Carleman estimates for parabolic equations with nonhomogeneous boundary conditions

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.

Keywords

Controllability / Parabolic equations / Carleman estimates

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., 1939, 26B: 1-9

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

[3] Evans L.. Partial Differential Equations. Graduate Studies in Mathematics, 1998, Providence, RI: A. M. S.

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

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

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

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

[8] Hörmander L.. The spectral function of elliptic operators. Acta Math., 1968, 121: 193-218

[9] Imanuvilov O.. Controllability of parabolic equations. Mat. Sb., 1995, 186(6): 109-132

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

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

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

[13] Imanuvilov O., 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., 2003, 39: 227-274

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

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

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

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

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