February  2021, 41(2): 849-898. doi: 10.3934/dcds.2020302

Well-posedness of some non-linear stable driven SDEs

1. 

Université de Paris, Laboratoire de Probabilités, Statistiques et Modélisation (LPSM), UMR 8001, F-75013 Paris, France

2. 

Laboratory of Stochastic Analysis, Higher School of Economics, Pokrovsky boulevard 11, Moscow, Russian federation

3. 

Laboratoire de Modélisation Mathématique d'Evry (LaMME), UMR CNRS 8070, Université d'Evry Val d'Essonne, Université Paris-Saclay, 23 Boulevard de France 91037 Evry, France

4. 

Laboratory of Stochastic Analysis, Higher School of Economics, Pokrovsky boulevard 11, Moscow, Russian federation

* Corresponding author: Stéphane Menozzi

Received  October 2019 Revised  June 2020 Published  February 2021 Early access  August 2020

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $ \alpha $-stable Lévy processes with values in $ {{{\mathbb R}}}^d $ under some mild Hölder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $ \alpha \in (0,1) $. New strong well-posedness results are also derived from the previous analysis.

Citation: Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
References:
[1]

R. F. Bass, Stochastic differential equations driven by symmetric stable processes, Séminaire de Probabilités XXXVI, 1801 (2004), 302–313. doi: 10.1007/978-3-540-36107-7_11.

[2]

R. F. BassK. Burdzy and Z.-Q. Chen, Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stochastic Processes and their Applications, 111 (2004), 1-15.  doi: 10.1016/j.spa.2004.01.010.

[3]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I, volume 83 of Probability Theory and Stochastic Modelling, Springer, Cham, 2018. Mean field FBSDEs, control, and games.

[4]

P.-E. Chaudru de Raynal, Strong well posedness of McKean-Vlasov stochastic differential equations with Hölder drift, Stochastic Processes and their Applications, 130 (2020), 79-107.  doi: 10.1016/j.spa.2019.01.006.

[5]

P.-E. Chaudru de Raynal and N. Frikha, Well-posedness for some non-linear diffusion processes and related PDE on the Wasserstein space, arXiv: 1811.06904, under revision for Journal de Mathématiques Pures et Appliquées, 2018.

[6]

P.-E. Chaudru de Raynal and N. Frikha, From the Backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for McKean-Vlasov SDEs, accepted publication for Journal de Mathématiques Pures et Appliquées, 2019.

[7]

P.-E. Chaudru de Raynal, I. Honoré and S. Menozzi, Sharp Schauder Estimates for some Degenerate Kolmogorov Equations, To appear in Ann. Scie. Scuola Norm. Superiore, 2020. https://arXiv.org/abs/1810.12227.

[8]

P.-E. Chaudru de Raynal, S. Menozzi and E. Priola, Schauder estimates for drifted fractional operators in the supercritical case, Journal of Functional Analysis, 278 (2020), 108425, 57 pp. doi: 10.1016/j.jfa.2019.108425.

[9]

Z. Q. Chen, X. Zhang and G. Zhao, Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts, https://arXiv.org/pdf/1709.04632.pdf, 2017.

[10]

T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Zeitschrift Für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 67 (1984), 331–348. doi: 10.1007/BF00535008.

[11]

J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Mathematische Nachrichten, 137 (1988), 197-248.  doi: 10.1002/mana.19881370116.

[12]

C. Graham, Nonlinear diffusion with jumps, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 393-402. 

[13]

L. Huang and S. Menozzi, A parametrix approach for some degenerate stable driven SDEs, Annales Instit. H. Poincaré, 52 (2016), 1925–1975. doi: 10.1214/15-AIHP704.

[14]

L. Huang, S. Menozzi and E. Priola, $L^p$ estimates for degenerate non-local Kolmogorov operators, Journal de Mathématiques Pures et Appliquées, 121 (2019), 162–215. doi: 10.1016/j.matpur.2017.12.008.

[15]

Z. Hao, Z. Wang and M. Wu, Schauder's estimates for nonlocal equations with singular Lévy measures, arXiv: 2002.09887, 2020.

[16]

Z. Hao, M. Wu and X. Zhang, Schauder's estimate for nonlocal kinetic equations and its applications, J. Math. Pures Appl. (9), 140 (2020), 139–184, arXiv: 1903.09967. doi: 10.1016/j.matpur.2020.06.003.

[17]

X. Huang and F. F. Yang, Distribution dependent SDEs with Hölder continuous drift and $\alpha$-stable noise, arXiv: 1910.03299, 2019.

