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Large self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field

Lorenzo Brandolese, Grzegorz Karch

TL;DR

The paper investigates the 3D Oberbeck–Boussinesq system with a Newtonian gravitational field in ${\mathbb{R}}^3$ and establishes the existence of large self-similar solutions for initial data homogeneous of degree $-1$ by reducing the time-dependent problem to a perturbed elliptic system in ${\mathbb{R}}^3$ and applying the Leray–Schauder fixed-point theorem. A key technical contribution is the development of uniform-in-$\lambda$ a priori bounds and a careful treatment of the singular gravity term via Hardy-type estimates and cut-offs, enabling a global existence theory in unbounded domains. The resulting self-similar solution lies in $C_w([0,\infty), {\bf L}^{3,\infty}(\mathbb{R}^3))$ and satisfies quantitative time-decay relative to the heat semigroup, with $\|u(t)-e^{t\Delta}u_0\|_2+\|\theta(t)-e^{t\Delta}\theta_0\|_2= c t^{1/4}$ and $\|\nabla u(t)-\nabla e^{t\Delta}u_0\|_2+\|\nabla\theta(t)-\nabla e^{t\Delta}\theta_0\|_2= c' t^{-1/4}$. The approach generalizes to additional forcing terms and aligns with prior large-self-similar work for Navier–Stokes, while leveraging the special Newtonian gravitational structure.

Abstract

The Navier-Stokes system for an incompressible fluid coupled with the equation for a heat transfer is considered in the whole three dimensional space. This system is invariant under a suitable scaling. Using the Leray-Schauder theorem and compactness arguments, we construct self-similar solutions to this system without any smallness assumptions imposed on homogeneous initial conditions.

Large self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field

TL;DR

The paper investigates the 3D Oberbeck–Boussinesq system with a Newtonian gravitational field in and establishes the existence of large self-similar solutions for initial data homogeneous of degree by reducing the time-dependent problem to a perturbed elliptic system in and applying the Leray–Schauder fixed-point theorem. A key technical contribution is the development of uniform-in- a priori bounds and a careful treatment of the singular gravity term via Hardy-type estimates and cut-offs, enabling a global existence theory in unbounded domains. The resulting self-similar solution lies in and satisfies quantitative time-decay relative to the heat semigroup, with and . The approach generalizes to additional forcing terms and aligns with prior large-self-similar work for Navier–Stokes, while leveraging the special Newtonian gravitational structure.

Abstract

The Navier-Stokes system for an incompressible fluid coupled with the equation for a heat transfer is considered in the whole three dimensional space. This system is invariant under a suitable scaling. Using the Leray-Schauder theorem and compactness arguments, we construct self-similar solutions to this system without any smallness assumptions imposed on homogeneous initial conditions.
Paper Structure (6 sections, 7 theorems, 90 equations)

This paper contains 6 sections, 7 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Analysis of the perturbed elliptic system
  3. A priori estimates for a perturbed elliptic system in bounded domains
  4. Existence of solutions to the perturbed elliptic system in bounded domains
  5. Existence of solutions of the perturbed elliptic system in the whole space
  6. Conclusion

Key Result

Theorem 1.1

Let $(u_0,\theta_0)\in {\bf L}^\infty_{\rm loc}({\mathbb{R}}^3\backslash\{0\})$ be homogeneous of degree $-1$, with $\hbox{$\nabla\cdot u_0$}=0$. Let the external force $f(x,t)$ be of the form as in Gf with the profile $F\in H^{-1}({\mathbb{R}}^3)^3$. Then there exists a self-similar solution $(u,p)

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 5 more