## Abstract and Applied Analysis

### On Unique Continuation for Navier-Stokes Equations

#### Abstract

We study the unique continuation properties of solutions of the Navier-Stokes equations. We take advantage of rotation transformation of the Navier-Stokes equations to prove the “logarithmic convexity” of certain quantities, which measure the suitable Gaussian decay at infinity to obtain the Gaussian decay weighted estimates, as well as Carleman inequality. As a consequence we obtain sufficient conditions on the behavior of the solution at two different times ${t}_{0}=0$ and ${t}_{1}=1$ which guarantee the “global” unique continuation of solutions for the Navier-Stokes equations.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2014), Article ID 597946, 16 pages.

Dates
First available in Project Euclid: 15 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1429104583

Digital Object Identifier
doi:10.1155/2015/597946

Mathematical Reviews number (MathSciNet)
MR3335432

Zentralblatt MATH identifier
1356.35158

#### Citation

Duan, Zhiwen; Han, Shuxia; Sun, Peipei. On Unique Continuation for Navier-Stokes Equations. Abstr. Appl. Anal. 2015, Special Issue (2014), Article ID 597946, 16 pages. doi:10.1155/2015/597946. https://projecteuclid.org/euclid.aaa/1429104583

#### References

• J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,” Acta Mathematica, vol. 63, no. 1, pp. 193–248, 1934.
• O. A. Ladyzhenskaya, Mathematical Problems of the Dynamics of Viscous Incompressible Fluids, Nauka, Moscow, Russia, 2nd edition, 1970.
• P. L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press, New York, NY, USA, 1996.
• R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, The Netherlands, 1984.
• B. Q. Dong and Z. Zhang, “On the weak-strong uniqueness of Koch-Tataru's solution for the Navier-Stokes equations,” Journal of Differential Equations, vol. 256, no. 7, pp. 2406–2422, 2014.
• M. Cannone, F. Planchon, and M. Schonbek, “Strong solutions to the incompressible Navier-Stokes equations in the half-space,” Communications in Partial Differential Equations, vol. 25, no. 5-6, pp. 903–924, 2000.
• H. Koch and D. Tataru, “Well-posedness for the Navier-Stokes equations,” Advances in Mathematics, vol. 157, no. 1, pp. 22–35, 2001.
• O. A. Ladyzhenskaya, “Uniqueness and smoothness of generalized solutions of Navier-Stokes equations,” Zapiski Nauchnyh Seminarov POMI, vol. 5, pp. 169–185, 1967.
• Z. Lei and F. Lin, “Global mild solutions of Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 64, no. 9, pp. 1297–1304, 2011.
• A. Biswas, “Gevrey regularity for a class of dissipative equations with applications to decay,” Journal of Differential Equations, vol. 253, no. 10, pp. 2739–2764, 2012.
• L. Caffarelli, R. Kohn, and L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 35, no. 6, pp. 771–831, 1982.
• M. Struwe, “On partial regularity results for the Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 4, pp. 437–458, 1988.
• C. Amrouche, V. Girault, M. E. Schonbek, and T. P. Schonbek, “Pointwise decay of solutions and of higher derivatives to Navier-Stokes equations,” SIAM Journal on Mathematical Analysis, vol. 31, no. 4, pp. 740–753, 2000.
• W. Borchers and T. Miyakawa, “Algebraic ${L}_{2}$ decay for Navier-Stokes flows in exterior domains,” Acta Mathematica, vol. 165, no. 3-4, pp. 189–227, 1990.
• W. Borchers and T. Miyakawa, “${L}_{2}$ decay for the Navier-Stokes flow in halfspaces,” Mathematische Annalen, vol. 282, no. 1, pp. 139–155, 1988.
• T. Miyakawa and H. Sohr, “On energy inequality, smoothness and large time behavior in ${L}^{2}$ for weak solutions of the Navier-Stokes equations in exterior domains,” Mathematische Zeitschrift, vol. 199, no. 4, pp. 455–478, 1988.
• M. E. Schonbek, “L$^{2}$ decay for weak solutions of the Navier-Stokes equations,” Archive for Rational Mechanics and Analysis, vol. 88, no. 3, pp. 209–222, 1985.
