Global Existence and Weak-Strong Uniqueness for Chemotaxis Compressible Navier–Stokes Equations Modeling Vascular Network Formation

A model of vascular network formation is analyzed in a bounded domain, consisting of the compressible Navier–Stokes equations for the density of the endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, which triggers the migration of the endothelial cells and the blood vessel formation. The coupling of the equations is realized by the chemotaxis force in the momentum balance equation. The global existence of finite energy weak solutions is shown for adiabatic pressure coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >8/5$$\end{document}γ>8/5. The solutions satisfy a relative energy inequality, which allows for the proof of the weak–strong uniqueness property.


Introduction
The formation of blood vessels is regulated by chemical signals triggering the movement of endothelial cells. The cells may self-assemble into a vascular network, which is known as vasculogenesis. In this paper, we analyze a mathematical model for the formation of vascular networks, based on mass and momentum balance equations including a chemotaxis force and coupled with a reaction-diffusion equation for the signal concentration. The existence of global weak solutions to the resulting chemotaxis compressible Navier-Stokes equations was proved in [1] for pressures with adiabatic exponent γ > 3. We extend the existence result to the range γ > 8/5 and prove a weak-strong uniqueness property. The proofs are based on a new relative energy inequality.
The dynamics of the density ρ(x, t) of the endothelial cells, their velocity v(x, t), and the concentration c(x, t) of the chemoattractant (e.g. the vascular endothelial growth factor VEGF-A [17]) is given by the equations where Ω ⊂ R 3 is a bounded domain, p(ρ) = ρ γ with the adiabatic exponent γ > 1 is the pressure, the Lamé viscosity constants µ, λ satisfy µ > 0 and 3λ + 2µ > 0, and ζ > 0 is a relaxation constant. We impose the initial and boundary conditions ρ(·, 0) = ρ 0 , v(·, 0) = v 0 , c(·, 0) = c 0 in Ω, (4) v = 0, ∇c · ν = 0 on ∂Ω, t > 0. (5) The boundary condition for the velocity v is the no-slip condition, and the no-flux boundary condition for c means that there is no inflow or outflow of the concentration. The momentum balance equation (2) includes viscous terms as in [1] (suggested in [2, p. 1862]) as well as the chemotaxis force ρf chem = −ρ∇c and the drag force ρf drag = −ρv/ζ. The reaction-diffusion equation (3) for the signal concentration models diffusion in the surrounding medium, degradation of the signal in finite time, and the release of the signal produced by the cells. We have set the physical constants in (1)-(3) equal to one, except ζ to distinguish terms originating from the drag force.
The existence of global finite energy weak solutions to (1)- (5) has been proved in [1] for γ > 3. This restriction comes from the estimation of the chemotaxis force; see Remark 4 on page 10. We extend the existence result to γ > 8/5 by rewriting the force term ρ∇c via (3) as (∂ t c − ∆c + c)∇c and exploiting the properties of the Bogovskii operator. Replacing the parabolic equation (3) for c by the elliptic one, we can even allow for γ > 3/2, which is the condition needed in the existence theory of the compressible Navier-Stokes equations [8]; see Remark 6. This may indicate that our condition γ > 8/5 for system (1)-(3) is not optimal. We discuss this issue further in Remark 5.
The idea of the existence proof in [1] is to derive a priori estimate via the energy-type functional where ψ(ρ) = ρ ρ 0 s −2 p(s)ds = ρ γ /(γ − 1) can be interpreted as the internal energy. Unfortunately, this functional is not bounded as t → ∞. Our idea is to use the physical (free) energy, (6) E(ρ, v, c) = Ω ψ(ρ) + 1 2 which is the sum of the kinetic energy 1 2 Ω ρ|v| 2 dx and the energy E(ρ, 0, c) of the parabolicparabolic Keller-Segel model. We show in Section 2 (see Lemma 3 on page 6) that dE dt (ρ, v, c) + Ω µ|∇v| 2 + (λ + µ)| div v| 2 dx + Ω |∂ t c| 2 dx ≤ 0, providing a bound for E((ρ, v, c)(t)) uniformly in time. Clearly, to infer a priori estimates, we need an upper bound for ρc. This is done by using the inequality which is due to Sugiyama [18] (see Lemma 9 on page 21), where C 1 (γ) > 0 and C 2 (γ) > 0 only depend on γ, and which (7) requires the condition γ > 8/5. The first two terms on the right-hand side of (7) can be absorbed by the energy, while the L 1 (Ω) norm of c can be bounded in terms of the initial data (ρ 0 , c 0 ). This provides a bound for the modified energy-type functional which allows us to prove the global existence of finite energy weak solutions such that H(ρ, v, c) is finite for all t > 0. This type of solutions is defined as follows. (1) holds for a.e. t ∈ (0, T ).
