Global weak solutions to a Navier–Stokes–Cahn–Hilliard system with chemotaxis and singular potential

He, Jingning

doi:10.1088/1361-6544/abc596

Cited by 12 publications

(23 citation statements)

References 66 publications

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“…We deduce from Poincaré–Wirtinger inequality () and (), (), and () that

\begin{matrix} right ‖ σ ‖ & left \leq ‖ σ - χ φ ‖ + ‖ χ φ ‖ \\ right & left \leq C_{P} ‖ \nabla (σ - χ φ) ‖ + | Ω | | \overset{true‾}{σ - χ φ} | + ‖ χ φ ‖ \\ right & left \leq C_{P} ‖ \nabla (σ - χ φ) ‖ + C_{1}, \end{matrix}

where the constant C 1 > 0 only depends on

χ, normalΩ, false| \overset{σ_{0}}{true} false|

. Again by (), we have (see He 31 , (3.32))

\begin{matrix} right \int_{Ω} \frac{1}{2} | σ |^{2} + χ σ (1 - φ) d x & left \geq \int_{Ω} (\frac{1}{2} | σ |^{2} - 2 | χ | | σ |) d x \\ right & left \geq \frac{1}{4} ‖ σ ‖^{2} - 4 χ^{2} | Ω |, \end{matrix}

which implies that

scriptE false(t false)

is indeed bounded from below. Next, we recall the following inequality due to (H1) (see, e.g., Giorgini et al 9 ):

normalΨ false(r false) \leq

…”

Section: Global Weak Solutionsmentioning

confidence: 85%

“…Multiplying () with −Δ φ and integrating over Ω, we have (see, e.g., He 31 , Section 3.2.1 ).

\begin{matrix} right A (Ψ^{''} (φ) \nabla φ, \nabla φ) + B {‖Δ φ‖}^{2} & left \leq ‖ \nabla μ ‖ ‖ \nabla φ ‖ + | χ | ‖ \nabla σ ‖ ‖ \nabla φ ‖ + ϵ (\partial_{t} φ, Δ φ) \\ right & left \leq C (‖\nabla μ‖ + ‖\nabla σ‖ + ‖ \partial_{t} φ ‖^{2}) + \frac{B}{2} {‖Δ φ‖}^{2} . \end{matrix}

…”

Section: Global Weak Solutionsmentioning

confidence: 99%

“…At the level of weak solutions, the additional viscous term ϵ∂ t φ does not play an essential role and the conclusion can be easily extended to the non‐viscous case with

ϵ = 0

(cf. He 31 ). For well‐posedness results on more general system with further mechanics (like tumor proliferation, apoptosis, nutrient consumption, etc.)…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the viscous Cahn–Hilliard–Oono system with chemotaxis and singular potential

2021

Math Methods in App Sciences

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We analyze a diffuse interface model that couples a viscous Cahn–Hilliard equation for the phase function φ with a diffusion‐reaction equation for the nutrient concentration σ. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport, and nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of a global weak solution to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solution when t > 0. In particular, with the aid of the viscous term ϵ∂tφ, we prove the so‐called instantaneous separation property of the phase variable φ such that it stays away from the pure states ±1 as long as t > 0. Next, we study longtime behavior of the system, by proving the existence of the global attractor in a suitable phase space and characterizing the ω‐limit set. Moreover, we show the existence of an exponential attractor, which implies that the global attractor is of finite fractal dimension.

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“…We deduce from Poincaré–Wirtinger inequality () and (), (), and () that

\begin{matrix} right ‖ σ ‖ & left \leq ‖ σ - χ φ ‖ + ‖ χ φ ‖ \\ right & left \leq C_{P} ‖ \nabla (σ - χ φ) ‖ + | Ω | | \overset{true‾}{σ - χ φ} | + ‖ χ φ ‖ \\ right & left \leq C_{P} ‖ \nabla (σ - χ φ) ‖ + C_{1}, \end{matrix}

where the constant C 1 > 0 only depends on

χ, normalΩ, false| \overset{σ_{0}}{true} false|

. Again by (), we have (see He 31 , (3.32))

\begin{matrix} right \int_{Ω} \frac{1}{2} | σ |^{2} + χ σ (1 - φ) d x & left \geq \int_{Ω} (\frac{1}{2} | σ |^{2} - 2 | χ | | σ |) d x \\ right & left \geq \frac{1}{4} ‖ σ ‖^{2} - 4 χ^{2} | Ω |, \end{matrix}

which implies that

scriptE false(t false)

is indeed bounded from below. Next, we recall the following inequality due to (H1) (see, e.g., Giorgini et al 9 ):

normalΨ false(r false) \leq

…”

