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Version: publishedVersion × Subject: Navier--Stokes ×

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Now showing 1 - 4 of 4
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    On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Lasarzik, Robert
    We define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. This concept is shown to be equivalent to weak solutions with energy conservation. Via a standard Galerkin discretization, we prove the existence of energy-variational solutions and thus weak solutions in any space dimension for the Navier--Stokes equations. In the limit of vanishing viscosity the same assertions are deduced for the incompressible Euler system. Via the selection criterion of maximal dissipation we deduce well-posedness for these equations.
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    Maximal dissipative solutions for incompressible fluid dynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lasarzik, Robert
    We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages.
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    On the spatially asymptotic structure of time-periodic solutions to the Navier--Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Eiter, Thomas
    The asymptotic behavior of weak time-periodic solutions to the Navier--Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.
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    Analysis of a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Lasarzik, Robert
    In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak-strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.