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’
Convex integration solutions to the transport equation with full dimensional concentration
Modena, Stefano
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2020-01-01
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’File in questo prodotto:
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