On non-uniqueness for the system $\bu_t+(\bu\cdot\nabla)\bu=μΔ{\bf u}$

TL;DR

This work analyzes the Cauchy problem for the $n$-dimensional vector Burgers (Cole) system ${\bf u}_t+({\bf u}\cdot\nabla){\bf u}=\mu\Delta{\bf u}$ by exploiting the Cole-Hopf transform from radial heat solutions. It constructs explicit radial, irrotational solutions from the heat equation to demonstrate non-uniqueness in supercritical function spaces: $L^p$, $W^{1,p}$, and $W^{2,p}$ with $1\le p<n$, $p<\frac{n}{2}$, and $p<\frac{n}{3}$ respectively. The principal example yields solutions that are classical for $t>0$ and converge to vanishing initial data in $L^p$, yet blow up in $L^\infty$ as $t\to0^+$, and the paper also presents self-similar and stationary constructions illustrating broader non-uniqueness phenomena. The results illuminate ill-posedness in supercritical regimes and clarify that the observed non-uniqueness is not simply a reflection of the heat equation’s Tikhonov phenomenon. Overall, the paper broadens understanding of regularity thresholds for parabolic systems related to Navier–Stokes, highlighting gaps between classical theory and explicit, singular constructions.

Abstract

Explicit irrotational solutions, obtained via the Cole-Hopf transform from the multi-d heat equation, give examples of non-uniqueness for the Cauchy problem in supercritical $L^p$, $W^{1,p}$, and $W^{2,p}$ regimes. We verify non-uniqueness of the trivial solution in the sense of $L^p(\RR^n)$, whenever $n\geq2$ and $1\leq p<n$. The same solutions give non-uniqueness in $W^{1,p}(\RR^n)$ and $W^{2,p}(\RR^n)$ for $1\leq p<\frac{n}{2}$ and $1\leq p<\frac{n}{3}$, respectively. The main example provides solutions which are classical for strictly positive times, and vanish in the stated norms, but explode in $L^\infty(\RR^n)$, as $t\to0+$. The non-uniqueness is unrelated to the Tikhonov non-uniqueness phenomenon for the heat equation.

Figures (3)

• Figure 1: Maple plot of the solution (\ref{['soln_ex1']}) in the case $n=3$, with $\mu=0.1$ and $a=1$. The plot is for $0.0001\leq r\leq 0.1$ and $0.00002\leq t\leq 0.001$.
• Figure 2: Maple plot of the solution (\ref{['u_ss']}) in the case $n=3$, with $\mu=0.005$ and $a=1$. The plot is for $0.00005\leq r\leq 0.0007$ and $0\leq t\leq 0.00005$.
• Figure 3: Maple plot of the solution (\ref{['u_nst']}) to the 3-d Cole equation (\ref{['c_rad']}) with $n=3$ and $\mu=0.01$. The plot is for $0.001\leq r\leq 0.3$ and $0.001\leq t\leq 0.2$.

Theorems & Definitions (13)

• Remark 1.1: Non-unique Cole solutions and the Tikhonov phenomenon
• Remark 1.2: Singularity formation
• Definition 2.1: $L^p$-acceptable solutions
• Remark 2.1
• Lemma 2.1
• Remark 2.2
• proof
• Lemma 2.2
• proof
• Lemma 2.3