arXiv subjects

Filippo Palma

Publications and source records attributed to Filippo Palma.

7 recordsLinked to original sources

arXiv2026

An $L^p$-theory for global weak solutions to the Navier-Stokes equations in exterior domains

In this paper, we consider the Navier-Stokes initial boundary value problem in exterior domains with initial data in the Lebesgue space $L^p$, $p\in (2,3)$, and show the existence of a global weak solution. For the quoted solution, we also furnish a structure theorem. In particular, we see that the weak solution becomes regular after a certain instant of time and it is also regular a. e. in time. Although a general $L^p$-theory for local strong/mild solutions is well-established in the literature, a corresponding theory for global weak solutions in exterior domains seems to be new. Of course, our results hold in the particular cases of the Cauchy problem and the initial boundary value problem in a half-space.

math.AP↗

arXiv2026

Continuous Data Assimilation for Semilinear Parabolic Equations with Multiplicative Observation Noise

The problem of continuous data assimilation for semilinear parabolic equations based on partial observations corrupted by noise is investigated. The noise is allowed to be multiplicative, with additive noise arising as a special case. In a general Gelfand triple framework, an abstract theory for the nudging equation is developed that covers both weak and strong formulations. Mean square convergence of the assimilation error is proved under suitable assumptions, and, under additional integrability conditions on the noise, a uniform almost sure convergence result is established. Finally, the framework is applied to several PDE models, including the 2D Navier-Stokes, 2D magnetohydrodynamics, 2D quasi-geostrophic, and 1D Allen-Cahn equations.

math.AP↗

arXiv2026

Continuous Data Assimilation for Semilinear Parabolic Equations: A General Approach by Evolution Equations

This article develops a general framework for continuous deterministic data assimilation for semilinear parabolic equations by means of evolution equations. Introducing a nudged model driven by partial observations, the global well-posedness of the reference and the approximating systems is established under natural assumptions. In addition, it is shown that the approximating solution converges exponentially to the solution of the reference system, provided the observational resolution and the nudging parameter are suitably chosen. The approach allows us to consider many systems, such as the Allen-Cahn, Cahn-Hilliard, Sellers-type energy balance, and bidomain systems, for the first time.

math.AP↗

arXiv2026

On the interaction between a rigid-body and a viscous-fluid: existence of a weak solution and a suitable Th\'eor\`eme de Structure

In this paper, we prove the existence and a partial regularity of a weak solution to the system governing the interaction between a rigid body and a viscous incompressible Newtonian fluid. The evolution of the system body-fluid is studied in a frame attached to the body. The choice of this special frame becomes critical from an analytical point of view due to the presence of the term $\omega\times x\cdot\nabla u$ in the balance of momentum equation for the fluid. As a consequence, we are forced to look for a technique that is different from the ones usually employed both for the existence and for the partial regularity of a weak solution to the Navier-Stokes problem. Hence, we prove the existence of a weak solution in an original way and give a new proof of the celebrated Th\'eor\`eme de Structure due to Leray. However, the regularity obtained for our weak solution is only for large times, hence our result is weaker compared to the one obtained by Leray.

math-ph↗

arXiv2025

A new proof of the Th\'eor\`eme de Structure related to a weak solution to the Navier-Stokes equations

It is well known that a Leray's weak solution to the Navier-Stokes Cauchy problem enjoys a partial regularity which is known in the literature as the Th\'eor\`eme de Structure of a Leray's weak solution. As well, this result has been extended by some authors to the case of the IBVP. In this note, we achieve the Th\'eor\`eme de Structure by means of a new proof. Our proof is based on a priori estimates for a suitable approximating sequence. In this way our result covers a more general setting in the sense that, e.g., we can also include the case of the weak solutions furnished by Hopf for an IBVP in bounded domains without requiring an energy inequality in a strong form, but just employing a priori estimates on the Galerkin approximation.

math.AP↗

arXiv2025

Time-periodic solutions to an energy balance model coupled with an active fluid under arbitrary large forces

This article concerns time-periodic solutions to a two-dimensional Sellers-type energy balance model coupled to the three-dimensional primitive equations via a dynamic boundary condition. It is shown that the underlying equations admit at least one strong time-periodic solution, provided the forcing term is time-periodic. The forcing term does not need to satisfy a smallness condition and is allowed to be arbitrarily large.

math.AP↗

arXiv2025

The motion of a rigid body in a viscous fluid: new results for strong solutions, uniqueness and integrability properties

In this note, we show two results in the setting of Galdi-Silvestre strong solutions for the rigid body-viscous fluid interaction. The former, under an additional integrability assumption on the gradient of the initial data, proves that the time derivative of the solution belongs to $L^2(0,T;L^2({\Omega}))$. The latter, thanks to a further assumption only on one solution, proves that the uniqueness holds in the quoted setting. However, our extra assumption for the uniqueness is certainly verified under the integrability assumption on the gradient of the initial data. Hence, the set of solutions enjoying the uniqueness is not empty.

math.AP↗