Local well-posedness and blow-up for the restricted fourth-order Prandtl equation

TL;DR

This paper establishes local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation, involving nonlocal nonlinearities, by deriving uniform kernel estimates and analyzing the associated semigroup.

Contribution

It introduces a novel analysis of the restricted fourth-order Prandtl equation, including uniform kernel bounds and blow-up solutions, advancing understanding of high-order boundary layer models.

Findings

Proved local existence and uniqueness of solutions.

Constructed solutions that blow up in finite time.

Established uniform $L^1$ bounds for the biharmonic heat kernel.

Abstract

We prove local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation posed on the half-line with clamped boundary conditions. The equation arises from a two-dimensional fourth-order Prandtl system via an ansatz reduction, and its nonlinearity involves a nonlocal integral term. To close a Duhamel fixed-point argument, we need uniform $L^1$ bounds for the associated half-line biharmonic heat kernel. We establish uniform $L^1$ estimates for the kernel and its derivatives, and we show that the semigroup preserves spatial regularity under appropriate compatibility conditions, using an alternative representation derived by integration by parts. These kernel estimates yield local existence and uniqueness for the restricted model and allow us to construct solutions that blow-up in finite time.