메뉴 건너뛰기




Volumn 9, Issue 1, 2007, Pages 31-65

On generalized solutions of two-phase flows for viscous incompressible fluids

Author keywords

Free boundary problems; Measure valued solutions; Surface tension; Two phase flow; Varifold solutions

Indexed keywords


EID: 34248181466     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/155     Document Type: Article
Times cited : (57)

References (36)
  • 1
    • 33644784642 scopus 로고    scopus 로고
    • q-Sobolev spaces
    • Zbl pre05056142 MR 2106120
    • q-Sobolev spaces. Adv. Differential Equations 10 (2005), 45-64. Zbl pre05056142 MR 2106120
    • (2005) Adv. Differential Equations , vol.10 , pp. 45-64
    • ABELS, H.1
  • 2
    • 34248149626 scopus 로고    scopus 로고
    • AMBROSIO, L., FUSCO, N., & PALLARA, D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr., Clarendon Press, Oxford (2000). Zbl 0957.49001 MR 1857292
    • AMBROSIO, L., FUSCO, N., & PALLARA, D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr., Clarendon Press, Oxford (2000). Zbl 0957.49001 MR 1857292
  • 3
    • 84980139502 scopus 로고
    • The initial value problem for the Navier-Stokes equations with a free surface
    • Zbl 0464.76028 MR 0611750
    • BEALE, J. T. The initial value problem for the Navier-Stokes equations with a free surface. Comm. Pure Appl. Math. 34 (1981), 359-392. Zbl 0464.76028 MR 0611750
    • (1981) Comm. Pure Appl. Math , vol.34 , pp. 359-392
    • BEALE, J.T.1
  • 4
    • 0021197315 scopus 로고
    • Large-time regularity of viscous surface waves
    • Zbl 0545.76029 MR 0721189
    • BEALE, J. T. Large-time regularity of viscous surface waves. Arch. Ration. Mech. Anal. 84 (1984), 307-352. Zbl 0545.76029 MR 0721189
    • (1984) Arch. Ration. Mech. Anal , vol.84 , pp. 307-352
    • BEALE, J.T.1
  • 5
    • 34248150112 scopus 로고    scopus 로고
    • BERGH, J., & LÖFSTRÖM, J. Interpolation Spaces. Springer, Berlin (1976). Zbl 0344.46071 MR 0482275
    • BERGH, J., & LÖFSTRÖM, J. Interpolation Spaces. Springer, Berlin (1976). Zbl 0344.46071 MR 0482275
  • 6
    • 22044442615 scopus 로고    scopus 로고
    • A multi-phase Mullins-Sekerka system: Matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem
    • Zbl 0924.35199 MR 1632810
    • BRONSARD, L., GARCKE, H., & STOTH, B. A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem. Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 481-506. Zbl 0924.35199 MR 1632810
    • (1998) Proc. Roy. Soc. Edinburgh Sect. A , vol.128 , pp. 481-506
    • BRONSARD, L.1    GARCKE, H.2    STOTH, B.3
  • 7
    • 34248225564 scopus 로고    scopus 로고
    • DIAMOND, P., & KLOEDEN, P. Metric Spaces of Fuzzy Sets. World Sci., River Edge, NJ (1994). Zbl 0873.54019 MR 1337027
    • DIAMOND, P., & KLOEDEN, P. Metric Spaces of Fuzzy Sets. World Sci., River Edge, NJ (1994). Zbl 0873.54019 MR 1337027
  • 8
    • 34248232713 scopus 로고    scopus 로고
    • DIESTEL, J., & UHL, J. J., Jr. Vector Measures. Amer. Math. Soc., Providence, RI (1977). Zbl 0369.46039 MR 0453964
    • DIESTEL, J., & UHL, J. J., Jr. Vector Measures. Amer. Math. Soc., Providence, RI (1977). Zbl 0369.46039 MR 0453964
  • 9
    • 0002306354 scopus 로고
    • Ordinary differential equations, transport theory and Sobolev spaces
    • Zbl 0696.34049 MR 1022305
    • DIPERNA, R. J., & LIONS, P.-L. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511-547. Zbl 0696.34049 MR 1022305
    • (1989) Invent. Math , vol.98 , pp. 511-547
    • DIPERNA, R.J.1    LIONS, P.-L.2
  • 10
    • 21344481406 scopus 로고
