October  2018, 11(5): 915-939. doi: 10.3934/dcdss.2018055

Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity

1. 

Yau Mathematical Sciences Center, Tsinghua University, No 1. Tsinghua Yuan, Beijing 100084, China

2. 

Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA

3. 

Department of Mathematics, University of Wisconsin at Madison, 480 Lincoln Drive, Madison, WI 53706, USA

* Corresponding author: Wenjia Jing

Received  February 2017 Revised  July 2017 Published  June 2018

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

Citation: Wenjia Jing, Panagiotis E. Souganidis, Hung V. Tran. Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 915-939. doi: 10.3934/dcdss.2018055
References:
[1]

O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations Mem. Amer. Math. Soc. , 204 (2010), vi+77pp. doi: 10.1090/S0065-9266-09-00588-2. Google Scholar

[2]

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S. N. Armstrong and P. Cardaliaguet, Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions, J. Eur. Math. Soc., 20 (2018), 797-864. doi: 10.4171/JEMS/777. Google Scholar

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S. N. Armstrong and P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments, J. Math. Pures Appl. (9), 97 (2012), 460-504. doi: 10.1016/j.matpur.2011.09.009. Google Scholar

[5]

S. N. Armstrong and P. E. Souganidis, Stochastic homogenization of level-set convex Hamilton-Jacobi equations, Int. Math. Res. Not., 2013 (2013), 3420-3449. doi: 10.1093/imrn/rns155. Google Scholar

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S. N. Armstrong and H. V. Tran, Stochastic homogenization of viscous Hamilton-Jacobi equations and applications, Anal. PDE, 7 (2014), 1969-2007. doi: 10.2140/apde.2014.7.1969. Google Scholar

[7]

S. N. ArmstrongH. V. Tran and Y. Yu, Stochastic homogenization of a nonconvex Hamilton-Jacobi equation, Calc. Var. Partial Differential Equations, 54 (2015), 1507-1524. doi: 10.1007/s00526-015-0833-2. Google Scholar

[8]

S. N. ArmstrongH. V. Tran and Y. Yu, Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension, J. Differential Equations, 261 (2016), 2702-2737. doi: 10.1016/j.jde.2016.05.010. Google Scholar

[9]

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, second ed., vol. 250 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1988. Translated from the Russian by Joseph Szücs [József M. Szűcs]. doi: 10.1007/978-1-4612-1037-5. Google Scholar

[10]

G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466. doi: 10.1007/s00526-007-0097-6. Google Scholar

[11]

P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM Control Optim. Calc. Var., 12 (2006), 350-370 (electronic). doi: 10.1051/cocv:2006002. Google Scholar

[12]

P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $\mathbb{ R}^2/\mathbb {Z}^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856. doi: 10.1016/j.anihpc.2009.11.015. Google Scholar

[13]

P. CardaliaguetP.-L. Lions and P. E. Souganidis, A discussion about the homogenization of moving interfaces, J. Math. Pures Appl. (9), 91 (2009), 339-363. doi: 10.1016/j.matpur.2009.01.014. Google Scholar

[14]

P. CardaliaguetJ. Nolen and P. E. Souganidis, Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199 (2011), 527-561. doi: 10.1007/s00205-010-0332-8. Google Scholar

[15]

P. Cardaliaguet and L. Silvestre, Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side, Comm. Partial Differential Equations, 37 (2012), 1668-1688. doi: 10.1080/03605302.2012.660267. Google Scholar

[16]

P. Cardaliaguet and P. E. Souganidis, Homogenization and enhancement of the G-equation in random environments, Comm. Pure Appl. Math., 66 (2013), 1582-1628. doi: 10.1002/cpa.21449. Google Scholar

[17]

A. CiomagaP. E. Souganidis and H. V. Tran, Stochastic homogenization of interfaces moving with changing sign velocity, J. Differential Equations, 258 (2015), 1025-1057. doi: 10.1016/j.jde.2014.09.019. Google Scholar

[18]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. Google Scholar

[19]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121. Google Scholar

[20]

W. Feldman and P. E. Souganidis, Homogenization and non-homogenization of certain non-convex hamilton-jacobi equations, J. Math. Pures Appl., 55 (2017), 751-782. Google Scholar

[21]

H. Gao, Random homogenization of coercive Hamilton-Jacobi equations in 1d, Calc. Var. Partial Differential Equations, 55 (2016), Art. 30, 39pp. doi: 10.1007/s00526-016-0968-9. Google Scholar

[22]

J. C. Hansen and P. Hulse, Subadditive ergodic theorems for random sets in infinite dimensions, Statist. Probab. Lett., 50 (2000), 409-416. doi: 10.1016/S0167-7152(00)00156-5. Google Scholar

[23]

C. Imbert and R. Monneau, Homogenization of first-order equations with (u/ε)-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89. doi: 10.1007/s00205-007-0074-4. Google Scholar

[24]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999). World Sci. Publ., River Edge, NJ, 2000,600-605. Google Scholar

