메뉴 건너뛰기




Volumn 14, Issue 3, 2013, Pages 42-76

Mathematics and turbulence: Where do we stand?

Author keywords

Euler equations; Navier Stokes equations; turbulence; viscosity limit

Indexed keywords

FINITE TIME; IT FOCUS; MATHEMATICAL APPROACH; MATHEMATICAL STATEMENT; MATHEMATICAL TOOLS; VANISHING VISCOSITY; WEAK SOLUTION;

EID: 84880260733     PISSN: None     EISSN: 14685248     Source Type: Journal    
DOI: 10.1080/14685248.2013.771838     Document Type: Article
Times cited : (113)

References (88)
  • 1
    • 0004185784 scopus 로고
    • Cambridge: Cambridge University Press
    • Frisch, U. 1995. Turbulence, Cambridge: Cambridge University Press.
    • (1995) Turbulence
    • Frisch, U.1
  • 4
    • 0001300072 scopus 로고    scopus 로고
    • What do the Navier-Stokes equations tell us about turbulence?
    • Contemporary Mathematics, Harmonic analysis and Nonlinear Differential Equations Providence, RI, M. L. Lapidus, L. H. Harper and A. J. Rumbos, eds.
    • Foias, C. 1997. " What do the Navier-Stokes equations tell us about turbulence? ". In Contemporary Mathematics, Vol. 208, Harmonic analysis and Nonlinear Differential Equations, Amer. Math. Soc., Providence, RI, M. L. Lapidus, L. H. Harper and A. J. Rumbos, eds. 151-180.
    • (1997) Amer. Math. Soc. , vol.208 , pp. 151-180
    • Foias, C.1
  • 6
    • 54149105152 scopus 로고    scopus 로고
    • Some specific mathematical constraints on 2 d turbulence
    • Dascaliuc, R., Foias, C., and, Jolly, M. 2008. Some specific mathematical constraints on 2 d turbulence. Physica D, 237: 3020-3029.
    • (2008) Physica D , vol.237 , pp. 3020-3029
    • Dascaliuc, R.1    Foias, C.2    Jolly, M.3
  • 7
    • 0036090156 scopus 로고    scopus 로고
    • Statistical estimates for the Navier-stokes equations and the Kraichnan theory of 2-D fully developed turbulence
    • DOI 10.1023/A:1015782025005
    • Foias, C., Jolly, M. S., Manley, O. P., and, Rosa, R. 2002. Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence. J. Statist. Phys., 108: 591-645. (Pubitemid 37133796)
    • (2002) Journal of Statistical Physics , vol.108 , Issue.3-4 , pp. 591-645
    • Foias, C.1    Jolly, M.S.2    Manley, O.P.3    Rosa, R.4
  • 8
    • 47049096148 scopus 로고    scopus 로고
    • Euler equations of incompressible ideal fluids
    • Bardos, C., and, Titi, E. S. 2007. Euler equations of incompressible ideal fluids. Uspekhi Mat. Nauk, 62: 409-451. (3) (375) (2007), pp. 5-46.
    • (2007) Uspekhi Mat. Nauk , vol.62 , Issue.3-375 , pp. 5-46
    • Bardos, C.1    Titi, E.S.2
  • 9
    • 35748978223 scopus 로고    scopus 로고
    • Also in Russian Math. Surveys 62(3) (2007), pp. 409-451.
    • Russian Math. Surveys , vol.62 , Issue.3 , pp. 409-451
  • 10
    • 33748858288 scopus 로고    scopus 로고
    • Euler equations, Navier-Stokes equations and turbulence
    • DOI 10.1007/11545989-1, Mathematical Foundation of Turbulent Viscous Flows: Lectures given at the C.I.M.E. Summer School
    • Constantin, P. 2006. " Euler equations, Navier-Stokes equations and turbulence ". In Mathematical Foundation of Turbulent Viscous Flows, Edited by: Mathematics, in Springer Lecture Notes in, 1871, Vol, School, Lectures given at the C.I.M.E. Summer, Franca, Martina, Cannone, Italy, M. and Miyakawa, T. 1-43. Berlin: Springer. (Pubitemid 44418415)
    • (2005) Lecture Notes in Mathematics , vol.1871 , pp. 1-43
    • Constantin, P.1
  • 11
    • 41849149355 scopus 로고    scopus 로고
    • On the euler equations of incompressible fluids
    • Constantin, P. 2007. On the Euler equations of incompressible fluids. Bull. Amer. Math. Soc., 44: 603-621.
