Irregular Diffusions and Loss of Regularity in Polyconvex Gradient Flows

TL;DR

The work reveals intrinsic irregularity in diffusion-type PDEs, including polyconvex gradient flows, by developing a tailored convex integration framework based on $\mathcal{T}_N$-configurations and the structural Condition $O_N$. The authors implement a staged construction that perturbs weak solutions along carefully chosen wave-cone directions to produce Lipschitz solutions that are nowhere $C^1$, both for general IBVPs and for gradient flows of strongly polyconvex energies. A key contribution is the introduction of $\mathcal{T}_N$-configurations, a simplified perturbation scheme, and nondegeneracy conditions ensuring the embedding of configurations into the diffusion graph while preserving polyconvex structure. The results demonstrate nonuniqueness and loss of regularity even in the polyconvex gradient-flow setting, highlighting fundamental limits of regularity theory for nonconvex diffusion models with first-order structure.

Abstract

We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial differential relations and adapting the method of convex integration, we develop a construction scheme based on new geometric structures, referred to as $\mathcal{T}_N$-configurations, together with a simplified structural hypothesis on the diffusion functions, termed Condition $O_N$. Under this condition, we show that the associated initial and boundary value problems with certain smooth initial-boundary data admit infinitely many Lipschitz weak solutions that are nowhere $C^1$. We further analyze specific $\mathcal{T}_N$-configurations and establish nondegeneracy conditions that are essential for verifying Condition $O_N$. As an application, we construct examples of strongly polyconvex energy functionals whose gradient flows generate irregular diffusion equations, thereby revealing a failure of regularity and uniqueness even within the class of polyconvex gradient flows.

Figures (2)

• Figure 1: Illustration of a $\mathcal{T}_N$-configuration $(\xi_1,\xi_2,\dots ,\xi_N)$ determined by $\pi_1=\pi_{N+1}=\rho$ and $\gamma_i=\pi_{i+1}-\pi_i\in \Gamma$$(1\le i\le N).$ The set $\mathcal{T}(\xi_1,\xi_2,\dots ,\xi_N)$ is the union of all the closed line segments shown.
• Figure 2: An illustration for Theorem \ref{['lem1']} (shown here for the case $N=5$). The point $Y$ lies on the (ultra-thick) solid line segment $[\zeta_3(\lambda',\rho), \, \pi_3(\rho')].$ The sets $\mathcal{T}(X_1,\dots,X_N)$ and $\mathcal{T}(X'_1,\dots,X'_N)$ are depicted by the solid line segments, respectively. The union of dashed line segments represents the reference set $\mathcal{T}(X^0_1,\dots,X^0_N),$ where $X^0_j=\zeta_j(\mu, 0)$ for $1\le j\le N$ (not shown). The points $X^0_j,$$X_j$ and $X'_j$ lie in the open set $S^r_j(\mu)$ for each $1\le j \le N,$ and all line segments shown lie in the open set $\Sigma^r(\mu).$ However, only the points $X_j$ and $X'_j$ ($1\le j\le N$) and the solid line segments are used in the proof.

Theorems & Definitions (56)

• Theorem 2.1
• Remark 2.1
• Theorem 2.2
• Corollary 2.3
• proof
• Lemma 3.1
• proof
• Definition 3.1: $\mathcal{T}_N$-configuration
• Remark 3.2
• Lemma 3.2