[25]

E. KosyginaF. Rezakhanlou and S. R. S. Varadhan, Stochastic homogenization of Hamilton-Jacobi-Bellman equations, Comm. Pure Appl. Math., 59 (2006), 1489-1521. doi: 10.1002/cpa.20137. Google Scholar

[26]

E. Kosygina and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium, Comm. Pure Appl. Math., 61 (2008), 816-847. doi: 10.1002/cpa.20220. Google Scholar

[27]

W. Li and K. Lu, Rotation numbers for random dynamical systems on the circle, Trans. Amer. Math. Soc., 360 (2008), 5509-5528. doi: 10.1090/S0002-9947-08-04619-9. Google Scholar

[28]

P. -L. Lions, G. C. Papanicolaou and S. Varadhan, Homogenization of Hamilton-Jacobi equations, Unpublished preprint, 1987.Google Scholar

[29]

P.-L. Lions and P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities-revisited, Commun. Math. Sci., 8 (2010), 627-637. doi: 10.4310/CMS.2010.v8.n2.a14. Google Scholar

[30]

A. J. Majda and P. E. Souganidis, Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), 1-30. doi: 10.1088/0951-7715/7/1/001. Google Scholar

[31]

J. Nolen and A. Novikov, Homogenization of the G-equation with incompressible random drift in two dimensions, Commun. Math. Sci., 9 (2011), 561-582. doi: 10.4310/CMS.2011.v9.n2.a11. Google Scholar

[32]

F. Rezakhanlou and J. E. Tarver, Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151 (2000), 277-309. doi: 10.1007/s002050050198. Google Scholar

[33]

R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N. J., 1970. Google Scholar

[34]

K. Schürger, Ergodic theorems for subadditive superstationary families of convex compact random sets, Z. Wahrsch. Verw. Gebiete, 62 (1983), 125-135. doi: 10.1007/BF00532166. Google Scholar

[35]

R. W. Schwab, Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media, Indiana Univ. Math. J., 58 (2009), 537-581. doi: 10.1512/iumj.2009.58.3455. Google Scholar

[36]

B. Simon, Convexity, vol. 187 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011. An analytic viewpoint. doi: 10.1017/CBO9780511910135. Google Scholar

[37]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11. Google Scholar

[38]

J. Xin and Y. Yu, Periodic homogenization of the inviscid G-equation for incompressible flows, Commun. Math. Sci., 8 (2010), 1067-1078. doi: 10.4310/CMS.2010.v8.n4.a14. Google Scholar

[39]

B. Ziliotto, Stochastic homogenization of nonconvex hamilton-jacobi equations: A counterexample, Comm. Pure Appl. Math., 70 (2017), 1798-1809. doi: 10.1002/cpa.21674. Google Scholar

show all references

References:
[1]

O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations Mem. Amer. Math. Soc. , 204 (2010), vi+77pp. doi: 10.1090/S0065-9266-09-00588-2. Google Scholar

[2]

M. Arisawa and P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413. Google Scholar

[3]

S. N. Armstrong and P. Cardaliaguet, Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions, J. Eur. Math. Soc., 20 (2018), 797-864. doi: 10.4171/JEMS/777. Google Scholar

[4]

S. N. Armstrong and P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments, J. Math. Pures Appl. (9), 97 (2012), 460-504. doi: 10.1016/j.matpur.2011.09.009. Google Scholar

[5]

S. N. Armstrong and P. E. Souganidis, Stochastic homogenization of level-set convex Hamilton-Jacobi equations, Int. Math. Res. Not., 2013 (2013), 3420-3449. doi: 10.1093/imrn/rns155. Google Scholar

[6]

S. N. Armstrong and H. V. Tran, Stochastic homogenization of viscous Hamilton-Jacobi equations and applications, Anal. PDE, 7 (2014), 1969-2007. doi: 10.2140/apde.2014.7.1969. Google Scholar

[7]

S. N. ArmstrongH. V. Tran and Y. Yu, Stochastic homogenization of a nonconvex Hamilton-Jacobi equation, Calc. Var. Partial Differential Equations, 54 (2015), 1507-1524. doi: 10.1007/s00526-015-0833-2. Google Scholar

[8]

S. N. ArmstrongH. V. Tran and Y. Yu, Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension, J. Differential Equations, 261 (2016), 2702-2737. doi: 10.1016/j.jde.2016.05.010. Google Scholar

[9]

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, second ed., vol. 250 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1988. Translated from the Russian by Joseph Szücs [József M. Szűcs]. doi: 10.1007/978-1-4612-1037-5. Google Scholar

[10]

G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466. doi: 10.1007/s00526-007-0097-6. Google Scholar

[11]

P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM Control Optim. Calc. Var., 12 (2006), 350-370 (electronic). doi: 10.1051/cocv:2006002. Google Scholar

[12]

P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $\mathbb{ R}^2/\mathbb {Z}^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856. doi: 10.1016/j.anihpc.2009.11.015. Google Scholar