    • (2007) Bull. Amer. Math. Soc. , vol.44 , pp. 603-621
    • Constantin, P.1
  • 12
    • 47049089302 scopus 로고    scopus 로고
    • The 3D euler equations: Where do we stand?
    • Gibbon, J. D. 2008. The 3D Euler equations: Where do we stand?. Physica, D 237: 1894-1904.
    • (2008) Physica, D , vol.237 , pp. 1894-1904
    • Gibbon, J.D.1
  • 13
    • 47049103057 scopus 로고    scopus 로고
    • 3D Euler in a 2D symmetry plane
    • Bustamente, M. D., and, Kerr, R. M. 2008. 3D Euler in a 2D symmetry plane. Physica, D 237: 1912-1920.
    • (2008) Physica, D , vol.237 , pp. 1912-1920
    • Bustamente, M.D.1    Kerr, R.M.2
  • 14
    • 47049115354 scopus 로고    scopus 로고
    • Blow-up or no blow-up? the interplay between theory & numerics
    • Hou, T. Y. 2008. Blow-up or no blow-up? The interplay between theory & numerics. Physica, D 237: 1937-1944.
    • (2008) Physica, D , vol.237 , pp. 1937-1944
    • Hou, T.Y.1
  • 15
    • 51249193194 scopus 로고
    • Sur le mouvement dun liquide visqueux emplissant lespace
    • Leray, J. 1934. Sur le mouvement dun liquide visqueux emplissant lespace. (French) Acta Math., 63 (1): 193-248.
    • (1934) (French) Acta Math. , vol.63 , Issue.1 , pp. 193-248
    • Leray, J.1
  • 17
    • 0003628399 scopus 로고    scopus 로고
    • Oxford: Incompressible Models,Oxford Lecture Series in Mathematics and its Applications
    • Lions, P. L. 1996. Mathematical Topics in Fluid Mechanics, Oxford: Incompressible Models,Oxford Lecture Series in Mathematics and its Applications. Vol. 1
    • (1996) Mathematical Topics in Fluid Mechanics , vol.1
    • Lions, P.L.1
  • 18
    • 0003545358 scopus 로고
    • Chicago, IL: Chicago Lectures in Mathematics University of Chicago Press
    • Constantin, P., and, Foias, C. 1988. Navier-Stokes Equations, Chicago, IL: Chicago Lectures in Mathematics University of Chicago Press.
    • (1988) Navier-Stokes Equations
    • Constantin, P.1    Foias, C.2
  • 19
    • 21144480338 scopus 로고
    • On the blow-up of solutions of the 3-D Euler equations in a bounded domain
    • Ferrari, A. 1993. On the blow-up of solutions of the 3-D Euler equations in a bounded domain. Commun. Math. Phys., 155: 277-294.
    • (1993) Commun. Math. Phys. , vol.155 , pp. 277-294
    • Ferrari, A.1
  • 20
    • 0009002721 scopus 로고
    • Über einige Existenzproblem der Hydrodynamik homogener unzusammendrückbarer, reibunglosser Flüssikeiten und die Helmholtzschen Wirbelsalitze
    • 26 (1927), pp. 193-323 32: 608
    • Lichtenstein, L. 1930. Über einige Existenzproblem der Hydrodynamik homogener unzusammendrückbarer, reibunglosser Flüssikeiten und die Helmholtzschen Wirbelsalitze. Mat. Zeit. Phys. 23 (1925), pp. 89-154; 26 (1927), pp. 193-323;, 32: 608
    • (1930) Mat. Zeit. Phys. , vol.23 , Issue.1925 , pp. 89-154
    • Lichtenstein, L.1
  • 21
    • 0004259252 scopus 로고    scopus 로고
    • Cambridge: Cambridge Texts in Applied Mathematics Cambridge University Press
    • Majda, A. J., and, Bertozzi, A. L. 2002. Vorticity and Incompressible Flow, Cambridge: Cambridge Texts in Applied Mathematics Cambridge University Press. Vol. 27
    • (2002) Vorticity and Incompressible Flow , vol.27
    • Majda, A.J.1    Bertozzi, A.L.2
  • 23
    • 79951824256 scopus 로고    scopus 로고
    • Loss of smoothness and energy conserving rough weak solutions for the 3 d Euler equations
    • Bardos, C., and, Titi, E. S. 2010. Loss of smoothness and energy conserving rough weak solutions for the 3 d Euler equations. Discrete Contin. Dyn. Syst. Ser., S 3 (2): 185-197.
