Local well-posedness of 1D degenerate drift diffusion equation

La-Su Mai; Suriguga; La-Su Mai; Suriguga

doi:10.3934/mine.2024007

Mathematics in Engineering

2024, Volume 6, Issue 1: 155-172. doi: 10.3934/mine.2024007

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Local well-posedness of 1D degenerate drift diffusion equation

La-Su Mai ^,,
Suriguga

School of Mathematics Sciences, Inner Mongolia University, Hohhot 010021, China

Received: 07 December 2023 Revised: 15 February 2024 Accepted: 18 February 2024 Published: 28 February 2024

This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.
- drift diffusion equation,
- free boundary,
- degeneracy,
- local well-posedness
Citation: La-Su Mai, Suriguga. Local well-posedness of 1D degenerate drift diffusion equation[J]. Mathematics in Engineering, 2024, 6(1): 155-172. doi: 10.3934/mine.2024007

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Abstract

This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.

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