[Paper Review] On the Theory of Discontinuous Solutions to Some Strongly Degenerate Parabolic Equations

Yu. G. Rykov|ArXiv.org|May 22, 2001

Navier-Stokes equation solutions9 references3 citations

TL;DR

This paper establishes the existence and uniqueness of generalized discontinuous solutions for a class of strongly degenerate parabolic equations of Burgers' type with bounded dissipation flux $ Q(u_x) $, where viscosity vanishes as $ |u_x| \to \infty $. The key contribution is a novel a priori estimate on $ Q(u_x) $ alone, enabling arbitrary local growth of the gradient while ensuring entropy-satisfying solutions and uniqueness under Oleinik's condition.

ABSTRACT

It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted. In the paper the definition of the generalized solution is given and the existence theorem is established in the classes of functions close to ones of bounded variation. The main feature of used a priori estimates is the fact that one needs to estimate only the diffusion flux, which allows to have in fact arbitrary local growth of the velocity gradient. The uniqueness theorem is proven for essentially narrower class of piecewise smooth functions with regular behavior of discontinuity lines.

Motivation & Objective

• To establish the existence of generalized solutions for a class of strongly degenerate parabolic equations with bounded dissipation flux $ Q(u_x) $, where $ Q' > 0 $ and $ \max|Q(s)| < \infty $.
• To define a generalized solution concept that allows for discontinuities, consistent with entropy conditions, even as the equation degenerates to hyperbolic form at infinite gradients.
• To prove uniqueness of solutions within a restricted class $ \mathcal{K} $ of piecewise smooth functions with regular discontinuity lines, under Oleïnik's entropy condition.
• To overcome the challenge of proving general uniqueness by introducing a novel a priori estimate that depends only on $ Q(u_x) $, not on $ u_x $ itself.

Proposed method

• Define a generalized solution via an integral inequality formulation, ensuring entropy conditions are satisfied at discontinuities.
• Use the small viscosity method to construct approximate solutions and derive a priori estimates focused solely on $ Q(u_x) $, not on $ u_x $, enabling arbitrary local gradient growth.
• Introduce level lines of $ u(t,x) $ as analogs of characteristic lines in hyperbolic equations, to analyze discontinuity propagation.
• Apply contour integration over domains bounded by level lines and discontinuity curves to derive integral identities that enforce entropy conditions.
• Use the monotonicity of $ Q $ and the structure of the flux to bound integrals over discontinuity lines, showing that $ u - v \to 0 $ as $ \alpha \to 0 $, implying uniqueness.
• Leverage the fact that $ Q_{\text{lim}}^u \leq Q_{\text{lim}}^v $ at discontinuities due to $ u_x \to -\infty $, which supports entropy satisfaction.

Experimental results

Research questions

• RQ1 Can generalized discontinuous solutions exist for strongly degenerate parabolic equations where the dissipation flux $ Q(u_x) $ is bounded and $ Q' > 0 $, even as the equation degenerates to hyperbolic form?
• RQ2 How can one construct a priori estimates that allow for arbitrary local growth of the velocity gradient $ u_x $ while ensuring solution regularity?
• RQ3 What conditions ensure the uniqueness of generalized solutions when discontinuities are present?
• RQ4 Can the entropy condition be rigorously enforced in the context of degenerate parabolic equations with bounded dissipation?
• RQ5 How does the interaction between discontinuities and the degenerate viscous term generate a boundary layer-like structure near discontinuities?

Key findings

• The existence of generalized solutions in $ L_1(\Pi_T) $ is proven for initial data in $ BV_{C^1}^+(\mathbb{R}) $, with solutions in $ BV_{C^1}(\mathbb{R}) $ for almost all $ t \in [0,T] $.
• A priori estimates depend only on $ Q(u_x) $, not on $ u_x $, allowing for arbitrary local growth of the velocity gradient.
• Uniqueness is established for solutions in the class $ \mathcal{K} $, which consists of piecewise smooth functions with regular discontinuity lines, under Oleïnik's entropy condition.
• The integral identity (21) over contours enclosing level sets of $ w = u - v $ leads to $ \int |u - v| dx \leq O(\alpha) $, implying $ u = v $ a.e. as $ \alpha \to 0 $, proving uniqueness.
• At discontinuities, the limit of $ Q(u_x) $ satisfies $ Q_{\text{lim}}^u \leq Q_{\text{lim}}^v $ due to $ u_x \to -\infty $, which supports the entropy condition and enables the uniqueness proof.
• The method avoids reliance on classical entropy inequalities by using contour integration and level-line analysis, providing a new route to uniqueness in degenerate settings.