    • (2010) Discrete Contin. Dyn. Syst. Ser., S , vol.3 , Issue.2 , pp. 185-197
    • Bardos, C.1    Titi, E.S.2
  • 25
    • 79951849382 scopus 로고    scopus 로고
    • On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations
    • Kukavica, I., and, Vicol, V. 2011. On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations. Nonlinearity, 24: 765-796. 3
    • (2011) Nonlinearity , vol.24 , Issue.3 , pp. 765-796
    • Kukavica, I.1    Vicol, V.2
  • 26
    • 34250136481 scopus 로고
    • Remarks on the breakdown of smooth solutions for the 3D Euler equations
    • Beale, J. T., Kato, T., and, Majda, A. 1984. Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys., 94: 61-66.
    • (1984) Commun. Math. Phys. , vol.94 , pp. 61-66
    • Beale, J.T.1    Kato, T.2    Majda, A.3
  • 28
    • 0000647078 scopus 로고
    • Remarks on a paper by J.T. Beale, T. Kato, and A. Majda
    • Ponce, G. 1985. Remarks on a paper by J.T. Beale, T. Kato, and A. Majda. Commun. Math. Phys., 98: 349-353.
    • (1985) Commun. Math. Phys. , vol.98 , pp. 349-353
    • Ponce, G.1
  • 29
    • 17444395218 scopus 로고    scopus 로고
    • Geometric properties and nonblowup of 3D incompressible euler flow
    • DOI 10.1081/PDE-200044488
    • Deng, J., Hou, T.Y., and, Yu, X. 2005. Geometric properties and nonblowup of 3D incompressible Euler Flow. Commun. Partial Diff. Equations 30,: 225-243. (Pubitemid 40549853)
    • (2005) Communications in Partial Differential Equations , vol.30 , Issue.1-3 , pp. 225-243
    • Deng, J.1    Hou, T.Y.2    Yu, X.3
  • 30
    • 0001751564 scopus 로고
    • Non stationary flow of an ideal incompressible liquid
    • Yudovich, V. I. 1963. Non stationary flow of an ideal incompressible liquid. Zh. Vych. Mat., 3: 1032-1066.
    • (1963) Zh. Vych. Mat. , vol.3 , pp. 1032-1066
    • Yudovich, V.I.1
  • 31
    • 84968509708 scopus 로고
    • Existence de nappes de tourbillon en dimension deux
    • Delort, J.-M. 1991. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc., 4: 553-586.
    • (1991) J. Amer. Math. Soc. , vol.4 , pp. 553-586
    • Delort, J.-M.1
  • 32
    • 0040438755 scopus 로고    scopus 로고
    • Existence of vortex sheets with reflection symmetry in two space dimensions
    • Filho, M.C. Lopes, Lopes, H.J. Nussenzveig, and, Xin, Z. 2001. Existence of vortex sheets with reflection symmetry in two space dimensions. Arch. Ration. Mech. Anal., 158: 235-257. 3
    • (2001) Arch. Ration. Mech. Anal. , vol.158 , pp. 235-257
    • Filho, M.C.L.1    Lopes, H.J.N.2    Xin, Z.3
  • 33
    • 0001496981 scopus 로고
    • Remarks on weak solutions for vortex sheets with a distinguished sign
    • Majda, A. 1993. Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J., 42: 921-939.
