Viscoplastic Burgers Equation Analysis

• The viscoplastic Burgers equation is a 1D conservation law that incorporates nonlinear advection with a singular, plastic stress term to model yield‐stress phenomena.
• It employs a variational regularization framework with uniform energy-dissipation estimates and convergence to BV solutions as the regularization vanishes.
• The model provides insights into geophysical applications such as sea-ice dynamics by relating shock formation and plateau behaviors to plastic yield conditions.

The viscoplastic Burgers equation is a one-dimensional scalar conservation law that extends the classical inviscid Burgers equation by incorporating a singular plastic stress, formally represented by the 1-Laplacian or, equivalently, the subdifferential of the total variation functional. This prototype arises in contexts with yield-stress effects and plasticity, notably in models of sea-ice dynamics where plastic flow constrains strain rates. The equation admits a variational regularization yielding well-posedness, and its singular limit is characterized by solutions in the space of functions of bounded variation (BV), with the non-smooth stress interpreted via convex analysis and monotone operator theory (Liu et al., 10 Jan 2026).

1. Formulation and Regularization

The standard viscous Burgers equation involves linear viscosity, whereas the viscoplastic Burgers equation supplements the inviscid dynamics with a nonlinear, singular plastic stress. For $\varepsilon > 0$, the regularized Cauchy problem is: $$\begin{cases} \partial_t u^\varepsilon + \partial_x \left( \frac{1}{2} (u^\varepsilon)^2 \right) = \partial_x \sigma^\varepsilon, \qquad (t, x) \in (0, \infty) \times \mathbb{R}, \ \sigma^\varepsilon = \partial_s \psi_\varepsilon ( \partial_x u^\varepsilon ), \qquad \psi_\varepsilon(s) = \sqrt{\varepsilon^2 + |s|^2}, \ u^\varepsilon(0,x) = u_{\text{in}}^\varepsilon(x), \qquad u^\varepsilon \to 0 \text{ as } |x| \to \infty. \end{cases}$$ This yields the explicit relation

$$\sigma^\varepsilon = \frac{\partial_x u^\varepsilon}{\sqrt{\varepsilon^2 + |\partial_x u^\varepsilon|^2}}.$$

Initial data $u_{\text{in}}^\varepsilon$ are smooth, $H^2(\mathbb{R})$-functions vanishing at infinity with uniform-in-$\varepsilon$ bounds.

2. Existence and Regularity of Solutions

For fixed $\varepsilon > 0$, the equation admits a unique global solution with the following properties [Thm 3.1, (Liu et al., 10 Jan 2026)]:

• $$u^\varepsilon \in L^\infty(0, \infty; H^2(\mathbb{R})) \cap L^2(0, \infty; H^2(\mathbb{R}))$$,
• $$\partial_t u^\varepsilon \in L^\infty(0, \infty; L^2(\mathbb{R})) \cap L^2(0, \infty; H^1(\mathbb{R}))$$,
• Weak (integral) formulation and global energy-dissipation balance for every $T > 0$: $$\frac{1}{2} \| u^\varepsilon(T) \|_{L^2}^2 + \varepsilon \int_0^T \| \partial_x u^\varepsilon \|_{L^2}^2 \, dt + \int_0^T \int_\mathbb{R} \frac{|\partial_x u^\varepsilon|^2}{\sqrt{\varepsilon^2 + |\partial_x u^\varepsilon|^2}} dx\, dt = \frac{1}{2} \| u^\varepsilon_{\text{in}} \|_{L^2}^2.$$ A series of a priori estimates, uniform in $\varepsilon$, hold for $u^\varepsilon$ and $\sigma^\varepsilon$, including $L^\infty$ and BV estimates ensuring compactness as $\varepsilon \to 0$.

