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Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator

Author

Listed:
  • Lyailya Zhapsarbayeva

    (Department of Fundamental Mathematics, L.N. Gumilyov Eurasian National University, 2, Satbaev St., Astana 010000, Kazakhstan)

  • Dongming Wei

    (Department of Mathematics, Nazarbayev University, 53, Kabanbay batyr Ave., Astana 010000, Kazakhstan)

  • Bagyzhan Bagymkyzy

    (Department of Fundamental Mathematics, L.N. Gumilyov Eurasian National University, 2, Satbaev St., Astana 010000, Kazakhstan)

Abstract

In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids ρ ( ∂ t u + u ∂ x u ) = μ ∂ x ∂ x u p − 2 ∂ x u , augmented with the initial condition u ( 0 , x ) = u 0 , 0 < x < L , and the boundary condition u ( t , 0 ) = u ( t , L ) = 0 , where ρ is the density, μ the viscosity, u the velocity of the fluid, 1 < p < 2 , L > 0 , and T > 0 . We show that this initial boundary problem has an unique solution in the Buchner space L 2 0 , T ; W 0 1 , p ( 0 , 1 ) for the given set of conditions. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL Multiphysics 6.0 at small and large Reynolds numbers to show the images of the solutions.

Suggested Citation

  • Lyailya Zhapsarbayeva & Dongming Wei & Bagyzhan Bagymkyzy, 2025. "Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator," Mathematics, MDPI, vol. 13(5), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:708-:d:1597136
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