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Carleman Estimates for Second Order Parabolic Operators and Applications, a Unified Approach

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Carleman Estimates for Second Order Partial Differential Operators and Applications

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Abstract

In this chapter, we establish three Carleman estimates with different weight functions for second order parabolic operators. The first one is Theorem 3.1, which is used to obtain controllability/observability results for parabolic equations in Sect. 3.2. The second one is Theorem 3.2, via which, we solve an inverse parabolic problem in Sect. 3.3. The third one is Theorem 3.3, and it yields the SUCP of parabolic equations in Sect. 3.4.

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References

  1. Apraiz, J., Escauriaza, L., Wang, G., Zhang, C.: Observability inequalities and measurable sets. J. Eur. Math. Soc. 16, 2433–2475 (2014)

    Article  MathSciNet  Google Scholar 

  2. Barbu, V., Răşcanu, A., Tessitore, G.: Carleman estimate and cotrollability of linear stochastic heat equatons. Appl. Math. Optim. 47, 97–120 (2003)

    Article  MathSciNet  Google Scholar 

  3. Biccari, U., Zuazua, E.: Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function. J. Differ. Equ. 261, 2809–2853 (2016)

    Article  MathSciNet  Google Scholar 

  4. Boulakia, M., Guerrero, S.: Local null controllability of a fluid-solid interaction problem in dimension \(3\). J. Eur. Math. Soc. 15, 825–856 (2013)

    Google Scholar 

  5. Bukhgeim, A.L., Klibanov, M.V.: Global uniqueness of class of multidimensional inverse problems. Sov. Math. Dokl. 24, 244–247 (1981)

    MATH  Google Scholar 

  6. Cannarsa, P., Martinez, P., Vancostenoble, J.: Global Carleman estimates for degenerate parabolic operators with applications. Mem. Am. Math. Soc. 239, (2016)

    Google Scholar 

  7. Chen, X.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311, 603–630 (1998)

    Article  MathSciNet  Google Scholar 

  8. Di Cristo, M., Francini, E., Lin, C., Vessella, S., Wang, J.: Carleman estimate for second order elliptic equations with Lipschitz leading coefficients and jumps at an interface. J. Math. Pures Appl. 108, 163–206 (2017)

    Article  MathSciNet  Google Scholar 

  9. Doubova, A, Osses, A., Puel, J.-P.: Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM: Control Optim. Calc. Var. 8, 621–661 (2002)

    Google Scholar 

  10. Doubova, A., Fernández-Cara, E., González-Burgos, M., Zuazua, E.: On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41, 798–819 (2002)

    Article  MathSciNet  Google Scholar 

  11. Duyckaerts, T., Zhang, X., Zuazua, E.: On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 1–41 (2008)

    Google Scholar 

  12. Ervedoza, S.: Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun. Partial Differ. Equ. 33, 1996–2019 (2008)

    Article  MathSciNet  Google Scholar 

  13. Escauriaza, L., Vega, L.: Carleman inequalities and the heat operator. II. Indiana Univ. Math. J. 50 (2001)

    Google Scholar 

  14. Escauriaza, L., Seregin, G., Šverák, V.: Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169, 147–157 (2003)

    Article  MathSciNet  Google Scholar 

  15. Escauriaza, L., Seregin, G., Šverák, V.: \(L^{3,\infty }\)-solutions of Navier-Stokes equations and backward uniqueness. Russ. Math. Surveys. 58, 211–250 (2003)

    Google Scholar 

  16. Escauriaza, L., Fernández, F., Vessella, S.: Doubling properties of caloric functions. Appl. Anal. 85, 205–223 (2006)

    Article  MathSciNet  Google Scholar 

  17. Fabre, C., Puel, J.P., Zuazua, E.: Approximate controllability of the semilinear heat equations. Proc. R. Soc. Edinb. 125, 31–61 (1995)

    Article  MathSciNet  Google Scholar 

  18. Fernández-Cara, E., Zuazua, E.: Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 583–616 (2000)

    Google Scholar 

  19. Fernández-Cara, E., Zuazua, E.: The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5, 465–514 (2000)

    MathSciNet  MATH  Google Scholar 

  20. Fragnelli, G., Mugnai, D.: Carleman estimates, observability inequalities and null controllability for interior degenerate nonsmooth parabolic equations. Mem. Am. Math. Soc. 242, (2016)

    Google Scholar 

  21. Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Seoul National University, Seoul, Korea (1996)

    Google Scholar 

  22. Gao, P., Chen, M., Li, Y.: Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations. SIAM J. Control Optim. 53, 475–500 (2015)

    Article  MathSciNet  Google Scholar 

  23. Guerrero, S.: Null controllability of some systems of two parabolic equations with one control force. SIAM J. Control Optim. 46, 379–394 (2007)

    Article  MathSciNet  Google Scholar 

  24. Imanuvilov, O.Y., Yamamoto, M.: Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Probl. 14, 1229–1245 (1998)

    Google Scholar 

  25. Imanuvilov, OYu.: Controllability of the parabolic equations. Sb. Math. 186, 879–900 (1995)

    Article  MathSciNet  Google Scholar 

  26. Imanuvilov, OYu., Yamamoto, M.: Carleman inequalities for parabolic equations in a Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. RIMS Kyoto Univ. 39, 227–274 (2003)