    • On global weak solutions of the nonstationary two phase Stokes flow
    • Zbl 0806.35137 MR 1271315
    • GIGA, Y., & TAKAHASHI, S. On global weak solutions of the nonstationary two phase Stokes flow. SIAM J. Math. Anal. 25(1994), 876-893. Zbl 0806.35137 MR 1271315
    • (1994) SIAM J. Math. Anal , vol.25 , pp. 876-893
    • GIGA, Y.1    TAKAHASHI, S.2
  • 11
    • 34248184233 scopus 로고    scopus 로고
    • GOMEZ, N., & ZOLÉSIO, J.-P. Existence of free-boundary for a two non-Newtonian fluids problem. Shape Optimization and Optimal Design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math. 216, Dekker, New York (2001), 289-300. Zbl 0994.76006 MR 1816849
    • GOMEZ, N., & ZOLÉSIO, J.-P. Existence of free-boundary for a two non-Newtonian fluids problem. Shape Optimization and Optimal Design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math. 216, Dekker, New York (2001), 289-300. Zbl 0994.76006 MR 1816849
  • 12
    • 34248148780 scopus 로고    scopus 로고
    • JOSEPH, D. D., & RENARDY, Y. Y. Fundamentals of Two-Fluid Dynamics. Part I: Mathematical Theory and Applications. Interdiscip. Appl. Math. 3, Springer, New York (1993). Zbl 0784.76002 MR 1200237
    • JOSEPH, D. D., & RENARDY, Y. Y. Fundamentals of Two-Fluid Dynamics. Part I: Mathematical Theory and Applications. Interdiscip. Appl. Math. 3, Springer, New York (1993). Zbl 0784.76002 MR 1200237
  • 13
    • 34248230410 scopus 로고    scopus 로고
    • LIONS, J.-L. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). Zbl 0189.40603 MR 0259693
    • LIONS, J.-L. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). Zbl 0189.40603 MR 0259693
  • 14
    • 34248229403 scopus 로고    scopus 로고
    • LUCKHAUS, S. Uniform rectifiability from mean curvature bounds. Recent Advances in Elliptic and Parabolic Problems, World Sci. (2005), 197-201. MR 2172575
    • LUCKHAUS, S. Uniform rectifiability from mean curvature bounds. Recent Advances in Elliptic and Parabolic Problems, World Sci. (2005), 197-201. MR 2172575
  • 15
    • 0000850415 scopus 로고
    • Implicit time discretization for the mean curvature flow equation
    • Zbl 0821.35003 MR 1386964
    • LUCKHAUS, S., & STURZENHECKER, T. Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differential Equations 3 (1995), 253-271. Zbl 0821.35003 MR 1386964
    • (1995) Calc. Var. Partial Differential Equations , vol.3 , pp. 253-271
    • LUCKHAUS, S.1    STURZENHECKER, T.2
  • 16
    • 34248170388 scopus 로고    scopus 로고
    • MÁLEK, J., NEČAS, J., ROKYTA, M., & RŮŽIČKA, M. Weak and Measure-Valued Solutions to Evolutionary PDEs. Appl. Math. Math. Comput. 13, Chapman & Hall, London (1996). Zbl 0851.35002 MR 1409366
    • MÁLEK, J., NEČAS, J., ROKYTA, M., & RŮŽIČKA, M. Weak and Measure-Valued Solutions to Evolutionary PDEs. Appl. Math. Math. Comput. 13, Chapman & Hall, London (1996). Zbl 0851.35002 MR 1409366
  • 17
    • 0013308784 scopus 로고
    • An existence theorem for the multifluid Navier-Stokes problem
    • Zbl 0842.35079 MR 1356130
    • NOURI, A., & POUPAUD, F. An existence theorem for the multifluid Navier-Stokes problem. J. Differential Equations 122 (1995), 71-88. Zbl 0842.35079 MR 1356130
    • (1995) J. Differential Equations , vol.122 , pp. 71-88
    • NOURI, A.1    POUPAUD, F.2
  • 18
    • 0031226919 scopus 로고    scopus 로고
    • An existence theorem for the multi-fluid Stokes problem
    • Zbl 0882.35091 MR 1466141
    • NOURI, A., POUPAUD, F., & DEMAY, Y. An existence theorem for the multi-fluid Stokes problem. Quart. Appl. Math. 55 (1997), 421-435. Zbl 0882.35091 MR 1466141
