Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear distortion of symmetry in solutions to the convection-diffusion equation of Burgers type

Masakazu Yamamoto

TL;DR

The paper analyzes nonlinear distortion of symmetry in solutions to the Burgers-type convection-diffusion equation $\partial_t u - \Delta u = \mathbf{a}\cdot \nabla(u^2)$ on $\mathbb{R}^n$, focusing on dimensions $2,3,4$. It develops dimension-dependent asymptotic expansions via renormalization, showing explicit logarithmic evolutions in even dimensions (2D and 4D) that encode nonlinear distortion, while odd dimensions (3D and higher) exhibit higher-order distortions without basic logarithmic terms at leading orders; in 3D a logarithmic correction emerges at the next order with a coefficient determined by $M_0$ and $\mathbf{a}$. The approach connects to linear diffusion profiles and to Navier–Stokes scaling, illustrating how bilinear nonlinearities shape long-time behavior and symmetry of solutions. The results provide concrete, dimension-aware templates for nonlinear distortions in bilinear convection-diffusion problems and suggest using Burgers-type models as indicators for more complex fluid-like systems.

Abstract

In this paper, the initial value problem of the convection-diffusion equation of Burgers type is treated. In the asymptotic profile of solutions, the nonlinearity of the equation is reflected. Regarding the solutions to this model, the Spanish school in the 1990s performed asymptotic expansions based on the linear diffusion. Those profiles exhibit symmetries characteristic of linear phenomena. In this paper, the distortion of symmetry arising from the nonlinear effects is described explicitly. Furthermore, it is demonstrated that the extent of this distortion differs significantly depending on the parity of the spatial dimension. This contradicts the conventional expectation that the manifestation of nonlinearity depends on the scale of the equation. This interpretation is supported by comparison with similar Navier--Stokes equations. The Burgers type is applicable as an indicator for considering several bilinear problems.

Nonlinear distortion of symmetry in solutions to the convection-diffusion equation of Burgers type

TL;DR

The paper analyzes nonlinear distortion of symmetry in solutions to the Burgers-type convection-diffusion equation on , focusing on dimensions . It develops dimension-dependent asymptotic expansions via renormalization, showing explicit logarithmic evolutions in even dimensions (2D and 4D) that encode nonlinear distortion, while odd dimensions (3D and higher) exhibit higher-order distortions without basic logarithmic terms at leading orders; in 3D a logarithmic correction emerges at the next order with a coefficient determined by and . The approach connects to linear diffusion profiles and to Navier–Stokes scaling, illustrating how bilinear nonlinearities shape long-time behavior and symmetry of solutions. The results provide concrete, dimension-aware templates for nonlinear distortions in bilinear convection-diffusion problems and suggest using Burgers-type models as indicators for more complex fluid-like systems.

Abstract

In this paper, the initial value problem of the convection-diffusion equation of Burgers type is treated. In the asymptotic profile of solutions, the nonlinearity of the equation is reflected. Regarding the solutions to this model, the Spanish school in the 1990s performed asymptotic expansions based on the linear diffusion. Those profiles exhibit symmetries characteristic of linear phenomena. In this paper, the distortion of symmetry arising from the nonlinear effects is described explicitly. Furthermore, it is demonstrated that the extent of this distortion differs significantly depending on the parity of the spatial dimension. This contradicts the conventional expectation that the manifestation of nonlinearity depends on the scale of the equation. This interpretation is supported by comparison with similar Navier--Stokes equations. The Burgers type is applicable as an indicator for considering several bilinear problems.

Paper Structure

This paper contains 11 sections, 4 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Notations.
  3. The even-dimensional cases
  4. The two-dimensional case
  5. The four-dimensional case
  6. The three-dimensional case
  7. The expansion up to second order
  8. The expansion up to third order
  9. Derivation of the logarithmic evolution
  10. The other odd-dimensional cases
  11. The normal convection-diffusion equation

Key Result

Theorem 1.1

Let $n = 2,~ u_0 \in L^1 (\mathbb{R}^2) \cap L^\infty (\mathbb{R}^2)$ and $|x| u_0 \in L^1 (\mathbb{R}^2)$. Then, the solution $u$ of cd fulfills that as $t \to +\infty$ for $1 \le q \le \infty$ and $\gamma_q = 1 - \frac{1}{q}$, where $M_0 = \int_{\mathbb{R}^2} u_0 (x) dx$.

Theorems & Definitions (4)

  • Theorem 1.1: cf.EZDrCrpoDrZZ
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4