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December  2016, 5(4): 647-659. doi: 10.3934/eect.2016023

A local asymptotic expansion for a solution of the Stokes system

Güher Çamliyurt 1, and  Igor Kukavica 1,

1. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, United States

Received  May 2016 Revised  September 2016 Published  October 2016

We consider solutions of the Stokes system in a neighborhood of a point in which the velocity $u$ vanishes of order $d$. We prove that there exists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficients of degree $d$ which approximates the solution $u$ of order $d+\alpha$ for certain $\alpha>0$. The polynomial $P$ satisfies a Stokes equation with a forcing term which is a sum of two polynomials in $x$ of degrees $d-1$ and $d$.
Keywords: Navier-Stokes equations, Stokes equations, Oseen equations, unique continuation..
Mathematics Subject Classification: Primary: 76D07, 34A12; Secondary: 35Q3.
Citation: Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023
References:
[1]

G. Alessandrini and L. Escauriaza, Null-controllability of one-dimensional parabolic equations, ESAIM Control Optim. Calc. Var., 14 (2008), 284-293. doi: 10.1051/cocv:2007055.  Google Scholar

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G. Alessandrini, A. Morassi, E. Rosset and S. Vessella, On doubling inequalities for elliptic systems, J. Math. Anal. Appl., 357 (2009), 349-355. doi: 10.1016/j.jmaa.2009.04.024.  Google Scholar

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G. Alessandrini and S. Vessella, Local behaviour of solutions to parabolic equations, Comm. Partial Differential Equations, 13 (1988), 1041-1058. doi: 10.1080/03605308808820567.  Google Scholar

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L. Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math., 8 (1955), 473-496. doi: 10.1002/cpa.3160080404.  Google Scholar

[5]

G. Camliyurt and I. Kukavica, Quantitative unique continuation for a parabolic equation,, (submitted)., ().   Google Scholar

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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), 491-535. doi: 10.1090/S0002-9947-01-02860-4.  Google Scholar

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L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.  Google Scholar

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L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II, Indiana Univ. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.  Google Scholar

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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), 222-240. doi: 10.1007/BF00281533.  Google Scholar

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C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.  Google Scholar

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C. Fabre and G. Lebeau, Régularité et unicité pour le problème de Stokes, Comm. Partial Differential Equations, 27 (2002), 437-475. doi: 10.1081/PDE-120002863.  Google Scholar

[13]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[14]

Q. Han, Schauder estimates for elliptic operators with applications to nodal sets, J. Geom. Anal., 10 (2000), 455-480. doi: 10.1007/BF02921945.  Google Scholar

[15]

Q. Han, On the Schauder estimates of solutions to parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 1-26 (1999).  Google Scholar

[16]

R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom., 30 (1989), 505-522.  Google Scholar

[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), 463-494. doi: 10.2307/1971205.  Google Scholar

[18]

C. E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, 69-90. doi: 10.1007/BFb0086794.  Google Scholar

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C. E. Kenig, Some recent applications of unique continuation, Recent developments in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 439 (2007), 25-56. doi: 10.1090/conm/439/08462.  Google Scholar

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I. Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240. doi: 10.1215/S0012-7094-98-09111-6.  Google Scholar

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I. Kukavica, Length of vorticity nodal sets for solutions of the 2D Navier-Stokes equations, Comm. Partial Differential Equations, 28 (2003), 771-793. doi: 10.1081/PDE-120020496.  Google Scholar

[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), 305-366. doi: 10.1080/03605300902740395.  Google Scholar

[23]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[25]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[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, Vol. V (Paris, 1981/1982), Res. Notes in Math., vol. 93, Pitman, Boston, MA, 1983, 260-275.  Google Scholar

[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), 178-254, 257, Boundary value problems of mathematical physics and related questions in the theory of functions, 9.  Google Scholar

[28]

S. Vessella, Carleman estimates, optimal three cylinder inequalities and unique continuation properties for parabolic operators, Progress in analysis, Vol. I, II (Berlin, 2001), World Sci. Publ., River Edge, NJ, 2003, 485-492.  Google Scholar

show all references

References:
[1]

G. Alessandrini and L. Escauriaza, Null-controllability of one-dimensional parabolic equations, ESAIM Control Optim. Calc. Var., 14 (2008), 284-293. doi: 10.1051/cocv:2007055.  Google Scholar

[2]

G. Alessandrini, A. Morassi, E. Rosset and S. Vessella, On doubling inequalities for elliptic systems, J. Math. Anal. Appl., 357 (2009), 349-355. doi: 10.1016/j.jmaa.2009.04.024.  Google Scholar

[3]

G. Alessandrini and S. Vessella, Local behaviour of solutions to parabolic equations, Comm. Partial Differential Equations, 13 (1988), 1041-1058. doi: 10.1080/03605308808820567.  Google Scholar

[4]

L. Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math., 8 (1955), 473-496. doi: 10.1002/cpa.3160080404.  Google Scholar

[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), 491-535. doi: 10.1090/S0002-9947-01-02860-4.  Google Scholar

[7]

H. Donnelly and C. Fefferman, Growth and geometry of eigenfunctions of the Laplacian, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 122 (1990), 635-655.  Google Scholar

[8]

L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.  Google Scholar

[9]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II, Indiana Univ. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.  Google Scholar

[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), 222-240. doi: 10.1007/BF00281533.  Google Scholar

[11]

C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.  Google Scholar

[12]

C. Fabre and G. Lebeau, Régularité et unicité pour le problème de Stokes, Comm. Partial Differential Equations, 27 (2002), 437-475. doi: 10.1081/PDE-120002863.  Google Scholar

[13]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[14]

Q. Han, Schauder estimates for elliptic operators with applications to nodal sets, J. Geom. Anal., 10 (2000), 455-480. doi: 10.1007/BF02921945.  Google Scholar

[15]

Q. Han, On the Schauder estimates of solutions to parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 1-26 (1999).  Google Scholar

[16]

R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom., 30 (1989), 505-522.  Google Scholar

[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), 463-494. doi: 10.2307/1971205.  Google Scholar

[18]

C. E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, 69-90. doi: 10.1007/BFb0086794.  Google Scholar

[19]

C. E. Kenig, Some recent applications of unique continuation, Recent developments in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 439 (2007), 25-56. doi: 10.1090/conm/439/08462.  Google Scholar

[20]

I. Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240. doi: 10.1215/S0012-7094-98-09111-6.  Google Scholar

[21]

I. Kukavica, Length of vorticity nodal sets for solutions of the 2D Navier-Stokes equations, Comm. Partial Differential Equations, 28 (2003), 771-793. doi: 10.1081/PDE-120020496.  Google Scholar

[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), 305-366. doi: 10.1080/03605300902740395.  Google Scholar

[23]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[25]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[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, Vol. V (Paris, 1981/1982), Res. Notes in Math., vol. 93, Pitman, Boston, MA, 1983, 260-275.  Google Scholar

[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), 178-254, 257, Boundary value problems of mathematical physics and related questions in the theory of functions, 9.  Google Scholar

[28]

S. Vessella, Carleman estimates, optimal three cylinder inequalities and unique continuation properties for parabolic operators, Progress in analysis, Vol. I, II (Berlin, 2001), World Sci. Publ., River Edge, NJ, 2003, 485-492.  Google Scholar

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