American Institute of Mathematical Sciences

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December  2018, 23(10): 4207-4222. doi: 10.3934/dcdsb.2018133

Carleman estimate for solutions to a degenerate convection-diffusion equation

Chunpeng Wang 1, , Yanan Zhou 1, , Runmei Du 2, and  Qiang Liu 3,,

1. 

School of Mathematics, Jilin University, Changchun 130012, China
2. 

School of Basic Science, Changchun University of Technology, Changchun 130012, China
3. 

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China

* Corresponding author: Qiang Liu

Received  September 2017 Published  April 2018

Fund Project: Supported by the National Natural Science Foundation of China (No. 11222106, 11571137 and 11401049), the Natural Science Foundation for Young Scientists of Jilin Province(20170520048JH), the Scientific and Technological project of Jilin Provinces Education Department in Thirteenth Five-Year(JJKH20170534KJ), NSF of Guangdong Province(2016A030313048) and Excellent Young Teachers Program of Guandon

This paper concerns a control system governed by a convection-diffusion equation, which is weakly degenerate at the boundary. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. The Carleman estimate is established by means of a suitable transformation, by which the diffusion and the convection are transformed into a complex union, and complicated and detailed computations. Then the observability inequality is proved and the control system is shown to be null controllable.

Keywords: Carleman estimate, null controllability, degeneracy, convection.
Mathematics Subject Classification: 93B05, 93C20, 35K65.
Citation: Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133
References:
[1]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

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U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.  Google Scholar

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P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 1-20.   Google Scholar

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P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715.  doi: 10.3934/nhm.2007.2.695.  Google Scholar

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P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616.  doi: 10.1007/s00028-008-0353-34.  Google Scholar

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P. Cannarsa, G. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl., 320 (2006), 804-818.  doi: 10.1016/j.jmaa.2005.07.006.  Google Scholar

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P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal., 3 (2004), 607-635.  doi: 10.3934/cpaa.2004.3.607.  Google Scholar

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P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

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P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

[12]

M. M. Cavalcanti, E. Fernández-Cara and A. L. Ferreira, Null controllability of some nonlinear degenerate 1D parabolic equations, J. Franklin Inst., 354 (2017), 6405-6421.  doi: 10.1016/j.jfranklin.2017.08.015.  Google Scholar

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E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514.   Google Scholar

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E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[15]

C. Flores and L. Teresa, Carleman estimates for degenerate parabolic equations with first order terms and applications, C. R. Math. Acad. Sci. Paris, 348 (2010), 391-396.  doi: 10.1016/j.crma.2010.01.007.  Google Scholar

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G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016), v+84 pp.  Google Scholar

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G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.   Google Scholar

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A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996.  Google Scholar

[19]

H. Gao, X. Hou and N. H. Pavel, Optimal control and controllability problems for a class of nonlinear degenerate diffusion equations, Panamer. Math. J., 13 (2003), 103-126.   Google Scholar

[20]

P. Lin, H. Gao and X. Liu, Some results on controllability of a nonlinear degenerate parabolic system by bilinear control, J. Math. Anal. Appl., 326 (2007), 1149-1160.  doi: 10.1016/j.jmaa.2006.03.079.  Google Scholar

[21]

X. Liu and H. Gao, Controllability of a class of Newtonian filtration equations with control and state constraints, SIAM J. Control Optim., 46 (2007), 2256-2279.  doi: 10.1137/060649951.  Google Scholar

[22]

P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation, SIAM J. Control Optim., 42 (2003), 709-728.  doi: 10.1137/S0363012902403547.  Google Scholar

[23]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6.  Google Scholar

[24]

C. Wang, Approximate controllability of a class of degenerate systems, Appl. Math. Comput., 203 (2008), 447-456.  doi: 10.1016/j.amc.2008.04.056.  Google Scholar

[25]

C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equ., 10 (2010), 163-193.  doi: 10.1007/s00028-009-0044-4.  Google Scholar

[26]

C. Wang and R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term, J. Differential Equations, 254 (2013), 3665-3689.  doi: 10.1016/j.jde.2013.01.038.  Google Scholar

[27]

C. Wang and R. Du, Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52 (2014), 1457-1480.  doi: 10.1137/110820592.  Google Scholar

[28]

J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. Differential Equations, 237 (2007), 421-445.  doi: 10.1016/j.jde.2007.03.012.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

[2]

V. Barbu, Controllability of parabolic and Navier-Stokes equations, Sci. Math. Jpn., 56 (2002), 143-211.   Google Scholar

[3]

U. Biccari, Boundary controllability for a one-dimensional heat equation with two singular inverse-square potentials, arXiv: 1509.05178. Google Scholar

[4]

U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.  Google Scholar

[5]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 1-20.   Google Scholar

[6]

P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715.  doi: 10.3934/nhm.2007.2.695.  Google Scholar

[7]

P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616.  doi: 10.1007/s00028-008-0353-34.  Google Scholar

[8]

P. Cannarsa, G. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl., 320 (2006), 804-818.  doi: 10.1016/j.jmaa.2005.07.006.  Google Scholar

[9]

P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal., 3 (2004), 607-635.  doi: 10.3934/cpaa.2004.3.607.  Google Scholar

[10]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[11]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

[12]

M. M. Cavalcanti, E. Fernández-Cara and A. L. Ferreira, Null controllability of some nonlinear degenerate 1D parabolic equations, J. Franklin Inst., 354 (2017), 6405-6421.  doi: 10.1016/j.jfranklin.2017.08.015.  Google Scholar

[13]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514.   Google Scholar

[14]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[15]

C. Flores and L. Teresa, Carleman estimates for degenerate parabolic equations with first order terms and applications, C. R. Math. Acad. Sci. Paris, 348 (2010), 391-396.  doi: 10.1016/j.crma.2010.01.007.  Google Scholar

[16]

G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016), v+84 pp.  Google Scholar

[17]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.   Google Scholar

[18]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996.  Google Scholar

[19]

H. Gao, X. Hou and N. H. Pavel, Optimal control and controllability problems for a class of nonlinear degenerate diffusion equations, Panamer. Math. J., 13 (2003), 103-126.   Google Scholar

[20]

P. Lin, H. Gao and X. Liu, Some results on controllability of a nonlinear degenerate parabolic system by bilinear control, J. Math. Anal. Appl., 326 (2007), 1149-1160.  doi: 10.1016/j.jmaa.2006.03.079.  Google Scholar

[21]

X. Liu and H. Gao, Controllability of a class of Newtonian filtration equations with control and state constraints, SIAM J. Control Optim., 46 (2007), 2256-2279.  doi: 10.1137/060649951.  Google Scholar

[22]

P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation, SIAM J. Control Optim., 42 (2003), 709-728.  doi: 10.1137/S0363012902403547.  Google Scholar

[23]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6.  Google Scholar

[24]

C. Wang, Approximate controllability of a class of degenerate systems, Appl. Math. Comput., 203 (2008), 447-456.  doi: 10.1016/j.amc.2008.04.056.  Google Scholar

[25]

C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equ., 10 (2010), 163-193.  doi: 10.1007/s00028-009-0044-4.  Google Scholar

[26]

C. Wang and R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term, J. Differential Equations, 254 (2013), 3665-3689.  doi: 10.1016/j.jde.2013.01.038.  Google Scholar

[27]

C. Wang and R. Du, Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52 (2014), 1457-1480.  doi: 10.1137/110820592.  Google Scholar

[28]

J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. Differential Equations, 237 (2007), 421-445.  doi: 10.1016/j.jde.2007.03.012.  Google Scholar

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