    Article  MathSciNet  Google Scholar 

  27. Isakov, V.: Inverse Source Problems. AMS Mathematical Monographs and Surveys, Providence, vol. 34 (1990)

    Google Scholar 

  28. Koch, H., Tataru, D.: Carleman estimates and unique continuation for second order elliptic equations with nonsmooth coefficients. Commun. Pure Appl. Math. 54, 339–360 (2001)

    Article  MathSciNet  Google Scholar 

  29. Le Rousseau, J., Robbiano, L.: Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Invent. Math. 183, 245–336 (2011)

    Article  MathSciNet  Google Scholar 

  30. Le Rousseau, J., Léautaud, M., Robbiano, L.: Controllability of a parabolic system with a diffuse interface. J. Eur. Math. Soc. 15, 1485–1574 (2013)

    Article  MathSciNet  Google Scholar 

  31. Li, W., Zhang, X.: Controllability of parabolic and hyperbolic equations: Toward a unified theory. In Control Theory of Partial Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 242, pp. 157–174. Chapan & Hall/CRC, Boca Raton, FL (2005)

    Google Scholar 

  32. Li, H., Lü, Q.: A quantitative boundary unique continuation for stochastic parabolic equations. J. Math. Anal. Appl. 402, 518–526 (2013)

    Article  MathSciNet  Google Scholar 

  33. Li, L., Sverák, V.: Backward uniqueness for the heat equation in cones. Commun. Partial Differ. Equ. 37, 1414–1429 (2012)

    Article  MathSciNet  Google Scholar 

  34. Lin, F.: A uniqueness theorem for parabolic equations. Commun. Pure Appl. Math. 43, 127–136 (1990)

    Article  MathSciNet  Google Scholar 

  35. Liu, X.: Global Carleman estimate for stochastic parabolic equations, and its application. ESAIM Control Optim. Calc. Var. 20, 823–839 (2014)

    Article  MathSciNet  Google Scholar 

  36. Lü, Q.: Strong unique continuation property for stochastic parabolic equations. arXiv:1701.02136

  37. Lü, Q.: Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems. Inverse Probl. 28, 045008 (2012)

    Article  MathSciNet  Google Scholar 

  38. Lü, Q., Yin, Z.: Unique continuation for stochastic heat equations. ESAIM Control Optim. Calc. Var. 21, 378–398 (2015)

    Article  MathSciNet  Google Scholar 

  39. Mizohata, S.: Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques. Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 31, 219–239 (1958)

    Article  MathSciNet  Google Scholar 

  40. Phung, K.D., Wang, L., Zhang, C.: Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. H. Poincar Anal. Non Linéaire 31, 477–499 (2014)

    Google Scholar 

  41. Poon, C.: Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21, 521–539 (1996)

    Article  MathSciNet  Google Scholar 

  42. Rosier, L., Zhang, B.-Y.: Null controllability of the complex Ginzburg-Landau equation. Ann. Inst. H. Poincaré Anal. Non Linéaire. 26, 649–673 (2009)

    Google Scholar 

  43. Saut, J.C., Scheurer, B.: Unique continuation for some evolution equations. J. Differ. Equ. 66, 118–139 (1987)

    Article  MathSciNet  Google Scholar 

  44. Tang, S., Zhang, X.: Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48, 2191–2216 (2009)

    Article  MathSciNet  Google Scholar 

  45. Vancostenoble, J., Zuazua, E.: Null controllability for the heat equation with singular inverse-square potentials. J. Funct. Anal. 254, 1864–1902 (2008)

    Article  MathSciNet  Google Scholar 

  46. Vessella, S.: Carleman estimates, optimal three cylinder inequality, and unique continuation properties for solutions to parabolic equations. Commun. Partial Differ. Equ. 28, 637–676 (2003)

    Article  MathSciNet  Google Scholar 

  47. Wang, G., Wang, L., Xu, Y., Zhang, Y.: Time Optimal Control of Evolution Equations. Progress in Nonlinear Differential Equations and their Applications. Subseries in Control, vol. 92. Birkhäuser/Springer, Cham (2018)

    Google Scholar 

  48. Yamabe, H.: A unique continuation theorem of a diffusion equation. Ann. Math. 69, 462–466 (1959)

    Article  MathSciNet  Google Scholar 

  49. Yamamoto, M.: Carleman estimates for parabolic equations and applications. Inverse Probl. 25, 123013 (2009)

    Article  MathSciNet  Google Scholar 

  50. Yuan, G.: Conditional stability in determination of initial data for stochastic parabolic equations. Inverse Probl. 33, 035014 (2017)

    Article  MathSciNet  Google Scholar 

  51. Zhang, X.: Unique continuation for stochastic parabolic equations. Differ. Integral Equ. 21, 81–93 (2008)

    MathSciNet  MATH  Google Scholar 

  52. Zuazua, E.: Controllability and observability of partial differential equations: some results and open problems. In: Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, pp. 527–621. Elsevier Science, Amsterdam (2006)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, China

    Xiaoyu Fu, Qi Lü & Xu Zhang

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  1. Xiaoyu Fu
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  2. Qi Lü
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  3. Xu Zhang
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Correspondence to Xiaoyu Fu .

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Fu, X., Lü, Q., Zhang, X. (2019). Carleman Estimates for Second Order Parabolic Operators and Applications, a Unified Approach. In: Carleman Estimates for Second Order Partial Differential Operators and Applications. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-29530-1_3

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