A nonlinear fractional Rayleigh\u2013Stokes equation under nonlocal integral conditions

Nguyen Hoang Luc,Van Thinh Nguyen,Le Dinh Long,Ho Thi Kim Van

doi:10.1186/s13662-021-03545-z

Abstract

In this paper, we study the fractional nonlinear Rayleigh–Stokes equation under nonlocal integral conditions, and the existence and uniqueness of the mild solution to our problem are considered. The ill-posedness of the mild solution to the problem recovering the initial value is also investigated. To tackle the ill-posedness, a regularized solution is constructed by the Fourier truncation method, and the convergence rate to the exact solution of this method is demonstrated.

Highlights

Most of fluids in the real world, such as in food products, in natural substances, in biology, in industry, in cosmetics are treated as non-Newtonian fluids
For a second grade fluid, the equation of motion is of higher order than the Navier–Stokes equation, because it exhibits all properties of viscoelastic fluids
The starting point of the fractional derivative model of a viscoelastic fluid is usually a classical differential equation which is modified by replacing the time derivative of an integer order with a fractional calculus operator

Summary

Introduction

Most of fluids in the real world, such as in food products (mayonnaise, mustard, chocolate, ketchup, butter, cheese, yogurt, etc.), in natural substances (honey, magma, lava, gums, etc.), in biology (blood, semen, synovia, mucus, etc.), in industry (paint, glue, lubricant, ink, molten polymer, etc.), in cosmetics (soap solution, toothpaste, cream, silicone, nail polish, etc.) are treated as non-Newtonian fluids. The starting point of the fractional derivative model of a viscoelastic fluid is usually a classical differential equation which is modified by replacing the time derivative of an integer order with a fractional calculus operator. This generalization allows one to define precisely noninteger order integrals or derivatives. The exact solution of the Rayleigh–Stokes problem for a generalized second grade fluid in a porous half-space with a heated flat plate was considered by Xue et al [22]. It is emphasized that we cannot apply Parseval’s equality to obtain stable estimates in the Lp space To tackle this limitation, we need to develop additional techniques and Sobolev embedding in our study.

Preliminary results We recall the spectral problem as follows:

Conclusion