    • (1993) Indiana Univ. Math. J. , vol.42 , pp. 921-939
    • Majda, A.1
  • 34
    • 77952502222 scopus 로고    scopus 로고
    • Global regularity and convergence of a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations Commun
    • Bardos, C., Linshiz, J., and, Titi, E. S. 2010. Global regularity and convergence of a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of. the 2D Euler equations Commun. Pure Appl. Math., 63(6) 697-746.
    • (2010) Pure Appl. Math. , vol.63 , Issue.6 , pp. 697-746
    • Bardos, C.1    Linshiz, J.2    Titi, E.S.3
  • 35
    • 70350655922 scopus 로고    scopus 로고
    • On admissibility criteria for weak solutions of the Euler equations
    • De Lellis, C., and, Székelyhidi, L. Jr. 2010. On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal., 195(1) 225-260.
    • (2010) Arch. Ration. Mech. Anal. , vol.195 , Issue.1 , pp. 225-260
    • De Lellis, C.1    Székelyhidi Jr., L.2
  • 36
    • 0001768220 scopus 로고
    • An inviscid flow with compact support in space-time
    • Scheffer, V. 1993. An inviscid flow with compact support in space-time. J. Geom. Anal., 3: 343-401.
    • (1993) J. Geom. Anal. , vol.3 , pp. 343-401
    • Scheffer, V.1
  • 37
    • 0031491102 scopus 로고    scopus 로고
    • On the nonuniqueness of weak solutions of the Euler equations
    • Shnirelman, A. 1997. On the nonuniqueness of weak solutions of the Euler equations. Commun. Pure Appl. Math., 50: 1261-1286.
    • (1997) Commun. Pure Appl. Math. , vol.50 , pp. 1261-1286
    • Shnirelman, A.1
  • 38
  • 39
    • 80052936237 scopus 로고    scopus 로고
    • Existence of weak solutions for the incompressible Euler equations
    • Wiedemann, E. 2011. Existence of weak solutions for the incompressible Euler equations. Ann. Inst. H. Poincaré Anal. Non Linaire, 28(5) 727-730.
    • (2011) Ann. Inst. H. Poincaré Anal. Non Linaire , vol.28 , Issue.5 , pp. 727-730
    • Wiedemann, E.1
  • 40
    • 80054039647 scopus 로고    scopus 로고
    • Weak solutions to the incompressible Euler equations with vortex sheet initial data
    • Székelyhidi, L. 2011. Weak solutions to the incompressible Euler equations with vortex sheet initial data. C.R. Math. Acad. Sci. Paris, 349(19-20) 1063-1066.
    • (2011) C.R. Math. Acad. Sci. Paris , vol.349 , Issue.19-20 , pp. 1063-1066
    • Székelyhidi, L.1
  • 41
    • 84867335602 scopus 로고    scopus 로고
    • Vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow
    • Bardos, C., Titi, E. S., and, Wiedemann, E. 2012. Vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow. C.R. Acad. Sci. Paris, 350(15-16) 757-760.
    • (2012) C.R. Acad. Sci. Paris , vol.350 , Issue.15-16 , pp. 757-760
    • Bardos, C.1    Titi, E.S.2    Wiedemann, E.3
  • 43
    • 0000863080 scopus 로고
    • Oscillations and concentrations in weak solutions of the incompressible fluid equations
    • DiPerna, R., and, Majda, A. 1987. Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys., 108(4) 667-689.
    • (1987) Commun. Math. Phys. , vol.108 , Issue.4 , pp. 667-689
    • Diperna, R.1    Majda, A.2
  • 44
    • 51249192666 scopus 로고
    • Statistical hydrodynamics
    • Onsager, L. 1949. Statistical hydrodynamics. Nuovo Cimento, 6(2) 279-287.
    • (1949) Nuovo Cimento , vol.6 , Issue.2 , pp. 279-287
    • Onsager, L.1
  • 45
    • 21844522201 scopus 로고
    • Onsagers conjecture on the energy conservation for solutions of Eulers equation
    • Constantin, P., E, W., and, Titi, E. S. 1994. Onsagers conjecture on the energy conservation for solutions of Eulers equation. Commun. Math. Phys., 165: 207-209.
