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

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

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 $d$.
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. 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)., ().

[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)., ().

[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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