Strong Solutions, Global Regularity, and Stability of a Hydrodynamic System Modeling Vesicle and Fluid Interactions

Wu, Hao; Xu, Xiang

doi:10.1137/11085952x

Cited by 16 publications

(29 citation statements)

References 40 publications

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“…When the phase function φ is considered, similar regularity criteria as (1.14) and (1.15) for the system (1.1)-(1.3) have been established in [25]. The first author of the present paper in [26] obtained that the Beale-Kato-Majda criterion (1.16) still holds for the system (1.1)-(1.5).…”

Section: Introductionsupporting

confidence: 54%

“…Recently, there have been many numerical and theoretical studies on the configurations and deformations of elastic vesicle membranes under external flow fields [3,4,6,7,8,9,17,19,24,25]. The single component vesicle membranes are possibly the simplest models for the biological cells and molecules and have widely studied in biology, biophysics and bioengineering.…”

Section: Introductionmentioning

confidence: 99%

“…Note that they have to restrict the working space with proper limited regularity due to some compatibility conditions at the boundary which is required in the fixed point strategy. Very recently, Wu and Xu [25] considered the system (1.1)-(1.3) with initial conditions (1.4) and periodic boundary conditions (1.5) to avoid troubles caused by the boundary terms when performing integration by parts. They proved that, for any given initial data (u 0 , φ 0 ) ∈ H 1 per (Q) × H 4 per (Q), there exists a positive time T such that the system (1.1)-(1.5) admits a unique smooth solution (u, φ) satisfying…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Logarithmically improved blow-up criteria for a phase field Navier–Stokes vesicle–fluid interaction model

Zhao

Liu

2013

Journal of Mathematical Analysis and Applications

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In this paper, we study a hydrodynamical system modeling the deformation of vesicle membrane under external incompressible viscous flow fields. The system is in the Eulerian formulation and is governed by the coupling of the incompressible Navier-Stokes equations with a phase field equation. In the three dimensional case, we establish two logarithmically improved blow-up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous Besov spaces.

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Section: Introductionsupporting

confidence: 54%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Logarithmically improved blow-up criteria for a phase field Navier–Stokes vesicle–fluid interaction model

Zhao

Liu

2013

Journal of Mathematical Analysis and Applications

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“…For applications of the Lojasiewicz-Simon gradient inequality to proofs of global existence, convergence, convergence rate, and stability of non-linear evolution equations arising in other areas of mathematical physics (including the Cahn-Hilliard, Ginzburg-Landau, Kirchoff-Carrier, porous medium, reaction-diffusion, and semi-linear heat and wave equations), we refer to the monograph by Huang [50] for a comprehensive introduction and to the articles by Chill [17,18], Chill and Fiorenza [19], Chill, Haraux, and Jendoubi [20], Chill and Jendoubi [21,22], Feireisl and Simondon [36], Feireisl and Takáč [37], Grasselli, Wu, and Zheng [40], Haraux [42], Haraux and Jendoubi [43,44,45], Haraux, Jendoubi, and Kavian [46], Huang and Takáč [51], Jendoubi [53], Rybka and Hoffmann [71,72], Simon [75], and Takáč [80]. For applications to fluid dynamics, see the articles by Feireisl, Laurençot, and Petzeltová [35], Frigeri, Grasselli, and Krejčí [38], Grasselli and Wu [39], and Wu and Xu [85].…”

mentioning

confidence: 99%

Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces

Feehan

Maridakis

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic function on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz [62] proved by Simon as [75, Theorem 3]. We prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the function is Morse-Bott, improving on similar results due to Chill [17, Corollary 3.12], [18, Corollary 4], Haraux and Jendoubi [44, Theorem 2.1], and Simon [77, Lemma 3.13.1]. In [33], we apply our abstract gradient inequalities to prove Lojasiewicz-Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities generalize those of Kwon [58, Theorem 4.2], Liu and Yang [60, Lemma 3.3], Simon [75, Theorem 3], [76, Equation (4.27)], and Topping [83, Lemma 1]. In [32], we prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang-Mills energy function due to the first author [26, Theorems 23.1 and 23.17] for base manifolds of arbitrary dimension and due to Råde [69, Proposition 7.2] for dimensions two and three. Contents

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“…is bounded in L 2 (Ω). Therefore using the H 2 -regularity of problem (P 0 ) (see (22)), one has that ψ(t n ) is bounded in H 2 1 .…”

Section: Existence Of Weak Solutions Satisfying Energy Inequality (38)mentioning

confidence: 99%

Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model

Climent-Ezquerra

Guillén-González

2019

Z. Angew. Math. Phys.

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In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the Navier-Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon's result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.Numerous studies have been devoted to this type of models and a detailed description of them can be seen in [9] and references therein. A phase function is also used in subsequent papers to model vesicle membranes as diffuse interfaces. In [15] and [22], a coupled Allen-Cahn and Navier-Stokes problem is studied approaching both constrains, area and volume, via a penalization functional. On the other hand, without taking into account the vesiclefluid interaction, a Cahn-Hilliard phase-field model for vesicle membranes is introduced in [2], and the well-posedness of a phase-field approximation satisfying area and volume constraints pointwisely is established in [7].In this paper, a Cahn-Hilliard-Navier-Stokes model will be considered. Since the volume constraint is implicitly satisfied for the Cahn-Hilliard equation, only the surface area constraint will be approximated via penalization. Moreover, the resulting problem will be thermodynamically consistent because there exists a free-energy (kinetic plus bending plus penalized one) dissipative in time along the trajectories. This fact is used to prove the existence of global in time weak solutions.On the other hand, the large-time behavior of the solutions will be analyzed following the way of [12], [16], [4]. Firstly, we prove that the ω-limit set for weak solutions is composed by critical points of the free-energy. After that, by using a modified Lojasiewicz-Simon's result we demonstrate the convergence of the whole trajectory to a single equilibrium. Finally, the convergence is improved for the phase assuming more regular data, but without using strong regularity for the velocity and pressure variables.The main novelties in this paper are the following:• The introduction of a new Navier-Stokes-Cahn-Hilliard problem modeling vesicle-fluid interactions.• The proof of a modified Lojasiewicz-Simon's result associated to weak solutions of a fourth order elliptic problem, which is adapted to the exigences of this new model. In fact, it was not possible to apply any already known Lojasiewicz-Simon's result to this problem.• Given a global weak solution, we choose a special regularized energy satisfying the energy's law inequality eit...

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Strong Solutions, Global Regularity, and Stability of a Hydrodynamic System Modeling Vesicle and Fluid Interactions

Cited by 16 publications

References 40 publications

Logarithmically improved blow-up criteria for a phase field Navier–Stokes vesicle–fluid interaction model

Logarithmically improved blow-up criteria for a phase field Navier–Stokes vesicle–fluid interaction model

Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces

Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model

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