Source material

• Initial research request
• Research instructions
• Audit instructions

@mathematical_research_goal (1).md @mathematical_audit_prompt_v3 (1).md I am attaching `mathematical_research_goal.md`.    Read that file completely and treat it as the governing research procedure.  The authoritative mathematical problem is the complete LaTeX source between  the two TARGET_PROBLEM markers below.    PROJECT_WINDOW_BUDGET: 6 hours    \<TARGET_PROBLEM_BEGIN> \paragraph{Problem (global existence for general vortex-sheet data).}\
Let $\mathcal M(\mathbb R^2)$ denote the space of finite signed\
Radon measures, equipped with the total variation norm, and set\
\\[\
&#x20;   K(x)=\frac{(-x_2,x_1)}{2\pi|x|^2}.\
\\]\
For every\
\\[\
&#x20;   \omega_0\in\mathcal M(\mathbb R^2)\cap H^{-1}(\mathbb R^2),\
&#x20;   \qquad u_0=K\*\omega_0,\
\\]\
with no sign restriction on $\omega_0$, does there exist a global\
weak solution of the incompressible Euler equations\
\\[\
&#x20;   u\in C_w([0,\infty);\
&#x20;            L^2\_{\mathrm{loc}}(\mathbb R^2;\mathbb R^2))\
\\]\
with $u(0)=u_0$ and $\operatorname{div}u=0$, whose vorticity\
$\omega(t):=\partial_1u_2(t)-\partial_2u_1(t)$ satisfies\
\\[\
\begin{gathered}\
&#x20;   \omega(t)\in\mathcal M(\mathbb R^2),\
&#x20;   \qquad u(t)=K\*\omega(t)\
&#x20;   \qquad\text{for every }t\geq0,\\\\\
&#x20;   \sup\_{0\leq t\leq T}\\|\omega(t)\\|\_{\mathcal M}<\infty\
&#x20;   \qquad\text{for every }T>0?\
\end{gathered}\
\\]\
Here the Euler equations are required in the sense that\
\\[\
&#x20;   \int_0^T\\!\\!\int\_{\mathbb R^2}\
&#x20;   \bigl(u\cdot\partial_t\varphi\
&#x20;         +(u\otimes u):\nabla\varphi\bigr)\\,dx\\,dt\
&#x20;   +\int\_{\mathbb R^2}u_0\cdot\varphi(0,\cdot)\\,dx=0\
\\]\
for every $T>0$ and every divergence-free\
$\varphi\in C_c^\infty([0,T)\times\mathbb R^2;\mathbb R^2)$.\
The notation $C_w$ means weak continuity in $L^2$ on every\
bounded spatial subdomain. \<TARGET_PROBLEM_END>    This is a bounded mathematical research project, not a request for an  immediate literature-status answer.    Begin the procedure in `mathematical_research_goal.md` now. As the first  persistence action, create and maintain `checkpoint.md`. Copy the exact marked  problem into that checkpoint before beginning heavy route work.    Do not terminate except under a stopping condition permitted by  `mathematical_research_goal.md`.

The following requests introduced the vanishing-viscosity problem on 8 October 2026. They are reproduced verbatim from the shared conversation.

Research request

But this positive criterion I think is too strong to be verified in general.

Let us reverse the question: Can you prove the existence of \omega_0 \in L^1 \cal L^p with zero average so that the vanishing viscosity limit does not converge along the full sequence \nu \downarrow 0, but has different accumulation points?

Follow-up request

I like the strategy



1. The active cascade survives diffusion, uniformly in its depth.
2. Smaller scales dissipate before transmitting a significant unstable-mode impulse.
3. Older seeds do not override the selected orientation



Pursue this strategy to construct an example of non-unique vanishing viscosity limit