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Sarka Necasova

Publications and source records attributed to Sarka Necasova.

At least 19 recordsLinked to original sources
arXiv2026

On the Vanishing Viscosity and Magnetic Diffusion Limit for Incompressible Magnetohydrodynamics

The article focuses on the inviscid and non-resistive limit of the incompressible magnetohydrodynamics (MHD) equations in a bounded domain. We derive sufficient conditions for the limit of Leray-Hopf solutions of the MHD system, which converges to the weak solutions of the ideal MHD system when $μ,ν\rightarrow0$, under the assumptions of uniform interior regularity and uniform equi-continuity at the boundary. Therefore, we establish an analogue of the Kato-type criterion to show that the energy dissipation rate is independent of the coefficients of viscosity and resistivity. Moreover, to analyze the different decay rates for the viscosity and resistivity, we obtain the limits of infinite and vanishing magnetic Prandtl number ($Pm=\fracμν$) for the incompressible MHD system. To the best of the authors' knowledge, this is the first rigorous result from a mathematical viewpoint on the limit of the magnetic Prandtl number.

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arXiv2025

Steady Self-Propelled Motion of a Rigid Body in a Viscous Fluid with Navier-Slip Boundary Conditions

We investigate the steady self-propelled motion of a rigid body immersed in a three-dimensional incompressible viscous fluid governed by the Navier-Stokes equations. The analysis is performed in a body-fixed reference frame, so that the fluid occupies an exterior domain and the propulsion mechanism is modeled through nonhomogeneous Navier-slip boundary conditions at the fluid-body interface. Such conditions provide a realistic description of propulsion in microfluidic and rough-surface regimes, where partial slip effects are significant. Under suitable smallness assumptions on the boundary flux and on the normal component of the prescribed surface velocity, we establish the existence of weak steady solutions to the coupled fluid-structure system. A key analytical ingredient is the derivation of a Korn-type inequality adapted to exterior domains with rigid-body motion and Navier-slip interfaces, which yields uniform control of both the fluid velocity and the translational and rotational velocities of the body. Beyond existence, we provide a necessary and sufficient condition under which a prescribed slip velocity on the body surface induces nontrivial translational or rotational motion of the rigid body. This is achieved through the introduction of a finite-dimensional thrust space, defined via auxiliary exterior Stokes problems with Navier boundary conditions, which captures the effective contribution of boundary-driven flows to the rigid-body motion. Our results clarify how boundary effects generate propulsion and extend the classical Dirichlet-based theory to the Navier-slip setting.

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arXiv2025

A multifluid model with chemically reacting components -- construction of weak solutions

We investigate the existence of weak solutions to a multi-component system, consisting of compressible chemically reacting components, coupled with the compressible Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reactions and assume a nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum. It is shown that there exists a global weak solution, satisfying the $L^\infty$ bounds for all the components. We obtain strong compactness of the sequence of densities in $L^p$ spaces, under the assumption that all components are strictly positive. The applied method captures the properties of models of high generality, which admit an arbitrary number of components. Furthermore, the framework that we develop can handle models that contain both diffusing and non-diffusing elements.

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arXiv2024

Energy equality for the compressible Primitive Equations with vacuum

The paper deals with the problem of the energy conservation for the weak solutions to the compressible Primitive Equations (CPE) system with degenerate viscosity. The sufficient conditions on the regularity of weak solutions for the energy equality are obtained even for the case when the solutions may include vacuum. In this paper, we show two theorems, the first one gives regularity in the classical isotropic Sobolev and Besov spaces. The second one states regularity in the anisotropic spaces. We obtain new regularity results in the second theorem due to the special structure of CPE system, which are in contrast to compressible Navier-Stokes equations.

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arXiv2024

Dissipative solutions to the model of a general compressible viscous fluid with the Coulomb friction law boundary condition

We study a model of a general compressible viscous fluid subject to the Coulomb friction law boundary condition. For this model, we introduce a dissipative formulation and prove the existence of dissipative solutions. The proof of this result consists of a three-level approximation method: A Galerkin approximation, the classical parabolic regularization of the continuity equation as well as convex regularizations of the potential generating the viscous stress and the boundary terms incorporating the Coulomb friction law into the dissipative formulation. This approach combines the techniques already known from the proof of the existence of dissipative solutions to a model of general compressible viscous fluids under inflow-outflow boundary conditions as well as the proof of the existence of a weak solution to the incompressible Navier-Stokes equations under the Coulomb friction law boundary condition. It is the first time that this type of boundary condition is considered in the case of compressible flow.

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arXiv2022

Singular limit for the compressible Navier--Stokes equations with the hard sphere pressure law on expanding domains

The article is devoted to the asymptotic limit of the compressible Navier-Stokes system with a pressure obeying a hard--sphere equation of state on a domain expanding to the whole physical space $R^3$. Under the assumptions that acoustic waves generated in the case of ill-prepared data do not reach the boundary of the expanding domain in the given time interval and a certain relation between the Reynolds and Mach numbers and the radius of the expanding domain, we prove that the target system is the incompressible Euler system on $R^3$. We also provide an estimate of the rate of convergence expressed in terms of characteristic numbers and the radius of domains.

