Abstract
This article studies the Stochastic Degasperis-Procesi equation on \( \mathbb {R}\) with an additive noise. Applying the kinetic theory, and considering the initial conditions in \(L^2(\mathbb {R})\cap L^{2+\delta }( \mathbb {R})\), for arbitrary small \(\delta >0\), we establish the existence of a global pathwise solution. Restricting to the particular case of zero noise, our result improves the deterministic solvability results that exist in the literature.
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References
Bessaih, H., Flandoli, F.: 2-D Euler equation perturbed by noise. Nonlinear Differ. Equ. Appl. 6, 35–54 (1999)
Coclite, G.M., Holden, H., Karlsen, K.H.: Well-posedness for a parabolic-elliptic system. Discrete Contin. Dyn. Syst. 13(3), 659–682 (2005)
Coclite, G.M., Karlsen, K.H.: On the well-posedness of the Degasperis-Procesi equation. J. Funct. Anal. 233, 60–91 (2006)
Coclite, G.M., Karlsen, K.H.: On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation. J. Differ. Equ. 234(1), 142–160 (2007)
Coclite, G.M., Karlsen, K.H.: Periodic solutions of the Degasperis-Procesi equation: well-posedness and asymptotics. J. Funct. Anal. 268(5), 1053–1077 (2015)
Chen, G.-Q., Frid, H.: On the theory of divergence-measure fields and its applications. Boletin Soc. Bras. Mat. 32(3), 1–33 (2001)
Chen, Y., Gao, H.: Global existence for the stochastic Degasperis-Procesi equation. Discrete Contin. Dyn. Syst. 35(11), 5171–5184 (2015)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge (1992)
Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G., (eds.) Symmetry and Perturbation Theory (Rome, 1998), pp. 23–37. World Sci. Publishing, River, Edge, NJ (1999)
Degasperis, A., Holm, D.D., Hone, A.N.I.: A new integrable equation with peakon solutions. Teoret. Mat. Fiz. 133(2), 170–183 (2002)
De Lellis, C.: Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio. Astérisque 317, Séminaire Bourbaki, exp. no 972, 175–203 (2008)
Escher, J., Liu, Y., Yin, Z.: Global weak solutions and blow-up structure for the Degasperis- Procesi equation. J. Funct. Anal. 241, 457–485 (2007)
Escher, J., Liu, Y., Yin, Z.: Shock waves and blow-up phenomena for the periodic Degasperis- Procesi equation. Indiana Univ. Math. J. 56, 87–117 (2007)
Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society-NSF, USA (2010)
Evans, L.C.: Weak convergence methods for nonlinear partial differential equations. American Mathematical Society-NSF, USA (1990)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions, Revised Edition. In: Textbooks in Mathematics. CRC Press, Taylor & Francis Group (2015)
Himonas, A., Holliman, C., Grayshan, K.: Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation. Commun. Partial Differ. Equ. 39(12), 2198–2215 (2014)
Khorbatlyab, B., Molinet, L.: On the orbital stability of the Degasperis-Procesi antipeakon-peakon profile. J. Differ. Equ. 269(6), 4799–4852 (2020)
Li, M., Yin, Z.: Global solutions and blow-up phenomena for a generalized Degasperis-Procesi equation. J. Math. Anal. Appl. 478(2), 604–624 (2019)
Liu, Y., Yin, Z.Y.: Global existence and blow-up phenomena for the Degasperis-Procesi equation. Commun. Math. Phys. 267, 801–820 (2006)
Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasi-linear equations of parabolic type. In: Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI (1968)
Lin, Z.W., Liu, Y.: Stability of peakons for the Degasperis-Procesi equation. Commun. Pure Appl. Math. 62, 125–146 (2009)
Lundmark, H., Szmigeilski, J.: Multi-peakon solutions of the Degasperis-Procesi equation. Inverse Probl. 19, 1241–1245 (2003)
Lundmark, H., Szmigeilski, J.: Degasperis-Procesi peakons and the discrete cubic string. Int. Math. Res. Pap. 2, 53–116 (2005)
Lundmark, H.: Formation and dynamics of shock waves in the DegasperisProcesi equation. J. Nonlinear Sci. 17, 169–198 (2007)
Rohde, C., Tang, H.: On the stochastic Dullin-Gottwald-Holm equation: global existence and wave-breaking phenomena. Nonlinear Differ. Equ. appl. - NODEA 28, 5 (2021)
Simon, J.: Compact sets in the space \( L^{p}(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)
Yin, Z.Y.: On the Cauchy problem for an intergrable equation with peakon solutions. Ill. J. Math. 47, 649–666 (2003)
Yin, Z.Y.: Global weak solutions for a new periodic integrable equation with peakon solutions. J. Funct. Anal. 212, 182–194 (2004)
Yin, Z.Y.: Global solutions to a new integrable equation with peakons. Indiana Univ. Math. J. 53, 1189–1209 (2004)
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Lynnyngs K. Arruda acknowledges support from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo - Brazil), Grant 2017/23751-2. The work of F. Cipriano was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matemática e Aplicações).
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Arruda, L.K., Chemetov, N.V. & Cipriano, F. Solvability of the Stochastic Degasperis-Procesi Equation. J Dyn Diff Equat 35, 523–542 (2023). https://doi.org/10.1007/s10884-021-10021-5
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DOI: https://doi.org/10.1007/s10884-021-10021-5