[18] N. Jacob, Pseudo Differential Operators and Markov Processes, volume 1, Imperial College Press, 2005.  doi: 10.1142/9781860947155.
[19]

B. JourdainS. Méléard and W. A. Woyczynski, A probabilistic approach for nonlinear equations involving the fractional Laplacian and a singular operator, Potential Anal., 23 (2005), 55-81.  doi: 10.1007/s11118-004-3264-9.

[20]

B. JourdainS. Méléard and W. A. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli, 11 (2005), 689-714.  doi: 10.3150/bj/1126126765.

[21]

B. JourdainS. Méléard and W. A. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 1-29. 

[22]

B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations, ESAIM: PS, 1 (1997), 339-355.  doi: 10.1051/ps:1997113.

[23]

M. Kac, Foundations of kinetic theory, In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics, Berkeley, Calif., 1956,171–197. University of California Press.

[24]

V. Konakov and S. Menozzi, Weak error for stable driven stochastic differential equations: expansion of the densities, J. Theoret. Probab., 24 (2011), 454-478.  doi: 10.1007/s10959-010-0291-x.

[25]

V. Kolokoltsov, Symmetric stable laws and stable-like jump diffusions, Proc. London Math. Soc., 80 (2000), 725–768. doi: 10.1112/S0024611500012314.

[26] V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, volume 182 of Cambridge Tracts in Mathematics., Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511760303.
[27]

N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22.  doi: 10.1080/03605300903424700.

[28]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12. AMS, 1996. doi: 10.1090/gsm/012.

[29]

D. Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab., 23 (2018), Paper No. 45, 11 pp. doi: 10.1214/18-ECP150.

[30]

J. Li and H. Min, Weak solutions of mean-field stochastic differential equations and application to zero-sum stochastic differential games, SIAM Journal on Control and Optimization, 54 (2016), 1826-1858.  doi: 10.1137/15M1015583.

[31]

H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences of the United States of America, 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.

[32]

H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), 1967, 41–57.

[33]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563.  doi: 10.1007/s11118-013-9359-4.

[34]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, Preprint, arXiv: 1603.02212, 2018.

[35]

K. Oelschlager, A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), 458-479.  doi: 10.1214/aop/1176993301.

[36]

E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka J. Math., 49 (2012), 421-447. 

[37]

E. Priola, Davie's type uniqueness for a class of SDEs with jumps, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 694–725. doi: 10.1214/16-AIHP818.

[38]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, arXiv e-prints, arXiv: 1809.02216, Sep 2018.

[39]

M. Scheutzow, Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 43 (1987), 246-256.  doi: 10.1017/S1446788700029384.

[40]

T. Shiga and H. Tanaka, Central limit theorem for a system of Markovian particles with mean field interactions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 69 (1985), 439–459. doi: 10.1007/BF00532743.

[41]

A.-S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX –- 1989, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169.

[42]

P. Sztonyk, Estimates of tempered stable densities, J. Theoret. Probab., 23 (2010), 127-147.  doi: 10.1007/s10959-009-0208-8.

[43]

H. TanakaM. Tsuchiya and S. Watanabe, Perturbation of drift-type for Lévy processes, J. Math. Kyoto Univ., 14 (1974), 73-92.  doi: 10.1215/kjm/1250523280.

[44]

C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]., Springer-Verlag, Berlin, 2009. Old and new. doi: 10.1007/978-3-540-71050-9.

[45]

T. Watanabe, Asymptotic estimates of multi-dimensional stable densities and their applications, Transactions of the American Mathematical Society, 359 (2007), 2851-2879.  doi: 10.1090/S0002-9947-07-04152-9.

[46]

X. Zhang and G. Zhao, Dirichlet problem for supercritical non-local operators, arXiv: 1809.05712, 2018.

show all references

References:
[1]

R. F. Bass, Stochastic differential equations driven by symmetric stable processes, Séminaire de Probabilités XXXVI, 1801 (2004), 302–313. doi: 10.1007/978-3-540-36107-7_11.

[2]

R. F. BassK. Burdzy and Z.-Q. Chen, Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stochastic Processes and their Applications, 111 (2004), 1-15.  doi: 10.1016/j.spa.2004.01.010.

[3]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I, volume 83 of Probability Theory and Stochastic Modelling, Springer, Cham, 2018. Mean field FBSDEs, control, and games.