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arXiv2021

Lp-strong solution to fluid-rigid body interaction system with Navier slip boundary condition

We study a fluid-structure interaction problem describing movement of a rigid body inside a bounded domain filled by a viscous fluid. The fluid is modelled by the generalized incompressible Naiver-Stokes equations which include cases of Newtonian and non-Newtonian fluids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and balance of forces at the fluid-rigid body interface. Our analysis also includes the case of the nonlinear slip condition. The main results assert the existence of strong solutions, in an $L^p-L^q$ setting, globally in time, for small data in the Newtonian case, while existence of strong solutions in $L^p$-spaces, locally in time, is obtained for non-Newtonian case. The proof for the Newtonian fluid essentially uses the maximal regularity property of the associated linear system which is obtained by proving the $\mathcal{R}$-sectoriality of the corresponding operator. The existence and regularity result for the general non-Newtonian fluid-solid system then relies upon the previous case. Moreover, we also prove the exponential stability of the system in the Newtonian case.

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arXiv2021

Homogenization problems for the compressible Navier-Stokes system in 2D perforated domains

In this paper, we study the homogenization problems for the stationary compressible Navier-Stokes system in a bounded 2D domain, where the domain is perforated with very tiny holes (or obstacles) whose diameters are much smaller than their mutual distances. We obtain that the process of homogenization doesn't change the motion of the fluids. From another point of view, we obtain the same system of equations in the asymptotic limit. It is the first result of homogenization problem in the compressible case in 2 dimensions.

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arXiv2021

Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids with general boundary data

We prove global existence of weak solutions for a version of one velocity Baer-Nunziato system with dissipation describing a mixture of two non interacting viscous compressible fluids in a piecewise regular Lipschitz domain with general inflow/outfow boundary conditions. The geometrical setting is general enough to comply with most current domains important for applications as, for example, (curved) pipes of picewise regular and axis-dependent cross sections. As far as the existence proof is concerned, we adapt to the system the nowaday's classical Lions-Feireisl approach to the compressible Navier-Stokes equations which is combined with a generalization of the theory of renormalized solutions to the transport equations in the spirit of Vasseur-Wen-Yu. The results related to the families of transport equations presented in this paper extend/improve some of statements of the theory of renormalized solutions, and they are therefore of independent interest.

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arXiv2021

Existence and stability of dissipative turbulent solutions to a simple bi-fluid model of compressible fluids

Following Abbatiello et al. [ DCCDS-A (41), 2020], we introduce dissipative turbulent solutions to a simple model of a mixture of two non interacting compressible fluids {\tc filling a bounded domain with general non zero inflow/outflow boundary conditions.} We prove existence of such solutions for all adiabatic coefficients $γ>1$, their compatibility with classical solutions, the relative energy inequality, and the weak strong uniqueness principle in this class. The class of dissipative turbulent solutions is so far the largest class of generalized solutions which still enjoys the weak strong uniqueness property.

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arXiv2021

On the hydrostatic approximation of compressible anisotropic Navier-Stokes equations -- rigorous justification

In this work, we obtain the hydrostatic approximation by taking the small aspect ratio limit to the Navier-Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraint in general large scale geophysical motions that the vertical scale is significantly smaller than horizontal. We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier-Stokes equations to the compressible Primitive Equations. This is the first work to use relative entropy inequality for proving hydrostatic approximation and derive the compressible Primitive Equations.

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arXiv2020

On a singular limit for the compressible rotating Euler system

The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated.

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arXiv2020

On weak-strong uniqueness and singular limit for the compressible Primitive Equations

This paper addresses the weak-strong uniqueness property and singular limit for the compressible Primitive Equations (PE). We show that a weak solution coincides with the strong solution emanating from the same initial data. On the other hand, we prove compressible PE will approach the incompressible inviscid PE equations in the regime of low Mach number and large Reynolds number in the case of well-prepared initial data. To the best of the authors' knowledge, this is the first work to bridge the link between the compressible PE with incompressible inviscid PE.

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arXiv2020

Low Mach and thin domain limit for the compressible Euler system

We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer $Ω_δ=(0,δ)\times\mathbb{R}^2, \ \ δ>0$. In the framework of dissipative measure-valued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and $δ\rightarrow0$.

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arXiv2019

A Pressure Associated with a Weak Solution to the Navier-Stokes Equations with Navier's Boundary Condition

We show that if u is a weak solution to the Navier-Stokes initial-boundary value problem with Navier's slip boundary conditions in $Q_T:=Ω\times(0,T)$, where $Ω$ is a domain in $R^3$, then an associated pressure $p$ exists as a distribution with a certain structure. Furthermore, we also show that if $Ω$ is a "smooth" domain in $R^3$ then the pressure is represented by a function in $Q_T$ with a certain rate of integrability. Finally, we study the regularity of the pressure in sub-domains of $Q_T$, where $u$ satisfies Serrin's integrability conditions.

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arXiv2019

Self-propelled motion of a rigid body inside a density dependent incompressible fluid

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole $\mathbb{R}^3$. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.

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