-
Previous Article
A generalization of the Babbage functional equation
- DCDS Home
- This Issue
-
Next Article
Asymptotic dynamics of a system of conservation laws from chemotaxis
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 |
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.
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. Bass, K. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[18] |
N. Jacob, Pseudo Differential Operators and Markov Processes, volume 1, Imperial College Press, 2005.
doi: 10.1142/9781860947155.![]() ![]() |
[19] |
B. Jourdain, S. 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. Jourdain, S. 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. Jourdain, S. 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. Google Scholar |
[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. Google Scholar |
[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. Tanaka, M. 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. Google Scholar |
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. Bass, K. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[18] |
N. Jacob, Pseudo Differential Operators and Markov Processes, volume 1, Imperial College Press, 2005.
doi: 10.1142/9781860947155.![]() ![]() |
[19] |
B. Jourdain, S. 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. Jourdain, S. 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. Jourdain, S. 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. Google Scholar |
[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. Google Scholar |
[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. Tanaka, M. 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. Google Scholar |
[1] |
Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435 |
[2] |
Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020384 |
[3] |
Andrea Malchiodi. Perturbative techniques for the construction of spike-layers. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3767-3787. doi: 10.3934/dcds.2020055 |
[4] |
Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004 |
[5] |
Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020405 |
[6] |
Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289 |
[7] |
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 |
[8] |
Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 |
[9] |
Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020320 |
[10] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[11] |
Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 |
[12] |
Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 |
[13] |
Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 |
[14] |
Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260 |
[15] |
Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 |
[16] |
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 |
[17] |
Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020032 |
[18] |
Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052 |
[19] |
Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348 |
[20] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]