    • (1994) Commun. Math. Phys. , vol.165 , pp. 207-209
    • Constantin, P.E.W.1    Titi, E.S.2
  • 50
    • 0001489148 scopus 로고
    • Mémoire sur les lois du mouvement des fluides
    • Navier, C. 1822. Mémoire sur les lois du mouvement des fluides. Mém. Acad. Sci. Inst. France, 6: 389-440.
    • (1822) Mém. Acad. Sci. Inst. France , vol.6 , pp. 389-440
    • Navier, C.1
  • 51
    • 0000119455 scopus 로고
    • On the theories of the internal friction of fluids in motion
    • Stokes, G. 1845. On the theories of the internal friction of fluids in motion. Trans. Cambridge Philos. Soc., 8: 257
    • (1845) Trans. Cambridge Philos. Soc. , vol.8 , pp. 257
    • Stokes, G.1
  • 53
    • 0018507036 scopus 로고
    • Slightly rarefied gas flow over a body with small accommodation coefficient
    • Aoki, K., Inamuro, T., and, Onishi, Y. 1979. Slightly rarefied gas flow over a body with small accommodation coefficient. J. Phys. Soc. Japan, 47: 663-671.
    • (1979) J. Phys. Soc. Japan , vol.47 , pp. 663-671
    • Aoki, K.1    Inamuro, T.2    Onishi, Y.3
  • 54
    • 84855459964 scopus 로고    scopus 로고
    • The incompressible limit of the Boltzmann equation with accommodation boundary condition
    • Bardos, C., Golse, F., and, Paillard, L. 2012. The incompressible limit of the Boltzmann equation with accommodation boundary condition. Commun. Math. Sci., 10(1) 159-190.
    • (2012) Commun. Math. Sci. , vol.10 , Issue.1 , pp. 159-190
    • Bardos, C.1    Golse, F.2    Paillard, L.3
  • 55
    • 0042843384 scopus 로고    scopus 로고
    • From the Boltzmann equation to the Stokes-Fourier system in a bounded domain
    • Masmoudi, N., and, Saint-Raymond, L. 2003. From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Commun. Pure Appl. Math., 56: 1263-1293.
    • (2003) Commun. Pure Appl. Math. , vol.56 , pp. 1263-1293
    • Masmoudi, N.1    Saint-Raymond, L.2
  • 56
    • 0001307388 scopus 로고
    • Un teorema di unicità per le equazioni di Navier-Stokes
    • Prodi, G. 1959. Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl., 48: 173-182.
    • (1959) Ann. Mat. Pura Appl. , vol.48 , pp. 173-182
    • Prodi, G.1
  • 57
    • 0000169262 scopus 로고
    • On the interior regularity of weak solutions of the Navier-Stokes equations
    • Serrin, J. 1962. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal., 9: 187-191.
    • (1962) Arch. Rational Mech. Anal. , vol.9 , pp. 187-191
    • Serrin, J.1
  • 58
    • 33646507669 scopus 로고    scopus 로고
    • Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations
    • DOI 10.1016/j.jde.2005.10.007, PII S0022039605003633
    • Germain, P. 2006. Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations. J. Diff. Equations, 226: 373-428. (Pubitemid 43705771)
    • (2006) Journal of Differential Equations , vol.226 , Issue.2 , pp. 373-428
    • Germain, P.1
  • 59
    • 8644289805 scopus 로고    scopus 로고
    • The sixth millennium problem: Navier-Stokes equations, existence and smoothness
    • Ladyzhenskaya, O. A. 2003. The sixth millennium problem: Navier-Stokes equations, existence and smoothness. (Russian) Uspekhi Mat. Nauk. 58(2) (2003), pp. 45-78.
    • (2003) (Russian) Uspekhi Mat. Nauk. , vol.58 , Issue.2 , pp. 45-78
    • Ladyzhenskaya, O.A.1
  • 60
    • 0041522791 scopus 로고    scopus 로고
    • translation in Russian Math. Surveys, 58(2) 251-286.
