A local asymptotic expansion for a solution of the Stokes system

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	G. Camliyurt and I. Kukavica, Quantitative unique continuation for a parabolic equation, (submitted).
[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.
[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.
[8]	L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.
[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.
[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.
[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.
[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.
[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.
[14]	Q. Han, Schauder estimates for elliptic operators with applications to nodal sets, J. Geom. Anal., 10 (2000), 455-480. 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), 1-26 (1999).
[16]	R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom., 30 (1989), 505-522.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[5]	G. Camliyurt and I. Kukavica, Quantitative unique continuation for a parabolic equation, (submitted).
[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.
[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.
[8]	L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.
[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.
[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.
[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.
[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.
[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.
[14]	Q. Han, Schauder estimates for elliptic operators with applications to nodal sets, J. Geom. Anal., 10 (2000), 455-480. 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), 1-26 (1999).
[16]	R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom., 30 (1989), 505-522.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[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.
[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.
[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.
[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.

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