Discontinuity, Nonlinearity, and Complexity
Study on Stochastic Quasi-Linear Partial Differential Equations of Evolution
Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 1--11 | DOI:10.5890/DNC.2020.03.001
A. Anguraj, K. Ramkumar
Department of Mathematics, PSG College of Arts & Science, Coimbatore-14
Download Full Text PDF
Abstract
In this article, the existence and uniqueness of local mild solution of a stochastic counterpart of Tosio Kato’s Quasi-linear partial differential equation with additive cylindrical wiener process in a separable Hilbert space is established using contraction mapping principle.
References
-
[1]  | Kato, T. (1970), Linear evolution equations of “hyperbolic type”, J. Fac.Sci.Univ.Tokyo sect.I, 17, 241-258. |
-
[2]  | Kato, T. (1973), Linear evolution equations of “hyperbolic” type II , J. Math Soc. Japan., 25, 648-666. |
-
[3]  | Kato, T. (1975), Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, 448, 25-70. |
-
[4]  | Pazy, A. (1983), Semigroups of linear operators and applications to partial differential equations, AppliedMathematical Sciences, 44. |
-
[5]  | Dawson, D.A. (1975), Stochastic evolution equations and related measure processes, J. Multivariate Anal., 5, 1-52. |
-
[6]  | Prato, G.D. and Zabczyk, J. (1990), Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, U.K.. |
-
[7]  | Prato, G.D. and Zabczyk, J. (1992), A note on stochastic convolution, Stochastic Anal. Appl., 10, 143-153. |
-
[8]  | Seidler, J. (1993), Da Prato-Zabcyk’s maximal inequality revisited, I. Math. Bohem., 118(1), 67-106. |
-
[9]  | Pronk, M. andVeraar, M. (2014), A new approach to stochastic evolution equations with adapted drift, J. Diff. Equations, 256, 3634-3683. |
-
[10]  | Veraar, M.C. (2010), Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ., 10(1), 85-127. |
-
[11]  | Fernando, B.P.W. and Sritharan, S.S. (2015), Stochastic quasilinear partial differential equations of evolution, Infinite Dimentional Analysis, Quantum Probability and Related Topics, 18(3). |
-
[12]  | Suvinthra, M. and Balachandran, K. (2017), Large deviations for nonlinear ito type stochastic integrodifferential equations, Journal of Applied nonlinear Dynamics, 6, 1-15. |
-
[13]  | Mabel Lizzy, R., Balachandran,K., and Suvinthra, M. (2017), Controllability of non linear stochastic fractional systems with Levy noise, Discontinuity, Nonlinearity, and Complexity, 6(3), 409-420. |
-
[14]  | Haseena, A., Suvinthra, M., and Annapoorani, N. (2017), On Large deviations of stochastic integrodifferential equations with Brownian Motion, Discontinuity, Nonlinearity, and Complexity, 6, 281-294. |
-
[15]  | Barbu, V. (1976), Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Springer-Verlag, Bucherest. |
-
[16]  | Fischer, A.E. andMarsden, J.E. (1972), The Einstein evolution equations as a first order quasi-linear symmetric hyperbolic systems I, Comm. Math. Phys., 28, 1-38. |
-
[17]  | Gross, L. (1966), The Cauchy problem for the coupled maxwell and dirac equations, Comm. Pure Appl. Math., 19, 1-15. |
-
[18]  | Ichikawa, A. (1986), Some inequalities for martingales and stochastic convolution, Stochastic Anal. Appl., 4(3), 329- 339. |
-
[19]  | Kato, T. and Ponce, G. (1988), Commutator estimates and the Euler and Navier-stokes equations, Comm. Pure Appl. Math., 41, 891-907. |
-
[20]  | Pronk, M. and Veraar,M. (2013), A new approach to stochastic evolution equations with adapted drift. |
-
[21]  | Sjoberg, A. (1979), On the Korteweg-de Vries Equation: existence and uniqueness, J. Math. Anal. Appl., 29, 569-579. |
-
[22]  | Taylor, M.E. (1991), Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, 100. |
-
[23]  | Temam, R. (1975), On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20(1), 32-43. |