[4]

P.-E. Chaudru de Raynal, Strong well posedness of McKean-Vlasov stochastic differential equations with Hölder drift, Stochastic Processes and their Applications, 130 (2020), 79-107.  doi: 10.1016/j.spa.2019.01.006.

[5]

P.-E. Chaudru de Raynal and N. Frikha, Well-posedness for some non-linear diffusion processes and related PDE on the Wasserstein space, arXiv: 1811.06904, under revision for Journal de Mathématiques Pures et Appliquées, 2018.

[6]

P.-E. Chaudru de Raynal and N. Frikha, From the Backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for McKean-Vlasov SDEs, accepted publication for Journal de Mathématiques Pures et Appliquées, 2019.

[7]

P.-E. Chaudru de Raynal, I. Honoré and S. Menozzi, Sharp Schauder Estimates for some Degenerate Kolmogorov Equations, To appear in Ann. Scie. Scuola Norm. Superiore, 2020. https://arXiv.org/abs/1810.12227.

[8]

P.-E. Chaudru de Raynal, S. Menozzi and E. Priola, Schauder estimates for drifted fractional operators in the supercritical case, Journal of Functional Analysis, 278 (2020), 108425, 57 pp. doi: 10.1016/j.jfa.2019.108425.

[9]

Z. Q. Chen, X. Zhang and G. Zhao, Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts, https://arXiv.org/pdf/1709.04632.pdf, 2017.

[10]

T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Zeitschrift Für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 67 (1984), 331–348. doi: 10.1007/BF00535008.

[11]

J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Mathematische Nachrichten, 137 (1988), 197-248.  doi: 10.1002/mana.19881370116.

[12]

C. Graham, Nonlinear diffusion with jumps, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 393-402. 

[13]

L. Huang and S. Menozzi, A parametrix approach for some degenerate stable driven SDEs, Annales Instit. H. Poincaré, 52 (2016), 1925–1975. doi: 10.1214/15-AIHP704.

[14]

L. Huang, S. Menozzi and E. Priola, $L^p$ estimates for degenerate non-local Kolmogorov operators, Journal de Mathématiques Pures et Appliquées, 121 (2019), 162–215. doi: 10.1016/j.matpur.2017.12.008.

[15]

Z. Hao, Z. Wang and M. Wu, Schauder's estimates for nonlocal equations with singular Lévy measures, arXiv: 2002.09887, 2020.

[16]

Z. Hao, M. Wu and X. Zhang, Schauder's estimate for nonlocal kinetic equations and its applications, J. Math. Pures Appl. (9), 140 (2020), 139–184, arXiv: 1903.09967. doi: 10.1016/j.matpur.2020.06.003.

[17]

X. Huang and F. F. Yang, Distribution dependent SDEs with Hölder continuous drift and $\alpha$-stable noise, arXiv: 1910.03299, 2019.

[18] N. Jacob, Pseudo Differential Operators and Markov Processes, volume 1, Imperial College Press, 2005.  doi: 10.1142/9781860947155.
[19]

B. JourdainS. Méléard and W. A. Woyczynski, A probabilistic approach for nonlinear equations involving the fractional Laplacian and a singular operator, Potential Anal., 23 (2005), 55-81.  doi: 10.1007/s11118-004-3264-9.

[20]

B. JourdainS. Méléard and W. A. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli, 11 (2005), 689-714.  doi: 10.3150/bj/1126126765.

[21]

B. JourdainS. Méléard and W. A. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 1-29. 

[22]

B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations, ESAIM: PS, 1 (1997), 339-355.  doi: 10.1051/ps:1997113.

[23]

M. Kac, Foundations of kinetic theory, In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics, Berkeley, Calif., 1956,171–197. University of California Press.

[24]

V. Konakov and S. Menozzi, Weak error for stable driven stochastic differential equations: expansion of the densities, J. Theoret. Probab., 24 (2011), 454-478.  doi: 10.1007/s10959-010-0291-x.

[25]

V. Kolokoltsov, Symmetric stable laws and stable-like jump diffusions, Proc. London Math. Soc., 80 (2000), 725–768. doi: 10.1112/S0024611500012314.

[26] V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, volume 182 of Cambridge Tracts in Mathematics., Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511760303.
[27]

N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22.  doi: 10.1080/03605300903424700.

[28]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12. AMS, 1996. doi: 10.1090/gsm/012.

[29]

D. Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab., 23 (2018), Paper No. 45, 11 pp. doi: 10.1214/18-ECP150.

