Publication

Sharp nonuniqueness in the transport equation with Sobolev velocity field

2026

Abstract

Given a divergence-free vector field u in L1_t W1,p_x(R^d) and a non‑negative initial datum rho_0 in L^r, the DiPerna–Lions theory establishes uniqueness of the weak solution in the class of L1_t L^r_x densities when the condition 1/p + 1/r is less than or equal to 1. This range was later improved by Brue, Colombo, and De Lellis (2021) to the condition 1/p + (d minus 1) divided by (d times r) is less than or equal to 1. We prove that this improved range is sharp by providing a counterexample to uniqueness when 1/p + (d minus 1) divided by (d times r) is greater than 1. To achieve this, we introduce a new flow mechanism. It is not based on convex integration, which has produced non‑optimal results in this context, nor on purely self‑similar techniques, but it shares features of both. In particular, it has a locally discrete self‑similar structure and an intermittent localization in space and frequency.

Official source

https://infoscience.epfl.ch/entities/publication/78ecc661-31d8-4cde-8931-6ac7bbf0a3f7

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/78ecc661-31d8-4cde-8931-6ac7bbf0a3f7

About this result

Related concepts (32)

Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz.

Solenoidal vector field

In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: A common way of expressing this property is to say that the field has no sources or sinks. The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero: where is the outward normal to each surface element.

Divergence

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region.

Related publications (1)

Sharp nonuniqueness in the transport equation with Sobolev velocity field

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