# Convex integration solutions to the transport equation with full dimensional concentration

## TL;DR

This paper constructs specific Sobolev vector fields on a periodic domain that demonstrate non-uniqueness of solutions to the transport and transport-diffusion equations under certain integrability conditions, highlighting limitations of solution uniqueness.

## Contribution

It introduces convex integration solutions that show non-uniqueness for the transport equation in Sobolev vector fields, extending understanding of solution behavior in these PDEs.

## Key findings

- Non-uniqueness of solutions in certain Sobolev spaces
- Construction of vector fields with full dimensional concentration
- Applicability to both transport and transport-diffusion equations

## Abstract

We construct infinitely many incompressible Sobolev vector fields $u \in C_t W^{1,\tilde p}_x$ on the periodic domain $\mathbb{T}^d$ for which uniqueness of solutions to the transport equation fails in the class of densities $\rho \in C_t L^p_x$, provided $1/p + 1/\tilde p > 1 + 1/d$. The same result applies to the transport-diffusion equation, if, in addition $p'<d$.