    • (2003) Russian Math. Surveys , vol.58 , Issue.2 , pp. 251-286
  • 61
    • 10844242585 scopus 로고    scopus 로고
    • Some developments on navier-stokes equations in the second half of the 20th century
    • Birkhauser, Basel
    • Temam, R. 2000. Some developments on Navier-Stokes equations in the second half of the 20th century. Development of Mathematics 1950-2000, Birkhauser, Basel,: 1049-1106.
    • (2000) Development of Mathematics 1950-2000 , pp. 1049-1106
    • Temam, R.1
  • 62
    • 59549105057 scopus 로고    scopus 로고
    • Regularity criteria for the three-dimensional Navier-Stokes equations
    • Cao, C., and, Titi, E. S. 2008. Regularity criteria for the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J., 57: 2643-2662.
    • (2008) Indiana Univ. Math. J. , vol.57 , pp. 2643-2662
    • Cao, C.1    Titi, E.S.2
  • 63
    • 81555209707 scopus 로고    scopus 로고
    • Global regularity criterion for the 3 d Navier-Stokes equations involving one entry of the velocity gradient tensor
    • Cao, C., and, Titi, E. S. 2011. Global regularity criterion for the 3 d Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch. Rational Mech. Anal., 202: 919-932.
    • (2011) Arch. Rational Mech. Anal. , vol.202 , pp. 919-932
    • Cao, C.1    Titi, E.S.2
  • 64
    • 0042597992 scopus 로고
    • On unique solvability of three-dimensional Cauchy problem for the navier-stokes equations under the axial symmetry
    • (In Russian)
    • Ladyzhenskaya, O. A. 1968. On unique solvability of three-dimensional Cauchy problem for the Navier-Stokes equations under the axial symmetry. Zap. Nauchn. Sem. LOMI, 7: 155-177. (In Russian)
    • (1968) Zap. Nauchn. Sem. LOMI , vol.7 , pp. 155-177
    • Ladyzhenskaya, O.A.1
  • 65
    • 0001684396 scopus 로고
    • Invariant helical subspaces for the navier-stokes equations
    • Mahalov, A., Titi, E. S., and, Leibovich, S. 1990. Invariant helical subspaces for the Navier-Stokes equations. Arch. Rat. Mech. Anal., 112: 193-222.
    • (1990) Arch. Rat. Mech. Anal. , vol.112 , pp. 193-222
    • Mahalov, A.1    Titi, E.S.2    Leibovich, S.3
  • 66
    • 70450177158 scopus 로고    scopus 로고
    • Global existence and uniqueness of weak solutions of 3-D Euler equations with helical symmetry in the absence of vorticity stretching
    • Ettinger, B., and, Titi, E. S. 2009. Global existence and uniqueness of weak solutions of 3-D Euler equations with helical symmetry in the absence of vorticity stretching. SIAM J. Math. Anal., 41(1) 269-296.
    • (2009) SIAM J. Math. Anal. , vol.41 , Issue.1 , pp. 269-296
    • Ettinger, B.1    Titi, E.S.2
  • 67
    • 84990576577 scopus 로고
    • Partial regularity of suitable weak solutions of the Navier-Stokes equations
    • Caffarelli, L., Kohn, R., and, Nirenberg, L. 1982. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math., 35: 771-831.
    • (1982) Commun. Pure Appl. Math. , vol.35 , pp. 771-831
    • Caffarelli, L.1    Kohn, R.2    Nirenberg, L.3
  • 68
    • 0036017441 scopus 로고    scopus 로고
    • Navier-Stokes equations with lower bounds on the pressure
    • DOI 10.1007/s002050200199
    • Seregin, G., and, Sverák, V. 2002. Navier-Stokes equations with lower bounds on the pressure. Arch. Rational Mech. Anal., 163: 65-86. (Pubitemid 36195110)
    • (2002) Archive for Rational Mechanics and Analysis , vol.163 , Issue.1 , pp. 65-86
    • Seregin, G.1    Sverak, V.2
  • 69
    • 0001807443 scopus 로고
    • The initial value problem for the Navier-Stokes equations
    • University of Wisconsin Press, Madison, WI
    • Serrin, J. 1963. The initial value problem for the Navier-Stokes equations. Nonlinear Problems (Proc. Symp., Madison, WI), University of Wisconsin Press, Madison, WI,: 69-98.