[30]

J. Li and H. Min, Weak solutions of mean-field stochastic differential equations and application to zero-sum stochastic differential games, SIAM Journal on Control and Optimization, 54 (2016), 1826-1858.  doi: 10.1137/15M1015583.

[31]

H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences of the United States of America, 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.

[32]

H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), 1967, 41–57.

[33]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563.  doi: 10.1007/s11118-013-9359-4.

[34]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, Preprint, arXiv: 1603.02212, 2018.

[35]

K. Oelschlager, A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), 458-479.  doi: 10.1214/aop/1176993301.

[36]

E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka J. Math., 49 (2012), 421-447. 

[37]

E. Priola, Davie's type uniqueness for a class of SDEs with jumps, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 694–725. doi: 10.1214/16-AIHP818.

[38]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, arXiv e-prints, arXiv: 1809.02216, Sep 2018.

[39]

M. Scheutzow, Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 43 (1987), 246-256.  doi: 10.1017/S1446788700029384.

[40]

T. Shiga and H. Tanaka, Central limit theorem for a system of Markovian particles with mean field interactions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 69 (1985), 439–459. doi: 10.1007/BF00532743.

[41]

A.-S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX –- 1989, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169.

[42]

P. Sztonyk, Estimates of tempered stable densities, J. Theoret. Probab., 23 (2010), 127-147.  doi: 10.1007/s10959-009-0208-8.

[43]

H. TanakaM. Tsuchiya and S. Watanabe, Perturbation of drift-type for Lévy processes, J. Math. Kyoto Univ., 14 (1974), 73-92.  doi: 10.1215/kjm/1250523280.

[44]

C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]., Springer-Verlag, Berlin, 2009. Old and new. doi: 10.1007/978-3-540-71050-9.

[45]

T. Watanabe, Asymptotic estimates of multi-dimensional stable densities and their applications, Transactions of the American Mathematical Society, 359 (2007), 2851-2879.  doi: 10.1090/S0002-9947-07-04152-9.

[46]

X. Zhang and G. Zhao, Dirichlet problem for supercritical non-local operators, arXiv: 1809.05712, 2018.

[1]

Xing Huang, Feng-Yu Wang. Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1667-1679. doi: 10.3934/dcds.2020336

[2]

Huyên Pham. Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 7-. doi: 10.1186/s41546-016-0008-x

[3]

Matthias Erbar, Max Fathi, Vaios Laschos, André Schlichting. Gradient flow structure for McKean-Vlasov equations on discrete spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6799-6833. doi: 10.3934/dcds.2016096

[4]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[5]

Franca Franchi, Barbara Lazzari, Roberta Nibbi. Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2111-2132. doi: 10.3934/dcdsb.2014.19.2111

[6]

Yu Liu, Ting Zhang. On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021307

[7]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181

[8]

Wenjing Zhao. Weak-strong uniqueness of incompressible magneto-viscoelastic flows. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2907-2917. doi: 10.3934/cpaa.2020127

[9]

Etienne Emmrich, Robert Lasarzik. Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4617-4635. doi: 10.3934/dcds.2018202

[10]

Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240

[11]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic and Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015

[12]

Toyohiko Aiki, Adrian Muntean. On uniqueness of a weak solution of one-dimensional concrete carbonation problem. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1345-1365. doi: 10.3934/dcds.2011.29.1345

[13]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[14]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam. Conference Publications, 2011, 2011 (Special) : 813-823. doi: 10.3934/proc.2011.2011.813

[15]

Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437

[16]

Yue Zheng, Zhongping Wan, Shihui Jia, Guangmin Wang. A new method for strong-weak linear bilevel programming problem. Journal of Industrial and Management Optimization, 2015, 11 (2) : 529-547. doi: 10.3934/jimo.2015.11.529

[17]

Andrea Malchiodi. Perturbative techniques for the construction of spike-layers. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3767-3787. doi: 10.3934/dcds.2020055

[18]

Bei Hu, Lishang Jiang, Jin Liang, Wei Wei. A fully non-linear PDE problem from pricing CDS with counterparty risk. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2001-2016. doi: 10.3934/dcdsb.2012.17.2001

[19]

Nanhee Kim. Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress. Evolution Equations and Control Theory, 2013, 2 (4) : 679-693. doi: 10.3934/eect.2013.2.679

[20]

Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : i-iv. doi: 10.3934/dcdss.201702i

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (205)
  • HTML views (225)
  • Cited by (1)

[Back to Top]