• L. Escauriaza, G. Seregin, and V. Šverák, “Backward uniqueness for parabolic equations,” Archive for Rational Mechanics and Analysis, vol. 169, no. 2, pp. 147–157, 2003.
• G. P. Galdi and B. Straughan, “Stability of solutions of the Navier-Stokes equations backward in time,” Archive for Rational Mechanics and Analysis, vol. 101, no. 2, pp. 107–114, 1988.
• D. Hoff and E. Tsyganov, “Time analyticity and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow,” Journal of Differential Equations, vol. 245, no. 10, pp. 3068–3094, 2008.
• L. Hörmander, The Analysis of Linear Partial Differential Operators III, IV, Springer, Berlin, Germany, 1990.
• N. Garofalo and Z. Shen, “Carleman estimates for a subelliptic operator and unique continuation,” Annales de l'Institut Fourier, vol. 44, no. 1, pp. 129–166, 1994.
• J. Saut and B. Scheurer, “Unique continuation for some evolution equations,” Journal of Differential Equations, vol. 66, no. 1, pp. 118–139, 1987.
• X. Chen, “A strong unique continuation theorem for parabolic equations,” Mathematische Annalen, vol. 311, no. 4, pp. 603–630, 1998.
• R. Regbaoui, “Strong unique continuation for Stokes equations,” Communications in Partial Differential Equations, vol. 24, no. 9-10, pp. 1891–1902, 1999.
• M. Ignatova and I. Kukavica, “Strong unique continuation for the Navier-Stokes equation with non-analytic forcing,” Journal of Dynamics and Differential Equations, vol. 25, no. 1, pp. 1–15, 2013.
• K. Masuda, “On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation,” Proceedings of the Japan Academy, vol. 43, pp. 827–832, 1967.
• A. D. Ionescu and C. E. Kenig, “${L}^{p}$-Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations,” Acta Mathematica, vol. 193, no. 2, pp. 193–239, 2004.
• Z. Duan and P. Li, “On unique continuation for the generalized Schrödinger equations,” Journal of Mathematical Analysis and Applications, vol. 420, no. 2, pp. 1719–1743, 2014.
• J. Bourgain, “On the compactness of the support of solutions of dispersive equations,” International Mathematics Research Notices, vol. 9, pp. 437–447, 1997.
• L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, “Hardy's uncertainty principle, convexity and Schrödinger evolutions,” Journal of the European Mathematical Society (JEMS), vol. 10, no. 4, pp. 883–907, 2008.
• C. E. Kenig, G. Ponce, and L. Vega, “On unique continuation for nonlinear Schrödinger equations,” Communications on Pure and Applied Mathematics, vol. 56, no. 9, pp. 1247–1262, 2003.
• R. Finn, “Stationary solutions of the Navier-Stokes equations,” Proceedings of Symposia in Applied Mathematics: American Mathematical Society, vol. 17, pp. 121–153, 1965.
• R. H. Dyer and D. E. Edmunds, “Asymptotic behaviour of solutions of the stationary Navier-Stokes equations,” Journal of the London Mathematical Society. Second Series, vol. 44, pp. 340–346, 1969.
• G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, I and II, Springer, New York, NY, USA, 1994.
• C. Lin, G. Uhlmann, and J. Wang, “Optimal three-ball inequalities and quantitative uniqueness for the Stokes system,” Discrete and Continuous Dynamical Systems A, vol. 28, no. 3, pp. 1273–1290, 2010.
• C. L. Lin, G. Uhlmann, and J. N. Wang, “Asymptotic behavior of solutions of the stationary Navier-Stokes equations in an exterior domainčommentComment on ref. [37?]: Please update the information of this reference, if possible.,” http://arxiv.org/abs/1008.3953.
• C.-L. Lin, G. Nakamura, and J.-N. Wang, “Optimal three-ball inequalities and quantitative uniqueness for the Lamé system with Lipschitz coefficients,” Duke Mathematical Journal, vol. 155, no. 1, pp. 189–204, 2010.
• A. Bonami and B. Demange, “A survey on uncertainty principles related to quadratic forms,” Collectanea Mathematica, pp. 1–36, 2006.
• L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998. \endinput