Convex integration solutions to the transport equation with full dimensional concentration

Abstract

We construct infinitely many incompressible Sobolev vector fields $u\in C_t{W}_x^{1,\tilde{p}}$ on the periodic domain ${\mathbb{T}}^d$ for which uniqueness of solutions to the transport equation fails in the class of densities $\rho \in C_t{L}_x^p$, provided $1/p+1/\tilde{p}>1+1/d$. The same result applies to the transport-diffusion equation, if, in addition $p^{\prime }<d$.

33 pages

References in corpus (4)

• Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations
• A quantitative theory for the continuity equation
• Stationary and discontinuous weak solutions of the Navier-Stokes equations
• Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation

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