Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity

Yu Tian; Zhaoyin Xiang

doi:10.1515/anona-2022-0228

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Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity

Yu Tian and Zhaoyin Xiang

Published/Copyright: August 19, 2022

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From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

In this article, we will develop an analytical approach to construct the global bounded weak solutions to the initial-boundary value problem of a three-dimensional chemotaxis-Stokes system with porous medium cell diffusion Δ n m for m ≥ 65 63 and general sensitivity. In particular, this extended the precedent results which asserted global solvability within the larger range m > 7 6 for general sensitivity (M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828) or m > 9 8 for scalar sensitivity (M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151). Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.

Keywords: chemotaxis-Stokes system; porous medium cell diffusion; general sensitivity; global boundedness

MSC 2010: 35K55; 35Q92; 35Q35; 92C17

Special Issue: – Nonlinear analysis: perspectives and synergies

1 Introduction

Chemotaxis-fluid model with linear cell diffusion. In Tuval et al. [22], the following chemotaxis-fluid model system has been proposed to describe the collective behavior of a suspension of oxytactic bacteria in an incompressible fluid under the assumption that the coupling of chemotaxis and fluid is realized through both the transport of bacteria and chemical substrates and the external force exerted on the fluid by bacteria (since the bacteria are about 10% denser than the fluid):

(1.1) n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n f ( c ) , x ∈ Ω , t > 0 , u t + κ ( u ⋅ ∇ ) u = Δ u + ∇ P + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 .

Here, the unknowns are n , c , u , and P , denoting the cell density, oxygen concentration, and velocity and pressure of the fluid, respectively. The known function S denotes the chemotactic sensitivity and f is the consumption rate of the substrate by the cells, while ϕ is a given potential function. Moreover, κ ∈ R is related to the strength of nonlinear fluid convection and in particular the Stokes flow ( κ = 0 ) is a reasonable simplification in the low-Reynolds-number flow, for instance, the Reynolds number is of order 1 0 − 4 for a single Bacillus subtilis while Reynolds numbers of order 1 0 − 2 are observed in bio-convective vortices (see, e.g., [22]).

A first local existence result for system (1.1) was obtained by Lorz [16] in a bounded domain with no-flux boundary conditions for n and c and no-slip boundary condition for u.^[1] Under some structural hypotheses on the scalar functions S and f , the global existence of three-dimensional (3D) Cauchy problem for system (1.1) with κ = 0 was shown by Duan et al. [6] for small initial concentrations of the substrates c . Similar assumptions are applied to the initial-boundary value problem by Winkler [26] to show the global existence of classical solutions to two-dimensional (2D) system (1.1) with κ = 1 and of weak solutions to 3D system (1.1) with κ = 0 . They are also used by Winkler [29] to deal with the global weak solutions of 3D version with κ = 1 . Then Winkler [27,30] further asserted that both the 2D classical solution and the 3D eventual energy solution will stabilize to the spatially uniform equilibrium ( n ¯ 0 , 0 , 0 , 0 ) in the sense that as t → ∞

n ( ⋅ , t ) → n ¯ 0 , c ( ⋅ , t ) → 0 , and u ( ⋅ , t ) → 0

hold with respect to the norm in L ∞ ( Ω ) , where n ¯ 0 ≔ 1 ∣ Ω ∣ ∫ Ω n ( x , 0 ) d x . In these works, an essential step forward in the analysis was marked by the observation that when S ≡ 1 and f ( c ) = c , system (1.1) admits for some natural quasi-Lyapunov functional of the form

(1.2) d d t ∫ Ω n ln n + 1 2 ∫ Ω ∣ ∇ c ∣ 2 c + δ ∫ Ω ∣ u ∣ 2 + 1 C ∫ Ω ∣ ∇ n ∣ 2 n + ∫ Ω ∣ ∇ c ∣ 4 c 3 + ∫ Ω ∣ ∇ u ∣ 2 ≤ C

for all t > 0 and some positive constants δ and C . In [4], Chae et al. relaxed the above structural conditions and obtained the global existence of classical solutions to Cauchy problem of system (1.1) with κ = 1 for some smooth small initial data.