We introduce for 1 < p, q < ∞ the space W 2−2/p,q ν (Ω) as the completion of the space of functions w ∈ C ∞ (Ω) satisfying ∇w · ν = 0 on ∂Ω in the norm of W 2−2/p,q (Ω). We can now state our first main result.
The condition on the initial datum c 0 ∈ W 2−2/γ,γ ν (Ω) can be rephrased in terms of interpolation or Besov spaces. Indeed, the condition is needed to apply the maximal regularity result of Theorem 10, and the regularity on the initial datum can be formulated in such spaces; see [6,Theorem 10.22]. The definition of the pressure can be relaxed to p ∈ C 0 ([0, ∞)) ∩ C 2 (0, ∞), p(0) = 0, p ′ (ρ) > 0 for ρ > 0, and ρ 1−γ p ′ (ρ) → a > 0 as ρ → ∞; see [7, (2.1)]. The proof of the theorem is based on the existence theory for the compressible Navier-Stokes equations [8]. More precisely, we add some artificial diffusion and an artificial pressure term, construct Faedo-Galerkin solutions to the approximate problem, prove an approximate energy inequality for these solutions, and pass to the deregularizing limit. Improved uniform bounds for the cell density in L γ+θ (Ω) for some θ > 0 are derived by testing the mass balance equation with a test function involving the Bogovskii operator. The novel part is the estimate of the chemotaxis force term.
Next, we formulate the weak-strong uniqueness property of the system, meaning that a weak and a strong solution emanating from the same initial data coincide as long as the latter exists.
The no-vacuum assumptionρ ≥ c p > 0 was also needed in [9] and in related contexts, e.g. for the weak-strong uniqueness property of Maxwell-Stefan systems [11]. The proof of Theorem 2 is based on the relative energy method. The relative energy, associated to the energy functional (6), is given by where ψ(ρ|r) = ψ(ρ) − ψ(r) − ψ ′ (r)(ρ − r) is the Bregman distance associated to ψ. We show in Lemma 7 on page 11 that where (ρ, v, c) is a finite energy weak solution to (1)-(5), (r, u, z) are smooth functions, and the remainder R(ρ, v, c|r, u, z) is defined in Lemma 7 below. Finite energy weak solutions to the compressible Navier-Stokes equations satisfying (10) have been called suitable weak solutions in [9]. It was shown in [7] that finite energy weak solutions in fact always satisfy the relative energy inequality (10) for smooth functions (r, u, z).
Defining the modified relative energy and giving another weak solution (r, u, z) = (ρ,v,c) satisfying the regularity (9), the idea of the proof is to show that and which leads to and which implies, by Gronwall's lemma, that H((ρ, v, c)(t)|(ρ,v,c)(t)) = 0, Consequently ρ(t) =ρ(t), v(t) =v(t), and c(t) =c(t) for t > 0. We finish the introduction by discussing the state of the art. The global existence of finite energy weak solutions to the compressible Navier-Stokes equations with adiabatic exponents γ > 3/2 was shown in [8]. The range of γ can be extended to γ > 1 for axisymmetric initial data [12] or for a class of density-dependent viscosity coefficients [14], for instance. Germain [10] proved a relative energy inequality and established the weakstrong uniqueness property for solutions to the compressible Navier-Stokes equations with an integrable spatial density gradient. Feireisl et al. [9] proved the existence of so-called suitable weak solutions satisfying a general relative energy inequality with respect to any sufficiently regular pair of functions and concluded the weak-strong uniqueness property.
Compressible Euler equations with chemotaxis force have been introduced in [17] to describe early stages of vascologenesis. As remarked in [2, Section 3], the fluid equations may also include viscous terms. This leads to chemotaxis compressible Navier-Stokes equations, which have been analyzed in [1] with the pressure function p(ρ) = max{0, ρ − ρ c } γ , where γ > 3 and ρ c > 0 is the so-called close-packing density. A viscoelastic mechanical interaction of the cells with the substratum was added to the compressible Euler equations in [20]. Related models are the incompressible Navier-Stokes equations coupled to the chemotaxis Keller-Segel system via the fluid velocity, proposed in [21] and analyzed in, e.g., [22].
The paper is organized as follows. Section 2 is devoted to the proof of Theorem 1. The technical relative energy inequality (10) is proved in Section 3. Based on this inequality, Theorem 2 is then shown in Section 4. Finally, some auxiliary results are presented in Appendix A.
