arXiv subjects

Tommaso Cortopassi

Publications and source records attributed to Tommaso Cortopassi.

6 recordsLinked to original sources

arXiv2025

A priori error estimates for the $\theta$-method for the flow of nonsmooth velocity fields

Velocity fields with low regularity (below the Lipschitz threshold) naturally arise in many models from mathematical physics, such as the inhomogeneous incompressible Navier-Stokes equations, and play a fundamental role in the analysis of nonlinear PDEs. The DiPerna-Lions theory ensures existence and uniqueness of the flow associated with a divergence-free velocity field with Sobolev regularity. In this paper, we establish a priori error estimates showing a logarithmic rate of convergence of numerical solutions, constructed via the $\theta$-method, towards the exact (analytic) flow for a velocity field with Sobolev regularity. In addition, we derive analogous a priori error estimates for Lagrangian solutions of the associated transport equation, exhibiting the same logarithmic rate of convergence. Our theoretical results are supported by numerical experiments, which confirm the predicted logarithmic behavior.

math.AP↗

arXiv2024

A generalised Nehari manifold method for a class of non linear Schr\"odinger systems in $\mathbb{R}^3$

We study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schr\"odinger equations (NLSEs), that is $-\epsilon^2 \Delta u + V(x) u= h_v(u,v), - \epsilon^2 \Delta v + V(x) v=h_u (u,v)$. Under certain hypotheses on the potential $V$ and the non linearity $h$, we manage to prove that there exists a solution $(u_\epsilon,v_\epsilon)$ that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as $\epsilon \to 0$. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more general non linearity and we restrict ourselves to the case where the domain is $\mathbb{R}^3$.

math.AP↗

arXiv2024

A current based approach for the uniqueness of the continuity equation

We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u \in [L^1 (0,T; W^{1,p}(\mathbb{T}^d)) \cap L^\infty ((0,T) \times \mathbb{T}^d)]^d$ with $\operatorname{div}(u) ^- \in L^1 (0,T; L^\infty (\mathbb{T}^d))$ and an initial datum $\rho_0 \in L^q (\mathbb{T}^d)$, where $\mathbb{T}^d$ is the $d$-dimensional torus and $ 1 \leq p,q \leq +\infty$ such that $1/p + 1/q =1$ without using the theory of renormalized solutions. We propose a more geometric approach which will however still rely on a strong $L^1$ estimate on the commutator (which is the key technical tool when using renormalized solutions, too), but other than that will be based on the theory of currents.

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arXiv2024

An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non cartesian grids

We prove a novel stability estimate in $L^\infty _t (L^p _x)$ between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, one probabilistic and one deterministic, using different approximations for both the flow and the initial datum. Such estimates for the solution of the continuity equation are derived on non Cartesian grids and without the need to assume a CFL condition.

math.AP↗

arXiv2023

A note on a $L^p$ stability estimate for regular Lagrangian flows

In this note, we prove a $L^p$ version of the well known stability estimate for regular Lagrangian flows derived by Gianluca Crippa and Camillo De Lellis in \cite{crippa2008estimates}. As far as we know, the only estimate of this kind readily available in the literature is for the case $p=1$. With minor modifications to the proof in \cite{crippa2008estimates}, we show that an analogous estimate holds in $L^p$ norm with $p \in (1, + \infty)$.

math.AP↗

arXiv2023

Existence, uniqueness and $L^2_t (H_x ^2) \cap L^\infty_t (H^1_x) \cap H^1_t (L^2_x) $ regularity of the gradient flow of the Ambrosio-Tortorelli functional

We consider the gradient flow of the Ambrosio-Tortorelli functional at fixed $\epsilon>0$, proving existence, uniqueness and $L^2 _t (H_x ^2) \cap L^\infty _t (H^1 _x) \cap H^1 _t (L^2 _x) $ regularity in dimension 2. In particular we improve a previous result where such regularity was known only up to a finite number of space time points, which diverged as $\epsilon \to 0$. By employing a different technique for the crucial $L^2 _t (H^2 _x)$ estimates we can see how in fact the desired regularity holds everywhere.

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