Calculus of Variations and Geometric Measure Theory
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S. Modena - G. Sattig

Convex integration solutions to the transport equation with full dimensional concentration

created by modena on 27 Feb 2019

[BibTeX]

preprint

Inserted: 27 feb 2019

Year: 2019

ArXiv: 1902.08521 PDF

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 $p'<d$.

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