---
title: Plastic Burgers Equation Dynamics
url: https://www.emergentmind.com/topics/plastic-burgers-equation
type: topic
---

# Plastic Burgers Equation Dynamics

The plastic Burgers equation is a nonlinear partial differential equation modeling the interplay between advective transport and plastic, rate-independent stress, formulated in one spatial dimension. Its core structure is the classical Burgers equation augmented by a singular p-Laplacian term with $p=1$, represented by the multivalued subdifferential of the total variation functional. This coupling yields a model that encapsulates both shock dynamics characteristic of inviscid flows and local plastic yield constraints, offering relevance to contexts such as the rheology of sea-ice where plastic effects dominate.

## 1. Definition and Mathematical Formulation

The plastic Burgers equation on $D = [0, \infty) \times \mathbb{R}$ governs an unknown velocity field $u = u(t, x)$:
\[
u_t + \partial_x\left(\frac{u^2}{2}\right) - \sigma = 0, \qquad \sigma \in \partial\psi(u_x)
\]
with $\psi(e) = |e|$. The subdifferential $\partial\psi(e)$ is the multivalued "Sign" operator: $\partial\psi(e) = \{ e/|e| \}$ for $e \neq 0$ and $[-1,1]$ for $e = 0$, so that the stress $\sigma(t,x)$ is a "1-Laplacian" and satisfies $|\sigma| \leq 1$. Equivalently, this can be expressed as:
\[
u_t + \partial_x\left(\frac{u^2}{2} - R\right) = 0, \qquad R \in \text{Sign}(u_x)
\]
This formulation incorporates nonlinear advection and a plastic yielding mechanism, with $\sigma$ governing the rate-independent stress response.

## 2. Viscoplastic Regularization: Smooth Approximation and Well-Posedness

To construct global-in-time solutions, a viscoplastic regularization is introduced with $\varepsilon > 0$. The nonsmooth plastic term is approximated by the functional $\psi_\varepsilon(e) = \sqrt{e^2 + \varepsilon^2}$, with derivative $g_\varepsilon(e) = e/\sqrt{e^2 + \varepsilon^2}$, satisfying $|g_\varepsilon| < 1$. The regularized Cauchy problem reads:
\[
u^\varepsilon_t + \partial_x\left(\frac{(u^\varepsilon)^2}{2}\right) - \partial_x\left(g_\varepsilon(u^\varepsilon_x) + \varepsilon u^\varepsilon_x\right) = 0
\]
along with initial data $u^\varepsilon(0, x) = u^\varepsilon_{\text{in}}(x)$ and decay conditions at spatial infinity. In non-divergence form:
\[
u^\varepsilon_t + u^\varepsilon u^\varepsilon_x - \varepsilon u^\varepsilon_{xx} - \partial_x\left(\frac{u^\varepsilon_x}{\sqrt{(u^\varepsilon_x)^2 + \varepsilon^2}}\right) = 0
\]
This setup introduces rate-dependent viscous dissipation, rendering the problem parabolic and admitting standard existence theory.

## 3. Existence, Uniform Estimates, and Energy Dissipation

For smooth initial data $u^\varepsilon_{\text{in}} \in H^2(\mathbb{R})$ vanishing at infinity, the regularized problem possesses a unique global solution:
\[
u^\varepsilon \in L^\infty(0,T; H^2(\mathbb{R})) \cap L^2(0,T; H^2(\mathbb{R})), \qquad u^\varepsilon_t \in L^\infty(0,T; L^2) \cap L^2(0,T; H^1)
\]
solving a weak PDE form for all test functions $\varphi \in C_c^1([0,T) \times \mathbb{R})$. Energy-dissipation laws and uniform bounds independent of $\varepsilon$ hold:
\[
\frac{1}{2}\| u^\varepsilon(T) \|_{L^2}^2 + \varepsilon \int_0^T \| u^\varepsilon_x \|_{L^2}^2 dt + \int_0^T \int \frac{|u^\varepsilon_x|^2}{\sqrt{(u^\varepsilon_x)^2 + \varepsilon^2}} dx dt = \frac{1}{2}\| u^\varepsilon_{\text{in}} \|_{L^2}^2
\]
Additional uniform-in-$\varepsilon$ bounds are established for $L^\infty_tL^2_x$, $L^\infty_tBV_x$, and maximum principle quantities, including an Oleinik-type shock constraint $u^\varepsilon(t,x) < 1/t$ for $t > 0$.

