Speck, J., Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations, 2016, Providence: Am. Math. Soc., Providence · Zbl 1373.35005 |
| [54] | Speck, J., Shock formation for \(2D\) quasilinear wave systems featuring multiple speeds: blowup for the fastest wave, with non-trivial interactions up to the singularity, Ann. PDE, 4, 1, 2018 · Zbl 1400.35182 |
| [55] | Synge, J. L., Synge. Relativistic hydrodynamics, Proc. Lond. Math. Soc., s2-43, 376-400, 1938 · Zbl 0018.18503 |
| [56] | Unruh, W. G., Experimental black-hole evaporation?, Phys. Rev. Lett., 46, 1351-1353, 1981 |
| [57] | Visser, M.: Acoustic propagation in fluids: an unexpected example of Lorentzian geometry (1993) |
| [58] | Yin, H., 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 · Zbl 1133.35374 |
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