3. Singular Limit and the Plastic Burgers Equation

Letting $\varepsilon \to 0$, compactness arguments (energy estimates, Oleinik one-sided bound, and the Aubin–Simon lemma) yield convergence to a limit $(u, R)$, where $u$ solves the plastic (or perfectly plastic) Burgers equation in the sense of distributions: $$\partial_t u + \partial_x \left( \frac{1}{2} u^2 \right) \in \partial_x \xi,$$ for $$\xi(t, \cdot) \in \partial \mathrm{TV}(u(t, \cdot))$$, where $\mathrm{TV}$ denotes total variation. The limit equation includes the following features:

• Global energy-dissipation: $$\frac{1}{2}\|u(t)\|_{L^2}^2 + \int_0^t \mathrm{TV}(u(\tau))\, d\tau \leq \frac{1}{2} \|u_{\text{in}}\|_{L^2}^2,$$
• Oleinik’s condition: $\partial_x u(\cdot, t) < 1/t$ in distributions,
• Existence of $(u, R)$ with $$u \in L^\infty(0, \infty; BV \cap L^\infty) \cap C([0, \infty); L^1)$$, $R \in L^\infty(0, \infty; BV \cap L^\infty)$, $|R| \leq 1$.

4. Total Variation and Its Subdifferential

For $v \in BV(\mathbb{R})$, the total variation is defined by

$$\mathrm{TV}(v) = \int_{\mathbb{R}} |D_x v| = \sup \left\{ \int_{\mathbb{R}} v\, \phi_x\, dx : \phi \in C_0^1,\, |\phi| \leq 1 \right\}.$$

Its convex subdifferential $\partial \mathrm{TV}(v)$ in $L^2$ is the set of $\xi \in L^2$ such that

$$\forall w \in L^2:~ \mathrm{TV}(w) \geq \mathrm{TV}(v) + \int_{\mathbb{R}} \xi (w-v)\, dx,$$

which, equivalently, satisfies $\|\xi\|_{L^\infty} \leq 1$ and $\xi(x) \in \mathrm{Sign}(v_x(x))$ almost everywhere, with $\int \xi v_x dx = \mathrm{TV}(v)$. Here, $\mathrm{Sign}(s) = s/|s|$ for $s \neq 0$, $[-1,1]$ for $s=0$.

5. Interplay of Nonlinear Advection and Plastic Stresses

The model combines nonlinear advection (Burgers nonlinearity) with the singular 1-Laplacian stress, producing a “damped Hamiltonian system.” Without the plastic term, Burgers shocks are governed by the Rankine–Hugoniot and Oleinik’s entropy conditions. The 1-Laplacian stress term, expressed as $\partial_x \xi$ with $\xi \in \mathrm{Sign}(\partial_x u)$, implements a “plastic” regularization which prevents excessive gradient growth, produces plateaus, and accommodates shock-like discontinuities. The equation is not a gradient flow, and uniqueness cannot be guaranteed in full generality. However, Oleinik’s one-sided condition provides partial control of admissible jump discontinuities.

6. Connection to Sea-Ice Dynamics and the Hibler Model

The viscoplastic Burgers equation is motivated by similarities to Hibler’s two-dimensional viscous–plastic model of sea-ice, where the stress tensor is governed by a convex, positively 1-homogeneous potential $\psi_H(e) = P \sqrt{ \ldots |e|^2 \ldots }$ and $\sigma_H \in \partial \psi_H(e)$. In the scalar “toy” model discussed here, the 1D strain rate $u_x$ substitutes for the full strain-rate tensor, and $\partial \psi_H$ is replaced by the subdifferential of the modulus. The analogy aligns

• The convective term $u u_x$ with sea-ice advection,
• The plastic stress $\partial_x \xi$, $\xi \in \mathrm{Sign}(u_x)$, with Hibler’s non-Newtonian yield condition.

Explicit numerical simulations are not provided, but the qualitative parallels are observed: shock formation is analogous to ridge formation in sea ice, and plateaus represent intact ice floes (Liu et al., 10 Jan 2026).

7. Mathematical and Physical Significance

The viscoplastic Burgers equation provides a rigorous, variationally-regularized framework for studying interacting nonlinear transport and plastic effects. The limit equation characterizes the interplay between singular diffusion and nonlinear advection—a scenario encountered in idealized materials with yield stress and in geophysical applications such as sea-ice dynamics. Existence theory leverages vanishing-viscosity methods and convex functional analysis, exploiting the structure of the total variation subdifferential. The results establish uniform estimates, existence of BV solutions, and a global energy-entropy structure, confirming the suitability of the model as a testbed for understanding more complex vectorial plastic flows (Liu et al., 10 Jan 2026).