• Previous Article
    Non-existence of global solutions to nonlinear wave equations with positive initial energy
  • CPAA Home
  • This Issue
  • Next Article
    Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval
May  2018, 17(3): 959-985. doi: 10.3934/cpaa.2018047

Stability of traveling waves of models for image processing with non-convex nonlinearity

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

* Corresponding author: Tong Li

Received  September 2017 Revised  September 2017 Published  January 2018

We establish the existence and stability of smooth large-amplitude traveling waves to nonlinear conservation laws modeling image processing with general flux functions. We innovatively construct a weight function in the weighted energy estimates to overcome the difficulties caused by the absence of the convexity of fluxes in our model. Moreover, we prove that if the integral of the initial perturbation decays algebraically or exponentially in space, the solution converges to the traveling waves with rates in time, respectively. Furthermore, we are able to construct another new weight function to deal with the degeneracy of fluxes in establishing the stability.

Citation: Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure and Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047
References:
[1]

M. Bertalmio, A. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics and image and video inpainting, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Kayai, HI, (2001), 355–362.

[2]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (2000), 417–424.

[3]

J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., PAMI-8 (1986), 679-698. 

[4]

J. GoodmanA. Kurganov and P. Rosenau, Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12 (1999), 247-268. 

[5]

J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68. 

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. 

[7]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. 

[8]

J. J. Koenderink, The structure of image, Bio. Cybernet, 50 (1984), 363-370. 

[9]

A. KurganovD. Levy and P. Rosenau, On Burgers-Type equations with nonmonotonic dissipative fluxes, Commun. Pure Appl. Math., 51 (1998), 0443-0473. 

[10]

T-P. Liu and L-A Ying, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), 519-528. 

[11]

T. Li, Rigorous asymptotic stability of a Chapman-Jouguet detonation wave in the limit of small resolved heat release, Combust. Theory Model, 1 (1997), 259-270. 

[12]

T. Li, Stability of strong detonation waves and rates of convergence, Electron. J. Differential Equations, 1998 (1998), 1-17. 

[13]

T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075. 

[14]

T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118. 

[15]

T. Li and J. Park, Stability of traveling wave solutions of nonlinear conservation laws for image processing, Commun. Math. Sci., 15 (2017), 1073-1106. 

[16]

D. Marr and E. Hildreth, Theory of edge detection, Proc. Roy. Soc. London Ser. B, 207 (1980), 187-217. 

[17]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. 

[18]

M. Mei, Remark on stability of shock profile for nonconvex scalar viscous conservation laws, Bull. Inst. Math. Acad. Sin., 27 (1999), 213-226. 

[19]

M. Mei and T. Yang, Convergence rates to traveling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh, 128A (1998), 1053-1068. 

[20]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'osay 78-02, Départment de Mathématique, Université de Paris-Sud, Orsay, France, 78 (1978).

[21]

M. Nishikata, Convergence rate to the traveling wave for viscous conservation laws, Funkcialoj Ekvacioj, 41 (1998), 107-132. 

[22]

J. Pan and H. Liu, Convergence rates to traveling waves for viscous conservation laws with dispersion, J. Diff. Eqs., 187 (2003), 337-358. 

[23]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE. Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639. 

[24]

J. Tumblin and G. Turk, A boundary hierarchy for detail-preserving contrast reduction, in Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (1990), 83–90.

[25]

A. Witkin, Scale-space filtering, in Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, (1983), 1019–1021.

[26]

Y. Wu, The stability of traveling fronts for some quasilinear Burgers-type equations, Adv. Math. (China), 31 (2002), 363-371. 

[27]

Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process, 9 (2000), 1723-1730. 

show all references

References:
[1]

M. Bertalmio, A. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics and image and video inpainting, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Kayai, HI, (2001), 355–362.

[2]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (2000), 417–424.

[3]

J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., PAMI-8 (1986), 679-698. 