    • (1997) Quart. Appl. Math , vol.55 , pp. 421-435
    • NOURI, A.1    POUPAUD, F.2    DEMAY, Y.3
  • 19
    • 34248205348 scopus 로고    scopus 로고
    • PLOTNIKOV, P. I. On a class of curves that arises in a problem with a free boundary for Stokes flows. Sibirsk. Mat. Zh. 36 (1995), 619-627 (in Russian). Zbl 0861.76017 MR 1404886
    • PLOTNIKOV, P. I. On a class of curves that arises in a problem with a free boundary for Stokes flows. Sibirsk. Mat. Zh. 36 (1995), 619-627 (in Russian). Zbl 0861.76017 MR 1404886
  • 20
    • 34248148779 scopus 로고
    • Generalized solutions to a free boundary problem of motion of a non-Newtonian fluid
    • Zbl 0814.76007 MR 1248797
    • PLOTNIKOV, P. I. Generalized solutions to a free boundary problem of motion of a non-Newtonian fluid. Siberian Math. J. 34 (1993), 704-716. Zbl 0814.76007 MR 1248797
    • (1993) Siberian Math. J , vol.34 , pp. 704-716
    • PLOTNIKOV, P.I.1
  • 21
    • 34248148258 scopus 로고    scopus 로고
    • PLOTNIKOV, P. I. Compressible Stokes flow driven by capillarity on a free surface. Navier-Stokes Equations and Related Nonlinear Problems (Palanga, 1997), H. Amann et al. (eds.), VSP, Utrecht (1998), 217-238. Zbl 0948.35101 MR 1690711
    • PLOTNIKOV, P. I. Compressible Stokes flow driven by capillarity on a free surface. Navier-Stokes Equations and Related Nonlinear Problems (Palanga, 1997), H. Amann et al. (eds.), VSP, Utrecht (1998), 217-238. Zbl 0948.35101 MR 1690711
  • 22
    • 0042922298 scopus 로고
    • A generalization of the Lions-Temam compact imbedding theorem
    • Zbl 0755.46013 MR 1090857
    • ROUBÍČEK, T. A generalization of the Lions-Temam compact imbedding theorem. Časopis Pěst. Mat. 115 (1990), 338-342. Zbl 0755.46013 MR 1090857
    • (1990) Časopis Pěst. Mat , vol.115 , pp. 338-342
    • ROUBÍČEK, T.1
  • 23
    • 34248187279 scopus 로고    scopus 로고
    • SALVI, R. On the existence of free surface problem for viscous incompressible flow. Topics in Mathematical Fluid Mechanics (Capo Miseno, 2000), P. G. Galdi et al. (eds.), Quad. Mat. 10, Aracne, Rome (2002), 247-275. Zbl 1050.35089 MR 2051777
    • SALVI, R. On the existence of free surface problem for viscous incompressible flow. Topics in Mathematical Fluid Mechanics (Capo Miseno, 2000), P. G. Galdi et al. (eds.), Quad. Mat. 10, Aracne, Rome (2002), 247-275. Zbl 1050.35089 MR 2051777
  • 24
    • 30444459643 scopus 로고    scopus 로고
    • q maximal regularity and viscous incompressible flows with free surface. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 151-155. MR 2189671
    • q maximal regularity and viscous incompressible flows with free surface. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 151-155. MR 2189671
  • 25
    • 85118286105 scopus 로고    scopus 로고
    • q -spaces for bounded and exterior domains. Mathematical Problems Relating to the Navier-Stokes Equation, Ser. Adv. Math. Appl. Sci. 11, World Sci., Singapore (1992), 1-35. Zbl 0791.35096 MR 1190728
    • q -spaces for bounded and exterior domains. Mathematical Problems Relating to the Navier-Stokes Equation, Ser. Adv. Math. Appl. Sci. 11, World Sci., Singapore (1992), 1-35. Zbl 0791.35096 MR 1190728
  • 26
    • 34250112428 scopus 로고
    • P(0, T; B)
    • Zbl 0629.46031 MR 0916688
    • P(0, T; B). Ann. Mat. Pura Appl. (4) 146 (1987), 65-96. Zbl 0629.46031 MR 0916688
    • (1987) Ann. Mat. Pura Appl. (4) , vol.146 , pp. 65-96
    • SIMON, J.1
  • 27
    • 34248179243 scopus 로고    scopus 로고