2.1. Regularized system. We solve first the following regularized system for δ > 0, ε > 0, and β > 4: subject to the initial and boundary conditions where ρ 0 δ is a smooth strictly positive function such that ρ 0 δ → ρ 0 strongly in L γ (Ω). The artificial viscosity term ε∆ρ is balanced by the term ε∇ρ · ∇v in the momentum equation to control the energy. The artificial pressure term δ∇ρ β is needed to derive an L γ+θ (Ω) estimate for the density with θ > 0.
The existence of strong solutions to (12)- (15) was shown in [8, Section 2] without the chemotaxis term ρ∇c. Here, we sketch the proof for the problem including the chemotaxis coupling. As in [8], we use the Faedo-Galerkin method. Let (ψ n ) be a sequence of eigenfunctions of the Laplacian with homogeneous Dirichlet boundary conditions and let X n = span{ψ 1 , . . . , ψ n }. Then, following the proof of [16,Section 7.7] or [5,Chapter 7], we obtain the existence of a unique local strong solution (ρ n , v n , c n ) on (0, T n ) such that v n ∈ C 1 ([0, T n ]; X n ) and ρ n , ∂ t ρ n , ∇ρ n , ∇ 2 ρ n , c n , ∂ t c n , ∇c n , ∇ 2 c n are Hölder continuous on Ω × [0, T n ], To obtain global solutions, i.e. T = T n , we derive an energy inequality for the approximate system.

2.2.
Energy inequality for the approximate system. An energy-type inequality has been derived in [1, Section 2.2]. Here, we use a different energy functional by including the H 1 (Ω) norm of c. We show an inequality for the energies E(ρ, v, c) and H(ρ, v, c), defined in (6) and (8), respectively. Lemma 3. Let (ρ n , v n , c n ) be a strong solution to (12)-(15) constructed in the previous subsection. Then there exists C > 0 independent of (n, δ, ε) such that for any 0 < t < T n , Proof.
Step 1: Energy inequality for E. We choose the test function ψ ′ (ρ n ) − 1 2 |v n | 2 + δβρ β−1 n /(β − 1) in the weak formulation of the first equation in (12) and the test function v n in the weak formulation of (13). Adding both equations and taking into account (11), some terms cancel, and we arrive after a standard computation at We estimate the right-hand side by integrating by parts and using equation (12) for ρ n : Taking into account the second equation in (12), the second term on the right-hand side is written as Because of ρ n c n ≥ 0, the last term on the right-hand side of (17) becomes where the last inequality follows from γ ≥ 2. (We observe that at this point, we can weaken the condition to γ > 8/5 by using the Gagliardo-Nirenberg inequality and the estimate for ∇ρ γ/2 n L 2 (Ω) from (16).) We insert these estimates into (17): Therefore, (16) leads to where C > 0 only depends on γ and meas(Ω) but is independent of n, δ, and ε. This proves the inequality for E(ρ n , v n , c n ).
Step 2: Energy inequality for H. We need to estimate Ω ρ n c n dx in E(ρ n , v n , c n ). By Lemma 9 in Appendix A, applied to m = γ, κ = 1/(2(γ − 1)), and ξ = 1/4, Equation (12) implies that the mass is conserved, ρ n (t) L 1 (Ω) = ρ 0 δ L 1 (Ω) for 0 < t < T n . Furthermore, by the second equation in (12), This is an ordinary differential equation for t → c n (t) L 1 (Ω) , and a comparison principle as well as the nonnegativity of c n imply that where C > 0 is independent of δ. Thus, we conclude from (19) and It follows from the definitions of E(ρ n , v n , c n ) and H(ρ n , v n , c n ) that We insert these estimates in (18) and integrate over (0, t) for 0 < t < T n : where we used Ω ρ γ n dx ≤ CH(ρ n , v n , c n ). An application of Gronwall's lemma finishes the proof.

Remark 5 (On the condition γ > 8/5). This restriction is needed to estimate the integral
Ω ρ n c n dx by means of Lemma 9. The idea is to obtain "small" terms that can be absorbed by the left-hand side of the energy inequality (18) and terms that can be controlled (the L 1 (Ω) norm of c n ). By the Hölder and Gagliardo-Nirenberg inequalities, we may estimate in a different way: where θ = 3/(5γ − 3) ∈ (0, 1) (which requires that γ > 6/5). It follows from the maximal regularity result of Theorem 10 that Ω ρ n c n dx ≤ C ρ n L γ (Ω) ( ρ n L γ (Ω) + 1) θ , where C > 0 depends on c 0 L 1 (Ω) . We can conclude if 1 + θ < γ, which is equivalent to γ > 8/5. Thus, even taking into account maximal regularity does not improve the range for γ.

Relative energy inequality
We show a relative energy inequality for smooth functions.