## 4. Singular Limit and BV Solutions

Passing to the limit $\varepsilon \to 0$ leverages the uniform estimates to extract subsequential limits:
\[
u^\varepsilon \rightarrow u \text{ in } L^2_{\text{loc}}, \quad u^\varepsilon_x \rightharpoonup u_x \text{ in } M_{\text{loc}}, \quad R^\varepsilon \rightharpoonup R \text{ weak-* in } L^\infty
\]
with $\varepsilon u^\varepsilon_x \to 0$. The limiting pair $(u, R)$ satisfies:
\[
u \in L^\infty_t (L^\infty_x \cap BV_x) \cap C_t L^1_x \cap L^\infty_t L^2_x, \quad R \in L^\infty_t (L^\infty_x \cap BV_x), \quad |R| \leq 1
\]
as well as the weak PDE $u_t + \partial_x(u^2/2 - R) = 0$ and the initial condition $u(0) = u_{\text{in}}$ in $L^1$. The energy-dissipation inequality in the limit is:
\[
\frac{1}{2}\|u(t)\|_{L^2}^2 + \int_0^t TV_x(u(s)) ds \leq \frac{1}{2}\|u_{\text{in}}\|_{L^2}^2
\]
showing existence and regularity of solutions of bounded variation for the singular equation.

## 5. Precise Identification of the Singular Stress Law

The limiting stress law is obtained via convex analysis of the time-integrated total variation functional $\Psi$, defined on $X = L^2(0,T;L^2(\mathbb{R}))$:
\[
\Psi(v) = \int_0^T TV_x[v(\cdot, t)] dt \quad \text{if } v \in L^1_t BV_x, \qquad \Psi(v) = +\infty \text{ otherwise}
\]
The subdifferential $\partial\Psi$ characterizes rate-independent, singular dissipation:
\[
\zeta \in \partial\Psi(u) \iff \langle \zeta, v \rangle_X = \Psi(v) \text{ for some } v
\]
A Minty-type argument proves that the limit of the regularized stress $-\partial_x R^\varepsilon$ converges to $-D_x R$, a distributional element of $\partial\Psi(u)$. This precisely recovers the "1-Laplacian" (total variation flow) law for the singular plastic stress.

## 6. Interplay of Advection, Shock Formation, and Plastic Constraints

The advective term $u_t + \partial_x(u^2/2)$ induces shock formation identical to the inviscid Burgers equation. The plastic stress $\sigma \in \text{Sign}(u_x)$ enforces local constraints $|\sigma| \leq 1$ with yielding at unity, producing spatial plateaus where $u_x = 0$ and $\sigma \in [-1, 1]$. Oleinik's entropy condition $u(t, x + h) - u(t, x) \leq h/t$ selects the unique entropic BV solution under plastic yielding. The system, unlike total variation (TV) flows $u_t = \text{div}(u_x/|u_x|)$, does not admit a gradient-flow structure; rather, the convective derivative results in a damped Hamiltonian system without inertia. This feature distinguishes the plastic Burgers equation in the landscape of degenerate parabolic and rate-independent evolutionary PDEs.

## 7. Physical Interpretation and Relevance to Sea-Ice Dynamics

The plastic Burgers equation is a one-dimensional analogue of the momentum balance model for sea-ice dynamics, specifically the Hibler model, where the stress tensor incorporates plastic effects via terms analogous to the 1-Laplacian. In this context, plateaus and yield constraints encode the rigid-plastic behavior of ice floes, and the PDE serves as a simplified framework to study the interaction of inertial transport and local yielding. A plausible implication is refined insights into rate-independent phenomena in geophysical flows, with explicit characterization of solution regularity and selection mechanisms rooted in energy dissipation and entropy admissibility [2601.06489].

Source: https://www.emergentmind.com/topics/plastic-burgers-equation