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On the Muskat problem
A local asymptotic expansion for a solution of the Stokes system
1. | Department of Mathematics, University of Southern California, Los Angeles, CA 90089, United States |
References:
[1] |
G. Alessandrini and L. Escauriaza, Null-controllability of one-dimensional parabolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 284.
doi: 10.1051/cocv:2007055. |
[2] |
G. Alessandrini, A. Morassi, E. Rosset and S. Vessella, On doubling inequalities for elliptic systems,, J. Math. Anal. Appl., 357 (2009), 349.
doi: 10.1016/j.jmaa.2009.04.024. |
[3] |
G. Alessandrini and S. Vessella, Local behaviour of solutions to parabolic equations,, Comm. Partial Differential Equations, 13 (1988), 1041.
doi: 10.1080/03605308808820567. |
[4] |
L. Bers, Local behavior of solutions of general linear elliptic equations,, Comm. Pure Appl. Math., 8 (1955), 473.
doi: 10.1002/cpa.3160080404. |
[5] |
G. Camliyurt and I. Kukavica, Quantitative unique continuation for a parabolic equation,, (submitted)., (). Google Scholar |
[6] |
B. Canuto, E. Rosset and S. Vessella, Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries,, Trans. Amer. Math. Soc., 354 (2002), 491.
doi: 10.1090/S0002-9947-01-02860-4. |
[7] |
H. Donnelly and C. Fefferman, Growth and geometry of eigenfunctions of the Laplacian,, Analysis and partial differential equations, 122 (1990), 635.
|
[8] |
L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions,, Appl. Anal., 85 (2006), 205.
doi: 10.1080/00036810500277082. |
[9] |
L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II,, Indiana Univ. Math. J., 50 (2001), 1149.
doi: 10.1512/iumj.2001.50.1937. |
[10] |
E. B. Fabes, B. F. Jones and N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in $L^p$,, Arch. Rational Mech. Anal., 45 (1972), 222.
doi: 10.1007/BF00281533. |
[11] |
C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes,, Comm. Partial Differential Equations, 21 (1996), 573.
doi: 10.1080/03605309608821198. |
[12] |
C. Fabre and G. Lebeau, Régularité et unicité pour le problème de Stokes,, Comm. Partial Differential Equations, 27 (2002), 437.
doi: 10.1081/PDE-120002863. |
[13] |
N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation,, Indiana Univ. Math. J., 35 (1986), 245.
doi: 10.1512/iumj.1986.35.35015. |
[14] |
Q. Han, Schauder estimates for elliptic operators with applications to nodal sets,, J. Geom. Anal., 10 (2000), 455.
doi: 10.1007/BF02921945. |
[15] |
Q. Han, On the Schauder estimates of solutions to parabolic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 1.
|
[16] |
R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations,, J. Differential Geom., 30 (1989), 505.
|
[17] |
D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, With an appendix by E. M. Stein,, Ann. of Math. (2), 121 (1985), 463.
doi: 10.2307/1971205. |
[18] |
C. E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation,, Harmonic analysis and partial differential equations (El Escorial, (1987), 69.
doi: 10.1007/BFb0086794. |
[19] |
C. E. Kenig, Some recent applications of unique continuation,, Recent developments in nonlinear partial differential equations, 439 (2007), 25.
doi: 10.1090/conm/439/08462. |
[20] |
I. Kukavica, Quantitative uniqueness for second-order elliptic operators,, Duke Math. J., 91 (1998), 225.
doi: 10.1215/S0012-7094-98-09111-6. |
[21] |
I. Kukavica, Length of vorticity nodal sets for solutions of the 2D Navier-Stokes equations,, Comm. Partial Differential Equations, 28 (2003), 771.
doi: 10.1081/PDE-120020496. |
[22] |
H. Koch and D. Tataru, Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients,, Comm. Partial Differential Equations, 34 (2009), 305.
doi: 10.1080/03605300902740395. |
[23] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hall/CRC Research Notes in Mathematics, (2002).
doi: 10.1201/9781420035674. |
[24] |
G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).
doi: 10.1142/3302. |
[25] |
J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations,, J. Differential Equations, 66 (1987), 118.
doi: 10.1016/0022-0396(87)90043-X. |
[26] |
J.-C. Saut and B. Scheurer, {Unique combination and uniqueness of the Cauchy problem for elliptic operators with unbounded coefficients,, Nonlinear partial differential equations and their applications. Collège de France seminar, (1981), 260.
|
[27] |
V. A. Solonnikov, Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations,, Zap. Naučn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI), 59 (1976), 178.
|
[28] |
S. Vessella, Carleman estimates, optimal three cylinder inequalities and unique continuation properties for parabolic operators,, Progress in analysis, (2001), 485.