[13]

P. CardaliaguetP.-L. Lions and P. E. Souganidis, A discussion about the homogenization of moving interfaces, J. Math. Pures Appl. (9), 91 (2009), 339-363. doi: 10.1016/j.matpur.2009.01.014. Google Scholar

[14]

P. CardaliaguetJ. Nolen and P. E. Souganidis, Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199 (2011), 527-561. doi: 10.1007/s00205-010-0332-8. Google Scholar

[15]

P. Cardaliaguet and L. Silvestre, Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side, Comm. Partial Differential Equations, 37 (2012), 1668-1688. doi: 10.1080/03605302.2012.660267. Google Scholar

[16]

P. Cardaliaguet and P. E. Souganidis, Homogenization and enhancement of the G-equation in random environments, Comm. Pure Appl. Math., 66 (2013), 1582-1628. doi: 10.1002/cpa.21449. Google Scholar

[17]

A. CiomagaP. E. Souganidis and H. V. Tran, Stochastic homogenization of interfaces moving with changing sign velocity, J. Differential Equations, 258 (2015), 1025-1057. doi: 10.1016/j.jde.2014.09.019. Google Scholar

[18]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. Google Scholar

[19]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121. Google Scholar

[20]

W. Feldman and P. E. Souganidis, Homogenization and non-homogenization of certain non-convex hamilton-jacobi equations, J. Math. Pures Appl., 55 (2017), 751-782. Google Scholar

[21]

H. Gao, Random homogenization of coercive Hamilton-Jacobi equations in 1d, Calc. Var. Partial Differential Equations, 55 (2016), Art. 30, 39pp. doi: 10.1007/s00526-016-0968-9. Google Scholar

[22]

J. C. Hansen and P. Hulse, Subadditive ergodic theorems for random sets in infinite dimensions, Statist. Probab. Lett., 50 (2000), 409-416. doi: 10.1016/S0167-7152(00)00156-5. Google Scholar

[23]

C. Imbert and R. Monneau, Homogenization of first-order equations with (u/ε)-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89. doi: 10.1007/s00205-007-0074-4. Google Scholar

[24]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999). World Sci. Publ., River Edge, NJ, 2000,600-605. Google Scholar

[25]

E. KosyginaF. Rezakhanlou and S. R. S. Varadhan, Stochastic homogenization of Hamilton-Jacobi-Bellman equations, Comm. Pure Appl. Math., 59 (2006), 1489-1521. doi: 10.1002/cpa.20137. Google Scholar

[26]

E. Kosygina and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium, Comm. Pure Appl. Math., 61 (2008), 816-847. doi: 10.1002/cpa.20220. Google Scholar

[27]

W. Li and K. Lu, Rotation numbers for random dynamical systems on the circle, Trans. Amer. Math. Soc., 360 (2008), 5509-5528. doi: 10.1090/S0002-9947-08-04619-9. Google Scholar

[28]

P. -L. Lions, G. C. Papanicolaou and S. Varadhan, Homogenization of Hamilton-Jacobi equations, Unpublished preprint, 1987.Google Scholar

[29]

P.-L. Lions and P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities-revisited, Commun. Math. Sci., 8 (2010), 627-637. doi: 10.4310/CMS.2010.v8.n2.a14. Google Scholar

[30]

A. J. Majda and P. E. Souganidis, Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), 1-30. doi: 10.1088/0951-7715/7/1/001. Google Scholar

[31]

J. Nolen and A. Novikov, Homogenization of the G-equation with incompressible random drift in two dimensions, Commun. Math. Sci., 9 (2011), 561-582. doi: 10.4310/CMS.2011.v9.n2.a11. Google Scholar

[32]

F. Rezakhanlou and J. E. Tarver, Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151 (2000), 277-309. doi: 10.1007/s002050050198. Google Scholar

[33]

R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N. J., 1970. Google Scholar

[34]

K. Schürger, Ergodic theorems for subadditive superstationary families of convex compact random sets, Z. Wahrsch. Verw. Gebiete, 62 (1983), 125-135. doi: 10.1007/BF00532166. Google Scholar

[35]

R. W. Schwab, Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media, Indiana Univ. Math. J., 58 (2009), 537-581. doi: 10.1512/iumj.2009.58.3455. Google Scholar

[36]

B. Simon, Convexity, vol. 187 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011. An analytic viewpoint. doi: 10.1017/CBO9780511910135. Google Scholar

[37]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11. Google Scholar

[38]

J. Xin and Y. Yu, Periodic homogenization of the inviscid G-equation for incompressible flows, Commun. Math. Sci., 8 (2010), 1067-1078. doi: 10.4310/CMS.2010.v8.n4.a14. Google Scholar

[39]

B. Ziliotto, Stochastic homogenization of nonconvex hamilton-jacobi equations: A counterexample, Comm. Pure Appl. Math., 70 (2017), 1798-1809. doi: 10.1002/cpa.21674. Google Scholar

Figure 1.  The construction of admissible paths
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