Section: Global Weak Solutionsmentioning

confidence: 85%

“…Multiplying () with −Δ φ and integrating over Ω, we have (see, e.g., He 31 , Section 3.2.1 ).

\begin{matrix} right A (Ψ^{''} (φ) \nabla φ, \nabla φ) + B {‖Δ φ‖}^{2} & left \leq ‖ \nabla μ ‖ ‖ \nabla φ ‖ + | χ | ‖ \nabla σ ‖ ‖ \nabla φ ‖ + ϵ (\partial_{t} φ, Δ φ) \\ right & left \leq C (‖\nabla μ‖ + ‖\nabla σ‖ + ‖ \partial_{t} φ ‖^{2}) + \frac{B}{2} {‖Δ φ‖}^{2} . \end{matrix}

…”

Section: Global Weak Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

On the viscous Cahn–Hilliard–Oono system with chemotaxis and singular potential

2021

Math Methods in App Sciences

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“…For every κ ∈ (0, a 0 ), the regularized problem (3.2)-(3.5) can be solved via a standard Galerkin scheme like in [12]. Then following the arguments in [22,Appendix], one can derive uniform estimates of the approximate solutions (ϕ κ , µ κ , σ κ ) with respect to the parameter κ and time on a given interval [0, T ]. In particular, the additional viscous term ǫ∂ t ϕ κ in (3.3) yields some additional regularity on ∂ t ϕ κ such that ǫ∂ t ϕ κ ∈ L 2 (0, T ; L 2 (Ω)).…”

Section: Existencementioning

confidence: 99%

“…First, under suitable assumptions on the singular potential function Ψ and coefficients of the system, we show the existence and uniqueness of global weak solutions to problem (1.1a)-(1.3) on the whole interval [0, +∞) (see Theorem 2.1). The proof is based on a suitable Galerkin approximation that mainly follows the argument in [22] for a more general system with fluid interaction. Thanks to the singular potential, we are allowed to remove certain restricted assumptions on the coefficients A and χ when a regular potential was adopted (cf.…”

Section: Introductionmentioning

confidence: 99%

On the Viscous Cahn-Hilliard-Oono System with Chemotaxis and Singular Potential

2021

Preprint

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We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport as well as nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of global weak solutions to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solutions when t > 0. In particular, with the aid of the viscous term, we prove the so-called instantaneous separation property of the phase variable such that it stays away from the pure states ±1 as long as t > 0. Furthermore, we study long-time behavior of the system, by proving the existence of a global attractor and characterizing its ω-limit set.

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A phase separation model for binary fluids with hereditary viscosity

Grasselli

Poiatti

2022

Math Methods in App Sciences

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We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier-Stokes-Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term incorporating hereditary effects coupled with the Cahn-Hilliard equation with Flory-Huggins potential. The resulting system is subject to no-slip condition for the (volume averaged) fluid velocity u and to no-flux boundary conditions for the order parameter 𝜙 and the chemical potential 𝜇. We first establish the well-posedness of the initial and boundary value problem. Then, taking advantage of an Ekman-type damping, we obtain a dissipative estimate. Thus, we can define a dissipative dynamical system on a suitable phase space. Also, we show that any weak solution is such that 𝜙 regularizes in finite time, and in dimension two, it stays uniformly away from pure phases; that is, the instantaneous strict separation property holds. Finally, we prove the existence of the global attractor, and exploiting the strict separation property, we demonstrate that an exponential attractor exists in dimension two.

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Global weak solutions to a Navier–Stokes–Cahn–Hilliard system with chemotaxis and singular potential

Cited by 12 publications

References 66 publications

On the viscous Cahn–Hilliard–Oono system with chemotaxis and singular potential

On the viscous Cahn–Hilliard–Oono system with chemotaxis and singular potential

On the Viscous Cahn-Hilliard-Oono System with Chemotaxis and Singular Potential

A phase separation model for binary fluids with hereditary viscosity

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