American Institute of Mathematical Sciences

Advanced
December  2006, 16(4): 783-842. doi: 10.3934/dcds.2006.16.783

A concept of solution and numerical experiments for forward-backward diffusion equations

G. Bellettini 1, , Giorgio Fusco 2, and  Nicola Guglielmi 3,

1. 

Dipartimento di Matematica, Università di Roma 'Tor Vergata', 00133, Roma
2. 

Dipartimento di Matematica Pura e Applicata, Università de L’Aquila, I-67100 L’Aquila, Italy
3. 

Dipartimento di Matematica Pura e Applicata, Università de L'Aquila, I-67100 L'Aquila

Received  January 2006 Revised  June 2006 Published  September 2006

We study the gradient flow associated with the functional $F_\phi(u)$ := $\frac{1}{2}\int_{I} \phi(u_x)~dx$, where $\phi$ is non convex, and with its singular perturbation $F_\phi^\varepsilon(u)$:=$\frac{1}{2}\int_I (\varepsilon^2 (u_{x x})^2 + \phi(u_x))dx$. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions $u^\varepsilon$ of the singularly perturbed equation $u_t = - \varepsilon^2 u_{x x x x} + \frac{1}{2} \phi''(u_x)u_{x x}$ for small values of $\varepsilon>0$. Our analysis leads to a reinterpretation of the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$, and to a well defined notion of a solution. We also examine the conjecture that this solution coincides with the limit of $u^\varepsilon$ as $\varepsilon\to 0^+$.
Keywords: singular perturbations, nonconvex functionals, Forward-backward parabolic equations, fourth order regularization, microstructures, stiff problems..
Mathematics Subject Classification: Primary: 35B25; Secondary: 47j06, 35K9.
Citation: G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783
[1]

Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1017-1036. doi: 10.3934/cpaa.2020047
[2]

Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks & Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941
[3]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
[4]

Jiongmin Yong. Forward-backward evolution equations and applications. Mathematical Control & Related Fields, 2016, 6 (4) : 653-704. doi: 10.3934/mcrf.2016019
[5]

Flavia Smarrazzo, Andrea Terracina. Sobolev approximation for two-phase solutions of forward-backward parabolic problems. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1657-1697. doi: 10.3934/dcds.2013.33.1657
[6]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[7]

Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations & Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023
[8]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[9]

Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
[10]

Xiao Ding, Deren Han. A modification of the forward-backward splitting method for maximal monotone mappings. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 295-307. doi: 10.3934/naco.2013.3.295
[11]

Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1893-1906. doi: 10.3934/cpaa.2021051
[12]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013
[13]

Z. B. Ibrahim, N. A. A. Mohd Nasir, K. I. Othman, N. Zainuddin. Adaptive order of block backward differentiation formulas for stiff ODEs. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 95-106. doi: 10.3934/naco.2017006
[14]

Andrea L. Bertozzi, Ning Ju, Hsiang-Wei Lu. A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1367-1391. doi: 10.3934/dcds.2011.29.1367
[15]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
[16]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
[17]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[18]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493
[19]

Fangshu Wan. Bôcher-type results for the fourth and higher order equations on singular manifolds with conical metrics. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2919-2948. doi: 10.3934/cpaa.2020128
[20]

Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe, Atsushi Yagi. An $L^p$-approach to singular linear parabolic equations with lower order terms. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 989-1008. doi: 10.3934/dcds.2008.22.989

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy