# American Institute of Mathematical Sciences

December  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), 284-293. 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

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
 [1] Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159 [2] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [3] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [4] Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349 [5] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [6] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [7] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [8] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [9] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [10] Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269 [11] Roberto Triggiani. Unique continuation of boundary over-determined Stokes and Oseen eigenproblems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 645-677. doi: 10.3934/dcdss.2009.2.645 [12] Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537 [13] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277 [14] Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045 [15] Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027 [16] Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 [17] Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715 [18] Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239 [19] Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 [20] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141

2019 Impact Factor: 0.953