    • (1963) Nonlinear Problems (Proc. Symp., Madison, WI) , pp. 69-98
    • Serrin, J.1
  • 70
    • 33846833851 scopus 로고    scopus 로고
    • Remarks about the inviscid limit of the Navier-Stokes system
    • Masmoudi, N. 2007. Remarks about the inviscid limit of the Navier-Stokes system. Commun. Math. Phys., 270: 777-788.
    • (2007) Commun. Math. Phys. , vol.270 , pp. 777-788
    • Masmoudi, N.1
  • 71
    • 77249102612 scopus 로고    scopus 로고
    • On the convergence rate of the Euler-α, inviscid second-grade fluid, model to the Euler equations
    • Linshiz, J., and, Titi, E. S. 2010. On the convergence rate of the Euler-α, inviscid second-grade fluid, model to the Euler equations. J. Stat. Phys., 1389(1) 305-332.
    • (2010) J. Stat. Phys. , vol.138 , Issue.1 , pp. 305-332
    • Linshiz, J.1    Titi, E.S.2
  • 72
    • 34250940791 scopus 로고
    • Un théorème sur lexistence du mouvement plan dun fluide parfait homogène incompressible, pendant un temps infiniment long
    • Wolibner, W. 1933. Un théorème sur lexistence du mouvement plan dun fluide parfait homogène incompressible, pendant un temps infiniment long. Math. Z., 37(1) 698-726.
    • (1933) Math. Z. , vol.37 , Issue.1 , pp. 698-726
    • Wolibner, W.1
  • 73
    • 0040652193 scopus 로고
    • Remarks on zero viscosity limit for nonstationary navier-stokes flows with boundary. seminar on nonlinear partial differential equations
    • Berkeley, Calif., 1983 Springer, New York
    • Kato, T. 1984. Remarks on Zero Viscosity Limit for Nonstationary Navier-Stokes Flows With Boundary. Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., Vol. 2, Springer, New York,: 85-98.
    • (1984) Math. Sci. Res. Inst. Publ. , vol.2 , pp. 85-98
    • Kato, T.1
  • 74
    • 0039841935 scopus 로고    scopus 로고
    • A Kato type theorem on zero viscosity limit of Navier-Stokes flows
    • Wang, X. 2001. A Kato type theorem on zero viscosity limit of Navier-Stokes flows. Indiana Univ. Math. J., 50: 223-241. (Pubitemid 33576039)
    • (2001) Indiana University Mathematics Journal , vol.50 , Issue.SUPPL. , pp. 223-241
    • Wang, X.1
  • 75
    • 84857558969 scopus 로고    scopus 로고
    • The 3-D inviscid limit result under slip boundary conditions. A negative answer
    • da Veiga, H. Beirão, and, Crispo, F. 2012. The 3-D inviscid limit result under slip boundary conditions. A negative answer. J. Math. Fluid Mech., 14: 55-59. 1
    • (2012) J. Math. Fluid Mech. , vol.14 , pp. 55-59
    • Da Veiga, H.B.1    Crispo, F.2
  • 76
    • 84861412706 scopus 로고    scopus 로고
    • A missed persistence property for the Euler equations and its effect on inviscid limits
    • da Veiga, H. Beirão, and, Crispo, F. 2012. A missed persistence property for the Euler equations and its effect on inviscid limits. Nonlinearity, 25: 1661-1669.
    • (2012) Nonlinearity , vol.25 , pp. 1661-1669
    • Da Veiga, H.B.1    Crispo, F.2
  • 77
    • 84870881813 scopus 로고    scopus 로고
    • Vanishing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condition
    • Wang, L., Xin, Lizhen, Z., and, Zang, A. 2012. Vanishing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condition. J. Math. Fluid Mech., 14(4) 791-825.
