Plastic Burgers Equation Dynamics

• Plastic Burgers Equation is a nonlinear PDE coupling advective shock dynamics with rate-independent plastic yielding via the singular 1-Laplacian term.
• The model employs viscoplastic regularization to construct global-in-time solutions with uniform energy dissipation and bounded variation properties.
• It offers practical insights into sea-ice dynamics by capturing both inviscid shock formation and local plastic constraints inherent in geophysical flows.

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 (Liu et al., 10 Jan 2026).