[4]

J. GoodmanA. Kurganov and P. Rosenau, Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12 (1999), 247-268. 

[5]

J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68. 

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. 

[7]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. 

[8]

J. J. Koenderink, The structure of image, Bio. Cybernet, 50 (1984), 363-370. 

[9]

A. KurganovD. Levy and P. Rosenau, On Burgers-Type equations with nonmonotonic dissipative fluxes, Commun. Pure Appl. Math., 51 (1998), 0443-0473. 

[10]

T-P. Liu and L-A Ying, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), 519-528. 

[11]

T. Li, Rigorous asymptotic stability of a Chapman-Jouguet detonation wave in the limit of small resolved heat release, Combust. Theory Model, 1 (1997), 259-270. 

[12]

T. Li, Stability of strong detonation waves and rates of convergence, Electron. J. Differential Equations, 1998 (1998), 1-17. 

[13]

T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075. 

[14]

T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118. 

[15]

T. Li and J. Park, Stability of traveling wave solutions of nonlinear conservation laws for image processing, Commun. Math. Sci., 15 (2017), 1073-1106. 

[16]

D. Marr and E. Hildreth, Theory of edge detection, Proc. Roy. Soc. London Ser. B, 207 (1980), 187-217. 

[17]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. 

[18]

M. Mei, Remark on stability of shock profile for nonconvex scalar viscous conservation laws, Bull. Inst. Math. Acad. Sin., 27 (1999), 213-226. 

[19]

M. Mei and T. Yang, Convergence rates to traveling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh, 128A (1998), 1053-1068. 

[20]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'osay 78-02, Départment de Mathématique, Université de Paris-Sud, Orsay, France, 78 (1978).

[21]

M. Nishikata, Convergence rate to the traveling wave for viscous conservation laws, Funkcialoj Ekvacioj, 41 (1998), 107-132. 

[22]

J. Pan and H. Liu, Convergence rates to traveling waves for viscous conservation laws with dispersion, J. Diff. Eqs., 187 (2003), 337-358. 

[23]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE. Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639. 

[24]

J. Tumblin and G. Turk, A boundary hierarchy for detail-preserving contrast reduction, in Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (1990), 83–90.

[25]

A. Witkin, Scale-space filtering, in Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, (1983), 1019–1021.

[26]

Y. Wu, The stability of traveling fronts for some quasilinear Burgers-type equations, Adv. Math. (China), 31 (2002), 363-371. 

[27]

Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process, 9 (2000), 1723-1730. 

[1]

. Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Networks and Heterogeneous Media, 2007, 2 (1) : 127-157. doi: 10.3934/nhm.2007.2.127

[2]

C. M. Elliott, B. Gawron, S. Maier-Paape, E. S. Van Vleck. Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration. Communications on Pure and Applied Analysis, 2006, 5 (1) : 181-200. doi: 10.3934/cpaa.2006.5.181

[3]

Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011

[4]

Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks and Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010

[5]

Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure and Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835

[6]

John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 349-366. doi: 10.3934/dcds.2004.10.349

[7]

Alfonso Castro, Rosa Pardo. A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 783-790. doi: 10.3934/dcdsb.2017038

[8]

Chaudry Masood Khalique, Muhammad Usman, Maria Luz Gandarais. Nonlinear differential equations: Lie symmetries, conservation laws and other approaches of solving. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : i-ii. doi: 10.3934/dcdss.2020415

[9]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331

[10]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

[11]

Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17 (1) : 101-128. doi: 10.3934/nhm.2021025

[12]

Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014

[13]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024

[14]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134

[15]

Margaret Beck. Stability of nonlinear waves: Pointwise estimates. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 191-211. doi: 10.3934/dcdss.2017010

[16]

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure and Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867

[17]

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931

[18]

Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004

[19]

Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure and Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031

[20]

Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (227)
  • HTML views (310)
  • Cited by (0)

Other articles
by authors

[Back to Top]