The Role of Pressure in the Theory of Weak Solutions to the Navier-Stokes Equations

Neustupa, Jiřı́

doi:10.1007/978-3-030-39639-8_4

Cited by 4 publications

(5 citation statements)

References 66 publications

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“…It is easy to show that F ∈ L 1 (0, T ; W −1,2 0 (Ω F )). Therefore, Theorem 3 from [28] implies that there exist functions p 0 and p 1 such that [28] Section 7.3.2.C and 7.3.2.E). Therefore, by density, the equation above is true for all ψ ∈ H 1 0 ((0, T ) × Ω F ).…”

Section: Now We Definementioning

confidence: 94%

A uniqueness result for 3D incompressible fluid-rigid body interaction problem

Muha¹,

Nečasová²,

Ana³

2019

Preprint

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We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations called Euler equations for the rigid body. The equations are fully coupled via dynamical and kinematic coupling conditions. We consider two different kinds of kinematic coupling conditions: no-slip and slip. In both cases we prove a generalization of the well-known weak-strong uniqueness result for the Navier-Stokes equations to the fluid-rigid body system. More precisely, we prove that weak solutions that additionally satisfy Prodi-Serrin L r − L s condition are unique in the class of Leray-Hopf weak solutions. * The research of B.M. leading to these results has been supported by Croatian Science Foundation under the project IP-2018-01-3706† The research of Š.N. leading to these results has received funding from the Czech Sciences Foundation (GA ČR), 19-04243S and RVO 67985840.‡ The research of A.R. leading to these results has been supported by Croatian Science Foundation under the project IP-2018-01-3706, and by the Czech Sciences Foundation (GA ČR), 19-04243S and RVO 67985840.

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Section: Now We Definementioning

confidence: 94%

A uniqueness result for 3D incompressible fluid-rigid body interaction problem

Muha¹,

Nečasová²,

Ana³

2019

Preprint

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“…The distribution is regular (i.e. it can be identified with a function with some rate of integrability in Q T ) if domain Ω is "smooth", see [31], [13] and [22]. In section 4 of this paper, we show that one can naturally assign a pressure, as a distribution, to a weak solution to the Navier-Stokes equations with Navier's boundary conditions (1.3), too.…”

Section: Introductionmentioning

confidence: 85%

“…Furthermore, identifying functions from W 1,q τ (Ω) n with their restrictions to Ω n , we can also consider F to be an element of W −1,q ′ 0 (Ω n ), vanishing on W 1,q 0,σ (Ω n ). Thus, due to Lemma 1.4 in [22], there exists c(n) > 0 and a unique function…”

Section: Introductionmentioning

confidence: 94%

“…b) If Ω is a bounded or exterior domain R 3 with the boundary of the class C 2+(h) for some h > 0 and u satisfies the no-slip boundary condition u = 0 on ∂Ω × (0, T ) then p and ∂ t u have all spatial derivatives (of all orders) in L q (t 1 + ǫ, t 2 − ǫ; L ∞ (Ω 2 )) for any q ∈ (1, 2), see [19], [18], [22] or [30].…”

Section: Application Ofmentioning

confidence: 99%

“…It is known from the theory of the Navier-Stokes equations with the no-slip boundary condition (1.5) that the pressure, associated with a weak solution, generally exists only as a distribution in Q T . (See [16], [34], [29], [11], [32], [35] and [22].) The distribution is regular (i.e.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The interior regularity of pressure associated with a weak solution to the Navier–Stokes equations with the Navier-type boundary conditions

Neustupa

Baba

2018

Journal of Mathematical Analysis and Applications

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On steady solutions to the MHD equations with inhomogeneous generalized impermeability boundary conditions for the magnetic field

Neustupa,

Perisetti,

Yang

2024

Math Methods in App Sciences

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The paper deals with the steady MHD equations for a viscous incompressible fluid in a bounded and generally multiply connected domain with the no‐slip boundary condition for the velocity and inhomogeneous generalized impermeability boundary conditions for the magnetic field . The main results concern an appropriate definition of the weak solution , its existence, continuous dependence on the data when the data tend to zero, and uniqueness in a “small” neighborhood of the zero solution .

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The Role of Pressure in the Theory of Weak Solutions to the Navier-Stokes Equations

Cited by 4 publications

References 66 publications

A uniqueness result for 3D incompressible fluid-rigid body interaction problem

A uniqueness result for 3D incompressible fluid-rigid body interaction problem

The interior regularity of pressure associated with a weak solution to the Navier–Stokes equations with the Navier-type boundary conditions

On steady solutions to the MHD equations with inhomogeneous generalized impermeability boundary conditions for the magnetic field

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