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Local Well-posedness and Blow-up for the Restricted Fourth-Order Prandtl Equation

Ik Hyun Choi

TL;DR

The paper studies local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation on the half-line with nonlocal nonlinearity, expressed as $a_t = -a_{yyyy} + a^2 - a_y \int_0^y a\,dz$, under clamped boundary conditions. It develops a kernel-based spectral framework, deriving uniform $L^1$ bounds for the non-translation-invariant half-line biharmonic heat kernel and an integration-by-parts representation to preserve spatial regularity, enabling a Duhamel fixed-point construction. The main contributions include rigorous $L^1$ kernel estimates, smoothing and regularity results under compatibility conditions, a local existence/uniqueness theory in a suitable function space, and an energy-based criterion proving finite-time blow-up for negative-energy data, complemented by a numerical demonstration. This work advances the mathematical understanding of higher-order boundary-layer models, providing tools for analyzing reduced Prandtl-type dynamics and illustrating blow-up mechanisms in half-line dissipative systems.

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.

Local Well-posedness and Blow-up for the Restricted Fourth-Order Prandtl Equation

TL;DR

The paper studies local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation on the half-line with nonlocal nonlinearity, expressed as , under clamped boundary conditions. It develops a kernel-based spectral framework, deriving uniform bounds for the non-translation-invariant half-line biharmonic heat kernel and an integration-by-parts representation to preserve spatial regularity, enabling a Duhamel fixed-point construction. The main contributions include rigorous kernel estimates, smoothing and regularity results under compatibility conditions, a local existence/uniqueness theory in a suitable function space, and an energy-based criterion proving finite-time blow-up for negative-energy data, complemented by a numerical demonstration. This work advances the mathematical understanding of higher-order boundary-layer models, providing tools for analyzing reduced Prandtl-type dynamics and illustrating blow-up mechanisms in half-line dissipative systems.

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 bounds for the associated half-line biharmonic heat kernel. We establish uniform 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.
Paper Structure (6 sections, 12 theorems, 110 equations, 1 figure)

This paper contains 6 sections, 12 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Heat Kernel Representation
  4. Existence and Uniqueness of the Solution
  5. Blow-up of the Solution
  6. Numerical Example

Key Result

Theorem 1

Let $f\in C(\mathbb{R}_+)\cap L^1(\mathbb{R}_+)$. Consider the clamped biharmonic heat equation on the half-line: We seek a classical solution $u$ such that, for every $t>0$ and every integer $m\ge0$, $\lim_{x\to\infty} \partial_x^{\,m} u(t,x)=0.$ Define, for $t>0$ and $x,y\ge 0$, and the kernel $K$, Set Then $u$ solves eq:prob.

Figures (1)

  • Figure 1: The evolution of the solution profile for the initial data \ref{['eq:example']}. Profiles are sampled at time instances $\{t_n\}$ such that $\|a(\cdot, t_n)\|_{L^\infty} = 2^n \|a(\cdot, 0)\|_{L^\infty}$.

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Lemma 2
  • proof
  • ...and 15 more