    • SIMON, L. Lectures on Geometric Measure Theory. Proc. Centre Math. Anal. Austral. Nat. Univ. 3, Centre Math. Anal., Austral. Nat. Univ., Canberra (1983). Zbl 0546.49019 MR 0756417
    • SIMON, L. Lectures on Geometric Measure Theory. Proc. Centre Math. Anal. Austral. Nat. Univ. 3, Centre Math. Anal., Austral. Nat. Univ., Canberra (1983). Zbl 0546.49019 MR 0756417
  • 28
    • 34248207385 scopus 로고    scopus 로고
    • SOHR, H. The Navier-Stokes Equations. Birkhäuser (2001). Zbl 0983.35004 MR 1928881
    • SOHR, H. The Navier-Stokes Equations. Birkhäuser (2001). Zbl 0983.35004 MR 1928881
  • 29
    • 34248200639 scopus 로고    scopus 로고
    • SOLONNIKOV, V. A. Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 178-254, 257 (in Russian). Zbl 0357.76026 MR 0460931
    • SOLONNIKOV, V. A. Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 178-254, 257 (in Russian). Zbl 0357.76026 MR 0460931
  • 30
    • 84956207366 scopus 로고
    • On the transient motion of an isolated volume of viscous incompressible fluid
    • Zbl 0850.76180 MR 0925094
    • SOLONNIKOV, V. A. On the transient motion of an isolated volume of viscous incompressible fluid. Math. USSR-Izv. 31 (1988), 381-405. Zbl 0850.76180 MR 0925094
    • (1988) Math. USSR-Izv , vol.31 , pp. 381-405
    • SOLONNIKOV, V.A.1
  • 31
    • 34249757566 scopus 로고
    • Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling
    • Zbl 0829.73010 MR 1346368
    • SONER, H. M. Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling. Arch. Ration. Mech. Anal. 131 (1995), 139-197. Zbl 0829.73010 MR 1346368
    • (1995) Arch. Ration. Mech. Anal , vol.131 , pp. 139-197
    • SONER, H.M.1
  • 32
    • 29344464359 scopus 로고    scopus 로고
    • A dynamical approach to large eddy simulation of turbulent flows: Existence of weak solutions
    • Zbl 1080.76035 MR 2185636
    • ŚWIERCZEWSKA, A. A dynamical approach to large eddy simulation of turbulent flows: existence of weak solutions. Math. Methods Appl. Sci. 29 (2006), 99-121. Zbl 1080.76035 MR 2185636
    • (2006) Math. Methods Appl. Sci , vol.29 , pp. 99-121
    • ŚWIERCZEWSKA, A.1
  • 33
    • 21844524710 scopus 로고
    • Large-time existence of surface waves in incompressible viscous fluids with or without surface tension
    • Zbl 0844.76025 MR 1346360
    • TANI, A., & TANAKA, N. Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Ration. Mech. Anal. 130 (1995), 303-314. Zbl 0844.76025 MR 1346360
    • (1995) Arch. Ration. Mech. Anal , vol.130 , pp. 303-314
    • TANI, A.1    TANAKA, N.2
  • 34
    • 34248190949 scopus 로고    scopus 로고
    • TRIEBEL, H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978). Zbl 0387.46033 MR 0503903
    • TRIEBEL, H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978). Zbl 0387.46033 MR 0503903
  • 35
    • 34248192526 scopus 로고    scopus 로고
    • On the Bernoulli free-boundary problem
    • Zbl 1022.35050 MR 1797931
    • WAGNER, A. On the Bernoulli free-boundary problem. Differential Integral Equations 14 (2001), 51-58. Zbl 1022.35050 MR 1797931
    • (2001) Differential Integral Equations , vol.14 , pp. 51-58
    • WAGNER, A.1
  • 36
    • 34248218785 scopus 로고    scopus 로고
    • ZEIDLER, E. Nonlinear Functional Analysis and its Applications. II/B. Springer, New York (1990). Zbl 0684.47029 MR 1033498
    • ZEIDLER, E. Nonlinear Functional Analysis and its Applications. II/B. Springer, New York (1990). Zbl 0684.47029 MR 1033498


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.