# On existence and uniqueness of solution for a hydrodynamic problem related to water artificial circulation in a lake

## TL;DR

This paper develops a mathematical model using modified Navier-Stokes equations to analyze water circulation in lakes, proving key theoretical results on the existence, uniqueness, and smoothness of solutions to aid in preventing eutrophication.

## Contribution

It introduces a novel, well-posed model for lake water circulation based on the Smagorinsky turbulence model, with rigorous analytical proofs of solution properties.

## Key findings

- Proved existence of solutions for the model.
- Established uniqueness of solutions.
- Demonstrated smoothness properties of solutions.

## Abstract

In this work we introduce a well-posed mathematical model for the processes involved in the artificial circulation of water, in order to avoid eutrophication phenomena, for instance, in a lake. This novel and general formulation is based on the modified Navier-Stokes equations following the Smagorinsky model of turbulence, and presenting a suitable nonhomogeneous Dirichlet boundary condition. For the analytical study of the problem, we prove several theoretical results related to existence, uniqueness and smoothness for the solution of this recirculation model.