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Carleman estimates for parabolic equations with nonhomogeneous boundary conditions

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Abstract

The authors prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On the basis of this estimate, improved Carleman estimates for the Stokes system and for a system of parabolic equations with a penalty term are obtained. This system can be viewed as an approximation of the Stokes system.

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References

  1. Carleman, T., Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendentes, Ark. Mat. Astr. Fys. 26B, 1939, 1–9.

    MathSciNet  Google Scholar 

  2. Dubrovin, B. A., Fomenko, A. T. and Novikov, S. P., Modern Geometry: Methods and Applications, Part II, The Geometry and Topology of Manifolds, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  3. Evans, L., Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, A. M. S., Providence, RI, 1998.

    Google Scholar 

  4. Fernandez-Cara, E., Guerrero, S., Imanuvilov, O. and Puel, J. P., Some controllability results for the N-dimensional Navier-Stokes and Boussinesq system, SIAM J. Cont. Optim., 45, 2006, 146–173.

    Article  MATH  MathSciNet  Google Scholar 

  5. Fernandez-Cara, E., Guerrero, S., Imanuvilov, O. and Puel, J. P., Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., 83(12), 2005, 1501–1542.

    MathSciNet  Google Scholar 

  6. Fabre, C. and Lebeau, G., Prolongement unique des solutions de l’équation de Stokes, Comm. Part. Diff. Eqs., 21, 1996, 573–596.

    Article  MATH  MathSciNet  Google Scholar 

  7. Hörmander, L., Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963.

    MATH  Google Scholar 

  8. Hörmander, L., The spectral function of elliptic operators, Acta Math., 121, 1968, 193–218.

    Article  MATH  MathSciNet  Google Scholar 

  9. Imanuvilov, O., Controllability of parabolic equations, Mat. Sb., 186(6), 1995, 109–132.

    MathSciNet  Google Scholar 

  10. Imanuvilov, O., On exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Cale. Var., 3, 1998, 97–131.

    MATH  MathSciNet  Google Scholar 

  11. Imanuvilov, O., Remarks on exact controllability for Navier-Stokes equations, ESAIM Control Optim. Cale. Var., 6, 2001, 39–72.

    MATH  MathSciNet  Google Scholar 

  12. Imanuvilov, O. and Puel, J. P., Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., 16, 2003, 883–913.

    Article  MathSciNet  Google Scholar 

  13. Imanuvilov, O. and Yamamoto, M., Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39, 2003, 227–274.

    Article  MATH  MathSciNet  Google Scholar 

  14. Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

  15. Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, New Jersey, 1993.

    MATH  Google Scholar 

  16. Taylor, M., Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Berlin, 1991.

    MATH  Google Scholar 

  17. Taylor, M., Pseudodifferential Operators, Princeton University Press, New Jersey, 1981.

    MATH  Google Scholar 

  18. Temam, R., Navier-Stokes Equatons, A. M. S., Providence, RI, 2001.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins CO, 80523, USA

    Oleg Yu Imanuvilov

  2. Laboratoire de Mathematiques de Versailles, Université de Versailles-St. Quentin, 45 Avenue des Etats Unis, 78035, Versailles, France

    Jean Pierre Puel

  3. Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153, Japan

    Masahiro Yamamoto

Authors
  1. Oleg Yu Imanuvilov
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  2. Jean Pierre Puel
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  3. Masahiro Yamamoto
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Corresponding author

Correspondence to Oleg Yu Imanuvilov.

Additional information

Project supported by NSF grant DMS (No. 0808130) and ANR Project (No. C-QUID 06-BLAN-0052).

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Imanuvilov, O.Y., Puel, J.P. & Yamamoto, M. Carleman estimates for parabolic equations with nonhomogeneous boundary conditions. Chin. Ann. Math. Ser. B 30, 333–378 (2009). https://doi.org/10.1007/s11401-008-0280-x

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  • DOI: https://doi.org/10.1007/s11401-008-0280-x

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