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On the existence of non-negative weak solutions for $1D$ fourth order equations of gradient flow type

Stefanos Georgiadis, Stefano Spirito

TL;DR

This work studies a family of 1D fourth-order gradient-flow PDEs derived from the Korteweg energy, unifying models such as the Quantum-Drift-Diffusion and Thin-Film equations. It proves the global-in-time existence of non-negative weak solutions for all $β > -3$ under mild initial-data assumptions, without requiring an a priori bound on the energy exponent. The core method combines gradient-flow energy and entropy dissipations with an ε-regularized approximating system that preserves a skew-adjoint structure to avoid derivative loss, enabling uniform bounds and compactness. A careful convergence analysis (via Aubin–Lions and strong/weak convergences of density powers) yields a limit that satisfies the weak formulation and energy/entropy inequalities, establishing global solvability in one dimension. The results extend the one-dimensional theory beyond previously known ranges and support broader applicability of these gradient-flow models in capillarity, quantum diffusion, and related diffusion processes.

Abstract

In this paper, we consider a family of one-dimensional fourth order evolution equations arising as gradient flows of the Korteweg energy, i.e. the $L^2$-norm of the first derivative of some power of the density. This family of equations generalizes the Quantum-Drift-Diffusion equation and the Thin-Film equation. We prove the global-in-time existence of {\em non-negative} weak solutions without requiring any upper bound on the exponent of the power of the density in the energy.

On the existence of non-negative weak solutions for $1D$ fourth order equations of gradient flow type

TL;DR

This work studies a family of 1D fourth-order gradient-flow PDEs derived from the Korteweg energy, unifying models such as the Quantum-Drift-Diffusion and Thin-Film equations. It proves the global-in-time existence of non-negative weak solutions for all under mild initial-data assumptions, without requiring an a priori bound on the energy exponent. The core method combines gradient-flow energy and entropy dissipations with an ε-regularized approximating system that preserves a skew-adjoint structure to avoid derivative loss, enabling uniform bounds and compactness. A careful convergence analysis (via Aubin–Lions and strong/weak convergences of density powers) yields a limit that satisfies the weak formulation and energy/entropy inequalities, establishing global solvability in one dimension. The results extend the one-dimensional theory beyond previously known ranges and support broader applicability of these gradient-flow models in capillarity, quantum diffusion, and related diffusion processes.

Abstract

In this paper, we consider a family of one-dimensional fourth order evolution equations arising as gradient flows of the Korteweg energy, i.e. the -norm of the first derivative of some power of the density. This family of equations generalizes the Quantum-Drift-Diffusion equation and the Thin-Film equation. We prove the global-in-time existence of {\em non-negative} weak solutions without requiring any upper bound on the exponent of the power of the density in the energy.

Paper Structure

This paper contains 13 sections, 9 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Main result and state of the art
  3. Strategy of the proof
  4. A priori estimates and weak solutions
  5. Energy estimate.
  6. Entropy estimate.
  7. Definition of weak solutions.
  8. Some preliminary remarks on the Korteweg energy
  9. The approximating system
  10. Proof of the main result
  11. Uniform bounds
  12. Convergence lemma
  13. Proof of the main theorem

Key Result

Theorem 1.1

Assume that $\beta>-3$. Let $\rho^0\in L^{1}(\mathbb{T})$ be such that $\rho^0\geq 0$ if $\beta>-2$, and $m^0:=\min_{x\in\mathbb{T}}\rho^0(x)>0$ if $-3<\beta\leq-2$. Assume also that Then, there exists a global-in-time non-negative weak solution of eq:gfb-eq:gfgid, in the sense of Definition def:ws.

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1
  • Definition 2.1
  • Remark 2
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Remark 3
  • proof
  • Theorem 4.1
  • ...and 12 more