Finite-time singularity formation for \(C^{1,\alpha}\) solutions to the incompressible Euler equations on \(\mathbb{R}^3\). (English) Zbl 1492.35199

Elgindi, Tarek

Ann. Math. (2) 194, No. 3, 647-727 (2021).

The author considers a long-time open question on the finite-time singulary formation of solutions to Euler equations, and partially answers this long standing open question in a certain case. The targeted equations are the incompressible Euler equations in \(R^3\): \begin{align*} &\partial_t u + u\cdot \nabla u + \nabla p = 0, \tag{1}\\ &\operatorname{div} (u)=0, \tag{2} \\ & u|_{t=0}=u_0. \tag{3} \end{align*} The author proves the following singularity formation result.
Theorem. There exists an \(\alpha>0\) and a divergence-free and odd \(u_0\in C^{1,\alpha}(R^3)\) with initial vorticity \(|\omega_0(x)|:=|\nabla \times u_0| \le \frac{C}{|x|^\alpha +1}\) for some constant \(C>0\) so that the unique local odd solution to 1–3 belonging to the class \(C^{1,\alpha}_{x,t} ([0,1]\times R^3)\) satisfies \[ \lim_{t\to 1} \int^1_0 |\omega(s)|_{L^\infty} ds =+\infty. \] The proof is technical and elegant, which is divided in 5 steps. After providing a basic analysis of the “Fundamental Model”, which encodes the leading order dynamics of the type of the solutions, and describing the coercivity of the linearization of the fundamental model around its self-similar solutions, the author gives the coercivity estimates for the linearization of the fundamental model along with the relevant angular transport term. Then, the author shows the elliptic estimates that allow to approximate the main non-local terms. Furthermore, after obtaining some useful information about the function spaces, the author sets up the exact equation for the perturbation to the solution of the fundamental model, and proves the relevant a priori estimates on the perturbation, and finally constructs the full self-similar solution.

Reviewer: Ming Mei (Montreal)

MSC:

35Q31	Euler equations
35B40	Asymptotic behavior of solutions to PDEs
35B35	Stability in context of PDEs
35B45	A priori estimates in context of PDEs
35B20	Perturbations in context of PDEs

Keywords:

Euler equations; singularity formation; asymptotic stability

References:

