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  December 2018 Early access  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 and 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[18]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[18]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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