Lemma 7 (Relative energy inequality). Let (ρ, v, c) be a smooth solution to (1)-(5) and let (r, u, z) be smooth functions satisfying r > 0 in Ω × [0, T ] and u = 0 on ∂Ω. Then the relative energy inequality (10) holds for 0 < t < T with We prove in Section 4 that the relative energy inequality (10) holds for finite energy weak solutions (ρ, v, c) and (ρ,v,c), where (ρ,v) satisfies (9). The proof of (10) follows the lines of [9, Section 3.2], but some steps are different due to the additional chemotaxis force. For this reason, and for the convenience of the reader, we present a full proof.
Proof. Let (r m , u m , z m ) m∈N be smooth functions satisfying r m > 0 in Ω × [0, T ], v m ∈ C 1 ([0, T ]; X m ), and v m = 0 on ∂Ω such that (r m , u m , z m ) → (r, z, u) as m → ∞ in a sense made precise in Step 3 below. Here, X m is the Faedo-Galerkin space defined in Section 2.1. We introduce Then (f m , g m , h m ) → (f, g, h) as m → ∞ in the sense of distributions, where (f, g, h) is defined in (23)-(24). Finally, let (ρ n , v n , c n ) be a Galerkin solution to (12)- (15). We compute in the following the approximate relative energy inequality.
Step 1: Time derivative of the relative kinetic energy. We derive an equation for the time evolution of the relative kinetic energy 1 2 Ω ρ n |v n − u m | 2 dx. It follows from the approximative mass balance equation (12) Since ρ n (∂ t v n + v n · ∇v n ) = ∂ t (ρ n v n ) + div(ρ n v n ⊗ v n ) − ε∆ρ n v n , the second and third terms on the right-hand side are written as We insert this expression into (27), integrate over Ω, and replace ∂ t (ρ n v n ) + div(ρ n v n ⊗ v n ) by the momentum equation (13): We wish to reformulate the last but one term in the previous equality. For this, we add and subtract r m and replace r m (∂ t u m + u m · ∇u m ) by (25): Then, after a computation, (28) becomes We rewrite the first, second, and sixth terms on the right-hand side of (28).
Step 2a: Reformulation of the pressure term. Observing that p ′ (z) = zψ ′′ (z) for z ≥ 0 (see (11)) and that ρ m − r m satisfies we can write the first term on the right-hand side of (29) as Taking into account that the evolution of the relative internal energy is given by the first term on the right-hand side of (30) is reformulated as where we used definition (26) of g m in the last step. Integrating by parts to get rid of the divergence, inserting the corresponding expression into (30), and observing that the integral over (ψ ′ (ρ n ) − ψ ′ (r m ))g m cancels with the corresponding expression in (30), we find that We claim that the second term on the right-hand side can be formulated in terms of the relative pressure p(ρ n |r m ) = p(ρ n ) − p(r m ) − p ′ (r m )(ρ n − r m ). It follows from (11) that ∇p(ρ n ) = ρ n ∇ψ ′ (ρ n ), ∇p ′ (r m ) = ∇ψ ′ (r m ) + r m ∇ψ ′′ (r m ) and hence, and consequently, Therefore, Step 2b: Reformulation of the chemotaxis term. We reformulate the sixth term on the right-hand side of (29) by integrating by parts and using the mass balances (12) and (26): In view of the second equation in (26), we have We insert this expression into the second term on the right-hand side of (32): Step 2c: Reformulation of the artificial pressure term. We rewrite the second term on the right-hand side of (29) by integrating by parts and using the mass balance equation (12): Step 2d: Collecting the reformulations. We include the reformulations (31), (33), and (34) into (29) to find that Step 3: Limit (n, m) → ∞ and (δ, ε) → 0. As mentioned in [9, Section 3.3], the limit in the approximate relative energy inequality (35) follows step by step the existence proof in [5,Chapter 7] or [16,Chapter 7]. In particular, we perform first the limit n → ∞ in the Faedo-Galerkin approximation (ρ n , v n , c n ) → (ρ ε,δ , v ε,δ , c ε,δ ). Then the functions (r m , u m , z m ) are replaced by smooth functions (r, u, z) using a density argument. Third, we pass to the limit (ρ ε,δ , v ε,δ , c ε,δ ) → (ρ δ , v δ , c δ ) as ε → 0 and (ρ δ , v δ , c δ ) → (ρ, v, c) as δ → 0.
In view of the bounds (20), we can pass to the limit n → ∞ and (δ, ε) → 0 in (35). We assume that (r m , u m , z m ) converges to (r, u, z) as m → ∞ in such a way that the limit m → ∞ in (35) is possible. Then some integrals in (35) disappear and we end up with This shows (10) and finishes the proof.

Weak-strong uniqueness
We split the proof in several steps.
It follows from [7,Theorem 2.4] that (10) still holds if (ρ, v, c) is a finite energy weak solution.