arXiv subjects

Tarek Zöchling

Publications and source records attributed to Tarek Zöchling.

10 recordsLinked to original sources

arXiv2026

Moisture Dynamics with Phase Changes in a Compressible Hydrostatic Atmosphere

In this article, a rigorous well-posedness result is established for a moist compressible primitive-equation system with phase changes. The model couples the compressible primitive equations to a bulk microphysics model for vapor water, cloud water, and rain water, including evaporation, condensation, auto-conversion, collection, and sedimentation effects. Global strong well-posedness is proved for initial data sufficiently close to constant equilibria.

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

Noise-Driven Free Boundaries In The Compressible Navier-Stokes Equations

A stochastic free-boundary problem for the three-dimensional barotropic compressible Navier--Stokes equations is studied. The main feature of the model is that the free boundary is transported by a Stratonovich stochastic flow, so that the noise enters the kinematic boundary condition and hence the evolution of the moving domain. An additional Itô forcing in the momentum equation is also allowed. The problem is transformed by a stochastic Lagrangian map generated by the velocity and the transport vector fields. In these coordinates the density is represented through the Jacobian of the flow, and the remaining system is solved by combining stochastic maximal regularity, deterministic %\rL^p%-%\rL^q$ estimates, and a localized contraction argument. Local pathwise well-posedness is obtained up to an a.s. positive stopping time, with strictly positive density and pathwise uniqueness.

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

Global Strong Well-posedness of the heat-conducting, compressible primitive Equations

The full heat-conducting compressible primitive equations are considered, extending the compressible primitive-equation framework by coupling the temperature through the ideal gas law and the thermal energy balance in the presence of gravity. Global strong well-posedness is established for small perturbations of an equilibrium state, thereby providing a result beyond the isothermal regime. The proof relies on a structural representation of the density in terms of the temperature and on a new Lagrangian transformation.

math.AP↗

arXiv2025

Moisture dynamics with phase changes coupled to heat-conducting, compressible fluids

It is shown that a model coupling the heat-conducting compressible Navier-Stokes equations to a micro-physics model of moisture in air is locally strongly well-posed for large data in suitable function spaces and strongly well-posed on $[0,τ]$ for every $τ> 0$ for small initial data. This seems to be the first result on $[0,τ]$ for arbitrary $τ> 0$ for a model coupling moisture dynamics to heat-conducting, compressible Navier-Stokes equations. A key feature of the micro-physics model is that it also includes phase changes of water in moist air. These phase changes are associated with large amounts of latent heat and thus result in a strong coupling to the thermodynamic equation. The well-posedness results are obtained by means of a Lagrangian approach, which allows to treat the hyperbolicity in the continuity equation. More precisely, optimal $\mathrm{L}^p$-$\mathrm{L}^q$ estimates are shown for the linearized system, leading to the local well-posedness result by a fixed point argument and suitable nonlinear estimates. For the well-posedness result on $[0,τ]$ for arbitrary $τ> 0$, a refined analysis of the linearized problem close to equilibria is carried out, and the roughness of the source term, induced by the phase changes, requires to establish delicate a priori bounds.

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

Dynamic boundary conditions with noise for energy balance models coupled to geophysical flows

This article investigates an energy balance model coupled to the primitive equations by a dynamic boundary condition with and without noise on the boundary. It is shown that this system is globally strongly well-posed both in the deterministic setting for arbitrary large data in $W^{2(1-\frac{1}{p}),p}$ for $p \in [2,\infty)$ and in the stochastic setting for arbitrary large data in $H^1$.

math.AP↗

arXiv2025

Global strong well-posedness of the CAO-problem introduced by Lions, Temam and Wang

Consider the CAO-problem introduced by Lions, Temam and Wang, which concerns a system of two fluids described by two primitive equations coupled by fully nonlinear interface conditions. They proved in their pioneering work the existence of a weak solution to the CAO-system; its uniqueness remained an open problem. In this article, it is shown that this coupled CAO-system is globally strongly well-posed for large data, even in critical Besov spaces. It is furthermore shown that, away from the boundary, the solution is even real analytic. The approach presented relies on an optimal data result for the boundary terms in the linearized system in terms of time-space Triebel-Lizorkin spaces. Boundary terms are then controlled by paraproduct methods in these spaces.

math.AP↗

arXiv2025

The Lagrangian approach to the compressible primitive equations

This article develops the hydrostatic Lagrangian approach to the compressible primitive equations. A fundamental aspect in the analysis is the investigation of the compressible hydrostatic Lamé and Stokes operators. Local strong well-posedness for large data and global strong well-posedness for small data are established under various assumptions on the pressure law, both in the presence and absence of gravity.

math.AP↗