|
show all references
References:
[1] |
G. Alessandrini and L. Escauriaza, Null-controllability of one-dimensional parabolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 284.
doi: 10.1051/cocv:2007055. |
[2] |
G. Alessandrini, A. Morassi, E. Rosset and S. Vessella, On doubling inequalities for elliptic systems,, J. Math. Anal. Appl., 357 (2009), 349.
doi: 10.1016/j.jmaa.2009.04.024. |
[3] |
G. Alessandrini and S. Vessella, Local behaviour of solutions to parabolic equations,, Comm. Partial Differential Equations, 13 (1988), 1041.
doi: 10.1080/03605308808820567. |
[4] |
L. Bers, Local behavior of solutions of general linear elliptic equations,, Comm. Pure Appl. Math., 8 (1955), 473.
doi: 10.1002/cpa.3160080404. |
[5] |
G. Camliyurt and I. Kukavica, Quantitative unique continuation for a parabolic equation,, (submitted)., (). Google Scholar |
[6] |
B. Canuto, E. Rosset and S. Vessella, Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries,, Trans. Amer. Math. Soc., 354 (2002), 491.
doi: 10.1090/S0002-9947-01-02860-4. |
[7] |
H. Donnelly and C. Fefferman, Growth and geometry of eigenfunctions of the Laplacian,, Analysis and partial differential equations, 122 (1990), 635.
|
[8] |
L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions,, Appl. Anal., 85 (2006), 205.
doi: 10.1080/00036810500277082. |
[9] |
L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II,, Indiana Univ. Math. J., 50 (2001), 1149.
doi: 10.1512/iumj.2001.50.1937. |
[10] |
E. B. Fabes, B. F. Jones and N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in $L^p$,, Arch. Rational Mech. Anal., 45 (1972), 222.
doi: 10.1007/BF00281533. |
[11] |
C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes,, Comm. Partial Differential Equations, 21 (1996), 573.
doi: 10.1080/03605309608821198. |
[12] |
C. Fabre and G. Lebeau, Régularité et unicité pour le problème de Stokes,, Comm. Partial Differential Equations, 27 (2002), 437.
doi: 10.1081/PDE-120002863. |
[13] |
N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation,, Indiana Univ. Math. J., 35 (1986), 245.
doi: 10.1512/iumj.1986.35.35015. |
[14] |
Q. Han, Schauder estimates for elliptic operators with applications to nodal sets,, J. Geom. Anal., 10 (2000), 455.
doi: 10.1007/BF02921945. |
[15] |
Q. Han, On the Schauder estimates of solutions to parabolic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 1.
|
[16] |
R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations,, J. Differential Geom., 30 (1989), 505.
|
[17] |
D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, With an appendix by E. M. Stein,, Ann. of Math. (2), 121 (1985), 463.
doi: 10.2307/1971205. |
[18] |
C. E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation,, Harmonic analysis and partial differential equations (El Escorial, (1987), 69.
doi: 10.1007/BFb0086794. |
[19] |
C. E. Kenig, Some recent applications of unique continuation,, Recent developments in nonlinear partial differential equations, 439 (2007), 25.
doi: 10.1090/conm/439/08462. |
[20] |
I. Kukavica, Quantitative uniqueness for second-order elliptic operators,, Duke Math. J., 91 (1998), 225.
doi: 10.1215/S0012-7094-98-09111-6. |
[21] |
I. Kukavica, Length of vorticity nodal sets for solutions of the 2D Navier-Stokes equations,, Comm. Partial Differential Equations, 28 (2003), 771.
doi: 10.1081/PDE-120020496. |
[22] |
H. Koch and D. Tataru, Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients,, Comm. Partial Differential Equations, 34 (2009), 305.
doi: 10.1080/03605300902740395. |
[23] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hall/CRC Research Notes in Mathematics, (2002).
doi: 10.1201/9781420035674. |
[24] |
G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).
doi: 10.1142/3302. |
[25] |
J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations,, J. Differential Equations, 66 (1987), 118.
doi: 10.1016/0022-0396(87)90043-X. |
[26] |
J.-C. Saut and B. Scheurer, {Unique combination and uniqueness of the Cauchy problem for elliptic operators with unbounded coefficients,, Nonlinear partial differential equations and their applications. Collège de France seminar, (1981), 260.
|
[27] |
V. A. Solonnikov, Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations,, Zap. Naučn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI), 59 (1976), 178.
|
[28] |
S. Vessella, Carleman estimates, optimal three cylinder inequalities and unique continuation properties for parabolic operators,, Progress in analysis, (2001), 485.
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