Solvability of the Stochastic Degasperis-Procesi Equation
Tóm tắt
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.
Tài liệu tham khảo
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)