[1]	Abidi, Hammadi; Hmidi, Taoufik; Keraani, Sahbi, On the global well-posedness for the axisymmetric {E}uler equations, Math. Ann.. Mathematische Annalen, 347, 15-41 (2010) · Zbl 1398.35159
[2]	Bardos, K.; Titi, \`E. S., Euler equations for an ideal incompressible fluid, Uspekhi Mat. Nauk. Uspekhi Matematicheskikh Nauk, 62, 5-46 (2007) · Zbl 1139.76010
[3]	Beale, J. T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the {\(3\)}-{D} {E}uler equations, Comm. Math. Phys.. Communications in Mathematical Physics, 94, 61-66 (1984) · Zbl 0573.76029
[4]	Buckmaster, Tristan; Vicol, Vlad, Convex integration and phenomenologies in turbulence, EMS Surv. Math. Sci.. EMS Surveys in Mathematical Sciences, 6, 173-263 (2019) · Zbl 1440.35231
[5]	Chae, Dongho, Nonexistence of self-similar singularities for the 3{D} incompressible {E}uler equations, Comm. Math. Phys.. Communications in Mathematical Physics, 273, 203-215 (2007) · Zbl 1157.35079
[6]	Chae, Dongho; Shvydkoy, Roman, On formation of a locally self-similar collapse in the incompressible {E}uler equations, Arch. Ration. Mech. Anal.. Archive for Rational Mechanics and Analysis, 209, 999-1017 (2013) · Zbl 1285.35070
[7]	Choi, Kyudong; Hou, Thomas Y.; Kiselev, Alexander; Luo, Guo; Sverak, Vladimir; Yao, Yao, On the finite-time blowup of a one-dimensional model for the three-dimensional axisymmetric {E}uler equations, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 70, 2218-2243 (2017) · Zbl 1377.35218
[8]	Constantin, Peter, On the {E}uler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.). American Mathematical Society. Bulletin. New Series, 44, 603-621 (2007) · Zbl 1132.76009
[9]	Constantin, Peter; Lax, P. D.; Majda, A., A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 38, 715-724 (1985) · Zbl 0615.76029
[10]	Constantin, Peter; Sun, Weiran, Remarks on {O}ldroyd-{B} and related complex fluid models, Commun. Math. Sci.. Communications in Mathematical Sciences, 10, 33-73 (2012) · Zbl 1291.35201
[11]	Constantin, Peter, The {E}uler equations and nonlocal conservative {R}iccati equations, Internat. Math. Res. Notices. International Mathematics Research Notices, 455-465 (2000) · Zbl 0970.76017
[12]	Constantin, Peter; Fefferman, Charles; Majda, Andrew J., Geometric constraints on potentially singular solutions for the {\(3\)}-{D} {E}uler equations, Comm. Partial Differential Equations. Communications in Partial Differential Equations, 21, 559-571 (1996) · Zbl 0853.35091
[13]	Danchin, R., Axisymmetric incompressible flows with bounded vorticity, Uspekhi Mat. Nauk. Uspekhi Matematicheskikh Nauk, 62, 73-94 (2007) · Zbl 1139.76011
[14]	De Lellis, Camillo; Sz\'{e}kelyhidi, Jr., L\'{a}szl\'{o}, On turbulence and geometry: from {N}ash to {O}nsager, Notices Amer. Math. Soc.. Notices of the American Mathematical Society, 66, 677-685 (2019) · Zbl 1436.76009
[15]	Deng, Jian; Hou, Thomas Y.; Yu, Xinwei, Geometric properties and nonblowup of 3{D} incompressible {E}uler flow, Comm. Partial Differential Equations. Communications in Partial Differential Equations, 30, 225-243 (2005) · Zbl 1142.35549
[16]	Denisov, Sergey A., Infinite superlinear growth of the gradient for the two-dimensional {E}uler equation, Discrete Contin. Dyn. Syst.. Discrete and Continuous Dynamical Systems. Series A, 23, 755-764 (2009) · Zbl 1156.76009
[17]	Denisov, Sergey A., Double exponential growth of the vorticity gradient for the two-dimensional {E}uler equation, Proc. Amer. Math. Soc.. Proceedings of the American Mathematical Society, 143, 1199-1210 (2015) · Zbl 1315.35150
[18]	Do, Tam, On vorticity gradient growth for the axisymmetric 3{D} {E}uler equations without swirl, Arch. Ration. Mech. Anal.. Archive for Rational Mechanics and Analysis, 234, 181-209 (2019) · Zbl 1444.76024
[19]	Elgindi, Tarek M.; Ghoul, Tej-Eddine; Masmoudi, Nader, On the stability of self-similar blow-up for \({C}^{1,\alpha}\) solutions to the incompressible {E}uler equations on {\( \mathbb{R}^3\)} (2019)
[20]	Elgindi, Tarek M.; Ghoul, Tej-Eddine; Masmoudi, Nader, Stable self-similar blow-up for a family of nonlocal transport equations, Anal. PDE. Analysis & PDE, 14, 891-908 (2021) · Zbl 1472.35277
