arXiv subjects

Luigi De Rosa

Publications and source records attributed to Luigi De Rosa.

26 records · Page 2Linked to original sources

arXiv2021

Dimension of the singular set of wild Hölder solutions of the incompressible Euler equations

For $β<\frac13$, we consider $C^β(\mathbb{T}^3\times [0,T])$ weak solutions of the incompressible Euler equations that do not conserve the kinetic energy. We prove that for such solutions the closed and non-empty set of singular times $\mathcal{B}$ satisfies $\dim_{\mathcal{H}}(\mathcal{B})\geq \frac{2β}{1-β}$. This lower bound on the Hausdorff dimension of the singular set in time is intrinsically linked to the Hölder regularity of the kinetic energy and we conjecture it to be sharp. As a first step in this direction, for every $β<β'<\frac{1}{3}$ we are able to construct, via a convex integration scheme, non-conservative $C^β(\mathbb{T}^3\times [0,T])$ weak solutions of the incompressible Euler system such that $\dim_{\mathcal{H}}(\mathcal{B})\leq \frac{1}{2}+\frac{1}{2}\frac{2β'}{1-β'}$. The structure of the wild solutions that we build allows moreover to deduce non-uniqueness of $C^β(\mathbb{T}^3\times [0,T])$ weak solutions of the Cauchy problem for Euler from every smooth initial datum.

math.AP↗

arXiv2021

Typicality results for weak solutions of the incompressible Navier--Stokes equations

In this work we show that, in the class of $L^\infty((0,T);L^2(\mathbb{T}^3))$ distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in some open interval of times are meagre in the sense of Baire category, and the Leray ones are a nowhere dense set.

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arXiv2020

Infinitely many Leray-Hopf solutions for the fractional Navier-Stokes equations

We prove the ill-posedness for the Leray-Hopf weak solutions of the incompressible and ipodissipative Navier-Stokes equations, when the power of the diffusive term $(-Δ)^γ$ is $γ< \frac{1}{3}$. We construct infinitely many solutions, starting from the same initial datum, which belong to $C_{x,t}^{\frac{1}{3}-}$ and strictly dissipate their energy in small time intervals. The proof exploits the "convex integration scheme" introduced by C. De Lellis and L. Székelyhidi for the incompressible Euler equations, joining these ideas with new stability estimates for a class of non-local advection-diffusion equations and a local (in time) well-posedness result for the fractional Navier-Stokes system. Moreover we show the existence of dissipative Hölder continuous solutions of Euler equations that can be obtained as a vanishing viscosity limit of Leray-Hopf weak solutions of a suitable fractional Navier-Stokes equations.

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arXiv2019

Regularity results for rough solutions of the incompressible Euler equations via interpolation methods

Given any solution $u$ of the Euler equations which is assumed to have some regularity in space - in terms of Besov norms, natural in this context - we show by interpolation methods that it enjoys a corresponding regularity in time and that the associated pressure $p$ is twice as regular as $u$. This generalizes a recent result by Isett [16] (see also Colombo and De Rosa [8]), which covers the case of Hölder spaces.

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arXiv2019

On the upper semicontinuity of a quasiconcave functional

In the recent paper \cite{SER}, the second author proved a divergence-quasiconcavity inequality for the following functional $ \mathbb{D}(A)=\int_{\mathbb{T}^n} det(A(x))^{\frac{1}{n-1}}\,dx$ defined on the space of $p$-summable positive definite matrices with zero divergence. We prove that this implies the weak upper semicontinuity of the functional $\mathbb{D}(\cdot)$ if and only if $p>\frac{n}{n-1}$.

math.AP↗

arXiv2019

On the Helicity conservation for the incompressible Euler equations

In this work we investigate the helicity regularity for weak solutions of the incompressible Euler equations. To prove regularity and conservation of the helicity we will threat the velocity $u$ and its $curl\, u$ as two independent functions and we mainly show that the helicity is a constant of motion assuming $u \in L^{2r}_t(C^θ_x)$ and $curl \,u \in L^κ_t(W^{α,1}_x)$ where $r,κ$ are conjugate Hölder exponents and $2θ+α\geq 1$. Using the same techniques we also show that the helicity has a suitable Hölder regularity even in the range where it is not necessarily constant.

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arXiv2018

Regularity in time of Hölder solutions of Euler and hypodissipative Navier-Stokes equations

In this work we investigate some regularization properties of the incompressible Euler equations and of the fractional Navier-Stokes equations where the dissipative term is given by $(-Δ)^α$, for a suitable power $α\in (0,\frac{1}{2})$ (the only meaningful range for this result). Assuming that the solution $u \in L^\infty _t(C^θ_x)$ for some $θ\in (0,1)$ we prove that $u \in C^θ_{t,x}$, the pressure $p\in C^{2θ-}_{t,x}$ and the kinetic energy $e \in C^{\frac{2θ}{1-θ}}_t$. This result was obtained for the Euler equations in [Is13] with completely different arguments and we believe that our proof, based on a regularization and a commutator estimate, gives a simpler insight on the result.

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arXiv2017

Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations

We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-Δ)^α$ with exponent $α< \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and László Székelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.

math.AP↗