    • (2012) J. Math. Fluid Mech. , vol.14 , Issue.4 , pp. 791-825
    • Wang, L.1    Xin, L.Z.2    Zang, A.3
  • 78
    • 0001987016 scopus 로고
    • Uber flussigkeits-bewegung bei sehr kleiner reibung
    • Heidelberg, Teubner, Leipzig
    • Prandtl, L. 1904. Uber flussigkeits-bewegung bei sehr kleiner reibung. In Actes du 3me Congrès international des Mathematiciens, Heidelberg, Teubner, Leipzig,: 484-491.
    • (1904) Actes du 3me Congrès International des Mathematiciens , pp. 484-491
    • Prandtl, L.1
  • 79
    • 77957908145 scopus 로고    scopus 로고
    • Examples of boundary layers associated with the incompressible Navier-Stokes equations
    • Wang, X. 2010. Examples of boundary layers associated with the incompressible Navier-Stokes equations. Chin. Ann. Math. Ser. B, 31 (5): 781-792.
    • (2010) Chin. Ann. Math. Ser. B , vol.31 , Issue.5 , pp. 781-792
    • Wang, X.1
  • 80
    • 0034403210 scopus 로고    scopus 로고
    • On the nonlinear instability of euler and Prandtlequations
    • Grenier, E. 2000. On the nonlinear instability of Euler and Prandtlequations. Commun. Pure Appl. Math., 53 (9): 1067-1109.
    • (2000) Commun. Pure Appl. Math. , vol.53 , Issue.9 , pp. 1067-1109
    • Grenier, E.1
  • 81
    • 0039283401 scopus 로고
    • On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid
    • Oleinik, O. 1966. On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid. J. Appl. Math. Mech., 30 (5): 952-974.
    • (1966) J. Appl. Math. Mech. , vol.30 , Issue.5 , pp. 952-974
    • Oleinik, O.1
  • 83
    • 77749295388 scopus 로고    scopus 로고
    • On the ill-posedness of the Prandtl equation
    • Gérard-Varet, D., and, Dormy, E. 2010. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc., 23: 591-609.
    • (2010) J. Amer. Math. Soc. , vol.23 , pp. 591-609
    • Gérard-Varet, D.1    Dormy, E.2
  • 84
    • 84859471486 scopus 로고    scopus 로고
    • Remarks on the ill-posedeness of the Prandtl equation
    • Gérard-Varet, D., and, Nguyen, T. 2012. Remarks on the ill-posedeness of the Prandtl equation. Asymptot. Anal., 77 (1-2): 71-88.
    • (2012) Asymptot. Anal. , vol.77 , Issue.1-2 , pp. 71-88
    • Gérard-Varet, D.1    Nguyen, T.2
  • 86
    • 0032371860 scopus 로고    scopus 로고
    • Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations
    • Sammartino, M., and, Caflisch, R. 1998. Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I-II. Commun. Math. Phys., 192 (2): 433-461. 463-473. (Pubitemid 128357143)
    • (1998) Communications in Mathematical Physics , vol.192 , Issue.2 , pp. 433-461
    • Sammartino, M.1    Caflisch, R.E.2
  • 87
    • 3142746818 scopus 로고    scopus 로고
    • M. Well-posedness of the boundary layer equations
    • Lombardo, M., Cannone, M., and, Sammartino, M. 2003. M. Well-posedness of the boundary layer equations. SIAM J. Math. Anal., 35 (4): 987-1004.
    • (2003) SIAM J. Math. Anal. , vol.35 , Issue.4 , pp. 987-1004
    • Lombardo, M.1    Cannone, M.2    Sammartino, M.3
  • 88
    • 84867697714 scopus 로고    scopus 로고
    • On the local existence of analytic solutions to the Prandtl boundary layer equations
    • Kukavica, I., and, Vicol, V. 2013. On the local existence of analytic solutions to the Prandtl boundary layer equations. Commun. Math. Sci., 11 (1): 269-292.
    • (2013) Commun. Math. Sci. , vol.11 , Issue.1 , pp. 269-292
    • Kukavica, I.1    Vicol, V.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.