[21]	Elgindi, Tarek M.; Jeong, In-Jee, Finite-time singularity formation for strong solutions to the {B}oussinesq system, Ann. PDE. Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences, 6, 5-50 (2020) · Zbl 1462.35287
[22]	Elgindi, Tarek M.; Jeong, In-Jee, On singular vortex patches, {I}: Well-posedness issues (2019) · Zbl 1454.35265
[23]	Elgindi, Tarek M.; Jeong, In-Jee, On the effects of advection and vortex stretching, Arch. Ration. Mech. Anal.. Archive for Rational Mechanics and Analysis, 235, 1763-1817 (2020) · Zbl 1434.35091
[24]	Elgindi, Tarek M., Remarks on functions with bounded {L}aplacian (2016)
[25]	Elgindi, Tarek M.; Jeong, In-Jee, Finite-time singularity formation for strong solutions to the axi-symmetric 3{D} {E}uler equations, Ann. PDE. Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences, 5, 16-51 (2019) · Zbl 1436.35055
[26]	Elgindi, Tarek M.; Masmoudi, Nader, {\(L^\infty \)} ill-posedness for a class of equations arising in hydrodynamics, Arch. Ration. Mech. Anal.. Archive for Rational Mechanics and Analysis, 235, 1979-2025 (2020) · Zbl 1507.35153
[27]	Gibbon, J. D., The three-dimensional {E}uler equations: where do we stand?, Phys. D. Physica D. Nonlinear Phenomena, 237, 1894-1904 (2008) · Zbl 1143.76389
[28]	Gibbon, J. D.; Moore, D. R.; Stuart, J. T., Exact, infinite energy, blow-up solutions of the three-dimensional {E}uler equations, Nonlinearity. Nonlinearity, 16, 1823-1831 (2003) · Zbl 1040.35069
[29]	Grafakos, Loukas; e Silva, Diogo Oliveira; Pramanik, Malabika; Seeger, Andreas; Stovall, Betsy, Some problems in harmonic analysis (2017)
[30]	Gunther, N., On the motion of fluid in a moving container, Izvestia Akad. Nauk USSR, Ser. Fiz.–Mat., 20, 1323-1348 (1927) · JFM 53.0786.08
[31]	Hou, Thomas Y.; Lei, Zhen, On the stabilizing effect of convection in three-dimensional incompressible flows, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 62, 501-564 (2009) · Zbl 1171.35095
[32]	Jia, H.; Stewart, S.; Sverak, V., On the {D}e {G}regorio modification of the {C}onstantin-{L}ax-{M}ajda model, Arch. Ration. Mech. Anal.. Archive for Rational Mechanics and Analysis, 231, 1269-1304 (2019) · Zbl 1408.35152
[33]	Kato, Tosio; Ponce, Gustavo, Commutator estimates and the {E}uler and {N}avier-{S}tokes equations, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 41, 891-907 (1988) · Zbl 0671.35066
[34]	Kato, Tosio, Remarks on the {E}uler and {N}avier-{S}tokes equations in {\({\bf R}^2\)}. Nonlinear Functional Analysis and its Applications, {P}art 2, Proc. Sympos. Pure Math., 45, 1-7 (1986) · Zbl 0598.35093
[35]	Kerr, R. M., Evidence for a singularity of the three-dimensional, incompressible {E}uler equations. Topological {A}spects of the {D}ynamics of {F}luids and {P}lasmas ({S}anta {B}arbara, {CA}, 1991), NATO Adv. Sci. Inst. Ser. E: Appl. Sci., 218, 309-336 (1992) · Zbl 0800.76081
[36]	Kerr, Robert M., Evidence for a singularity of the three-dimensional, incompressible {E}uler equations, Phys. Fluids A. Physics of Fluids A. Fluid Dynamics, 5, 1725-1746 (1993) · Zbl 0800.76083
[37]	Kiselev, Alexander, Small scales and singularity formation in fluid dynamics (2018) · Zbl 1448.35398
[38]	Kiselev, Alexander; Ryzhik, Lenya; Yao, Yao; Zlato\v{s}, Andrej, Finite time singularity for the modified {SQG} patch equation, Ann. of Math. (2). Annals of Mathematics. Second Series, 184, 909-948 (2016) · Zbl 1360.35159
[39]	Kiselev, Alexander; \v{S}ver\'{a}k, Vladimir, Small scale creation for solutions of the incompressible two-dimensional {E}uler equation, Ann. of Math. (2). Annals of Mathematics. Second Series, 180, 1205-1220 (2014) · Zbl 1304.35521
[40]	Larios, Adam; Petersen, Mark R.; Titi, Edriss S.; Wingate, Beth, A computational investigation of the finite-time blow-up of the 3{D} incompressible {E}uler equations based on the {V}oigt regularization, Theor. Comput. Fluid Dyn.. Theoretical and Computational Fluid Dynamics, 32, 23-34 (2018)
[41]	Larios, Adam; Titi, Edriss S., Global regularity versus finite-time singularities: some paradigms on the effect of boundary conditions and certain perturbations. Recent Progress in the Theory of the {E}uler and {N}avier-{S}tokes Equations, London Math. Soc. Lecture Note Ser., 430, 96-125 (2016) · Zbl 1408.35166
