Analysis of a thermodynamically consistent Navier–Stokes–Cahn–Hilliard model

Lasarzik, Robert

doi:10.1016/j.na.2021.112526

Cited by 7 publications

(7 citation statements)

References 30 publications

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“…where it was used that P n is an orthogonal projection. This together with (21) imply that the consistency error vanishes, i.e.,…”

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confidence: 84%

“…Hence, this gives hope that the new approach may allows to prove the existence of energy-variational and thus weak solutions to some PDE systems, where this seems to be out of reach with other available techniques. This includes multidimensional conservation laws [3], liquid crystals [18], heat-conducting complex fluids [21], or GENERIC systems in general (see [14] and [20]).…”

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confidence: 99%

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On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics

Lasarzik

2021

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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.The Navier-Stokes and Euler equations are the standard models for incompressible fluid dynamics. Both are a recurrent tools in computational fluid dynamics for weather forecast, micro fluidic devices [26] or industrial processes like steel production [1]. There exists a vast literature concerning the Navier-Stokes and Euler equations. In case of the Navier-Stokes equation, we only mention here the existence proof for weak solutions in three dimension by Leray [22] and the weak-strong uniqueness result due to Serrin [25]. In the context of the Euler equations, the existence of weak solutions in any space dimension is already known (see [9]) also fulfilling the energy inequality (see [10]). This result was proven via the convex integration technique. This technique grants the existence of infinitely many and also non-physical weak solutions. Additionally, it was proven for the Navier-Stokes equations via similar techniques that there exist infinitely many weak solutions that do not fulfill the energy

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“…where it was used that P n is an orthogonal projection. This together with (21) imply that the consistency error vanishes, i.e.,…”

mentioning

confidence: 84%

mentioning

confidence: 99%

On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics

Lasarzik

2021

Self Cite

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“…In Refs. [19][20][21][22], the Allen-Cahn equation describing the phase-field process has been investigated applying the relative energy, see also Giesselmann et al [23] for further reading. Application of this technique to viscoelastic equations can be found in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…In Refs. [19–22], the Allen–Cahn equation describing the phase‐field process has been investigated applying the relative energy, see also Giesselmann et al. [23] for further reading.…”

Section: Introductionmentioning

confidence: 99%

Relative energy and weak–strong uniqueness of a two‐phase viscoelastic phase separation model

Brunk

Lukáčová-Medviďová

2022

Z Angew Math Mech

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The aim of this paper is to analyze a viscoelastic phase separation model. We derive a suitable notion of the relative energy taking into account the nonconvex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak–strong uniqueness principle. We will provide the estimates for the full model in two space dimensions. For a reduced model, we present the estimates in three space dimensions and derive conditional relative energy estimates.

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“…Due to the phase separation behavior that we study in the present paper we need a new approach taking into account a non-convex energy. In [12,21,24,22] the Allen-Cahn equation describing the phase-field process has been investigated applying the relative energy, see also [16] for further reading.…”

Section: Introductionmentioning

confidence: 99%

Relative energy and weak-strong uniqueness of the two-phase viscoelastic phase separation model

Brunk¹,

Lukáčová-Medviďová²

2021

Preprint

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The aim of this paper is to analyze a viscoelastic phase separation model. We derive a suitable notion of the relative energy taking into account the non-convex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak-strong uniqueness principle. We will provide the estimates for the full model in two space dimensions. For a reduced model we present the estimates in three space dimensions and derive conditional relative energy estimates.Application of this technique to viscoelastic equations can be found in [26,4] . Maximal dissipative solutions applying the weakstrong uniqueness were analysed in [23] .The aim of this paper is to develop a relative energy concept for the viscoelastic phase separation model. We will further derive the weak-strong-uniqueness principle for the viscoelastic phase separation model. For the two-dimensional case the weak-strong uniqueness result is unconditional. Furthermore, we state the corresponding results in three space dimensions requiring the existence of a global weak solution. Finally, by combination with the results of [4] we derive a conditional weak-strong uniqueness principle for the full model in three space dimensions.

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Analysis of a thermodynamically consistent Navier–Stokes–Cahn–Hilliard model

Cited by 7 publications

References 30 publications

On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics

On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics

Relative energy and weak–strong uniqueness of a two‐phase viscoelastic phase separation model

Relative energy and weak-strong uniqueness of the two-phase viscoelastic phase separation model

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