[42]	Lei, Zhen; Liu, Jie; Ren, Xiao, On the {C}onstantin-{L}ax-{M}ajda model with convection, Comm. Math. Phys.. Communications in Mathematical Physics, 375, 765-783 (2020) · Zbl 1439.35402
[43]	Lichtenstein, Leon, \"{U}ber einige {E}xistenzprobleme der {H}ydrodynamik homogener, unzusammendr\"{u}ckbarer, reibungsloser {F}l\"{u}ssigkeiten und die {H}elmholtzschen {W}irbels\"{a}tze, Math. Z.. Mathematische Zeitschrift, 23, 89-154 (1925) · JFM 51.0658.01
[44]	Luo, Guo; Hou, Thomas Y., Potentially singular solutions of the 3{D} axisymmetric {E}uler equations, Proc. Nat. Acad. Sci., 111, 12968-12973 (2014) · Zbl 1431.35115
[45]	Luo, Guo; Hou, Thomas Y., Toward the finite-time blowup of the 3{D} axisymmetric {E}uler equations: a numerical investigation, Multiscale Model. Simul.. Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12, 1722-1776 (2014) · Zbl 1316.35235
[46]	Majda, Andrew J.; Bertozzi, Andrea L., Vorticity and Incompressible Flow, Cambridge Texts Appl. Math., 27, xii+545 pp. (2002) · Zbl 0983.76001
[47]	Nadirashvili, N. S., Wandering solutions of the two-dimensional {E}uler equation, Funktsional. Anal. i Prilozhen.. Akademiya Nauk SSSR. Funktsional\cprime ny\u{\i} Analiz i ego Prilozheniya, 25, 70-71 (1991) · Zbl 0769.35048
[48]	Necas, J.; Ruzicka, M.; Sver{\'a}k, V., On {L}eray’s self-similar solutions of the {N}avier-{S}tokes equations, Acta Math.. Acta Mathematica, 176, 283-294 (1996) · Zbl 0884.35115
[49]	Saint Raymond, X., Remarks on axisymmetric solutions of the incompressible {E}uler system, Comm. Partial Differential Equations. Communications in Partial Differential Equations, 19, 321-334 (1994) · Zbl 0795.35063
[50]	Sarria, Alejandro; Saxton, Ralph, Blow-up of solutions to the generalized inviscid {P}roudman–{J}ohnson equation, J. Math. Fluid Mech.. Journal of Mathematical Fluid Mechanics, 15, 493-523 (2013) · Zbl 1277.35077
[51]	Serfati, Philippe, R\'{e}gularit\'{e} stratifi\'{e}e et \'{e}quation d’{E}uler {\(3\)}{D} \`a temps grand, C. R. Acad. Sci. Paris S\'{e}r. I Math.. Comptes Rendus de l’Acad\'{e}mie des Sciences. S\'{e}rie I. Math\'{e}matique, 318, 925-928 (1994) · Zbl 0805.76009
[52]	Shirota, Taira; Yanagisawa, Taku, Note on global existence for axially symmetric solutions of the {E}uler system, Proc. Japan Acad. Ser. A Math. Sci.. Japan Academy. Proceedings. Series A. Mathematical Sciences, 70, 299-304 (1994) · Zbl 0831.35141
[53]	Stuart, J. T., Nonlinear {E}uler partial differential equations: singularities in their solution. Applied Mathematics, Fluid Mechanics, Astrophysics, 81-95 (1988)
[54]	Tao, Terence, On the universality of the incompressible {E}uler equation on compact manifolds, {II}. {N}on-rigidity of {E}uler flows, Pure Appl. Funct. Anal.. Pure and Applied Functional Analysis, 5, 1425-1443 (2020) · Zbl 1470.35273
[55]	Tao, Terence, On the universality of the incompressible {E}uler equation on compact manifolds, Discrete Contin. Dyn. Syst.. Discrete and Continuous Dynamical Systems. Series A, 38, 1553-1565 (2018) · Zbl 1397.35193
[56]	Tao, Terence, Finite time blowup for {L}agrangian modifications of the three-dimensional {E}uler equation, Ann. PDE. Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences, 2, 9-79 (2016) · Zbl 1397.35181
[57]	Tsai, Tai-Peng, On {L}eray’s self-similar solutions of the {N}avier-{S}tokes equations satisfying local energy estimates, Arch. Rational Mech. Anal.. Archive for Rational Mechanics and Analysis, 143, 29-51 (1998) · Zbl 0916.35084
[58]	Ukhovskii, M. R.; Iudovich, V. I., Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech.. Journal of Applied Mathematics and Mechanics, 32, 52-61 (1968)
[59]	Yudovich, V. I., On the loss of smoothness of the solutions of the {E}uler equations and the inherent instability of flows of an ideal fluid, Chaos. Chaos. An Interdisciplinary Journal of Nonlinear Science, 10, 705-719 (2000) · Zbl 0982.76014
[60]	Zlato\v{s}, Andrej, Exponential growth of the vorticity gradient for the {E}uler equation on the torus, Adv. Math.. Advances in Mathematics, 268, 396-403 (2015) · Zbl 1308.35194