More experimental findings suggest that chemotactic migration may have rotational components, especially near the physical boundary of the domain, and thus that accordingly the chemotactic sensitivity S in system (1.1) should actually be matrix-valued function in R N × N (see Xue and Othmer [39] and Xue [38]). In this general situation, it is difficult to derive inequalities of type (1.2) which in a subtle way cancels contributions stemming from cross-diffusive interaction. Under certain smallness conditions on the initial data, however, Cao and Lankeit [2] established the global existence and exponential decay rates of classical solutions to system (1.1) with κ = 1 in both 2D and 3D bounded domains. For the 3D system (1.1) with κ = 0 , the aforementioned small assumptions can be replaced by the volume-filling effect (see [11]), while in the 2D case, Winkler [34] further proved that the global generalized solution will stabilize toward the unique spatially constant steady state for general initial data.

Chemotaxis-fluid system with nonlinear diffusion. To adequately account for the finite size of bacteria, Di Francesco et al. [5] modified the original model (1.1) by assuming that the random movement of cells is nonlinearly enhanced at large densities. This leads to the following chemotaxis-fluid system with nonlinear diffusion:

(1.3) n t + u ⋅ ∇ n = ∇ ⋅ ( D ( n ) ∇ n ) − ∇ ⋅ ( n S ( x , n , c ) ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n f ( c ) , x ∈ Ω , t > 0 , u t + κ ( u ⋅ ∇ ) u = Δ u + ∇ P + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0

with D ( n ) = n m − 1 ( m > 1 ) in the prototypical case of porous medium-type diffusion. In the case of S ≡ 1 , f ( c ) = c , and D ( n ) = n m − 1 ( m > 1 ) , Di Francesco et al. [5] proved that system (1.3) with κ = 0 features a global weak solution in a bounded domain Ω ⊂ R 2 for m ∈ ( 3 2 , 2 ] , and then Tao and Winkler [18] used a combined entropy-type estimate on ∫ Ω n γ with suitably large γ and ∫ Ω ∣ ∇ c ∣ 2 to establish the global existence of bounded weak solutions to the 2D system (1.3) with κ = 1 for all reasonably regular initial data in the whole range

m > 1

(see also Winkler [32] for mass-preserving generalized solutions in the case of κ = 0 and matrix-valued S ). For the 3D system (1.3), even for S ≡ 1 and f ( c ) = c and at the level of mere existence theory, the nonlinear diffusion brings about significant analytical challenges and very little is known in quite basic aspects. Accordingly, in his remarkable work [26] (on Page 323), Winkler wrote:

In the case N = 3 , a complete classification of all m > 1 which allow for global solutions is still lacking; ⋯ gives rise to the conjecture that for any m > 1 global weak solutions exist, at least in the simplified chemotaxis-Stokes system when κ = 0 .

Noting that a similar quasi-energy-type functional as (1.2) of the form

(1.4) d d t ∫ Ω n ln n + 1 2 ∫ Ω ∣ ∇ c ∣ 2 c + δ ∫ Ω ∣ u ∣ 2 + 1 C ∫ Ω ∣ ∇ n ∣ 2 n 2 − m + ∫ Ω ∣ ∇ c ∣ 4 c 3 + ∫ Ω ∣ ∇ u ∣ 2 ≤ C

still holds for system (1.3) with scalar sensitivity S , Winkler [31] invoked a combination of energy-based structure (1.4) and maximal Sobolev regularity theory and revealed that the condition

m > 9 8

is sufficient for the global existence and boundedness of weak solutions to the initial-boundary value problem of the 3D system (1.3) with κ = 0 . The latter partially extended the precedent global solvability within the larger values m ≥ 4 3 in Liu and Lorz [15] and also m > 8 7 in Tao and Winkler [20] (see also [5]). For the smaller values of m > 1 , up to now the global existence result is limited to classes of possibly unbounded solutions (see e.g., [7]). On the other hand, despite loss of energy structure in the case of matrix-valued function S in R N × N , Winkler [28] developed an alternative a priori estimation method involving a two-step bootstrap argument to assert the global existence of weak solutions to the 3D system (1.3) with κ = 0 under the constraint

m > 7 6 .

To be clear, we refer to Table 1 for an illustration of the main known results on the global existence of weak solutions to the 3D chemotaxis-Stokes system (1.3) with porous medium-type cell diffusion.^[2]

Table 1

On the 3D chemotaxis-Stokes system with porous medium-type cell diffusion

Ref.	Ω	Admissible range for m	Sensitivity S	Boundedness of solutions
[5]	Whole space R 3	2	Scalar S	Locally bounded
[15]	Whole space R 3	4 3	Scalar S	Locally bounded
[7]	Whole space R 3	( 1 , + ∞ )	Scalar S	Locally bounded
[5]	Bounded domain	7 + 217 12 , 2	Scalar S	Globally bounded
[15]	Bounded domain	4 3 , 2	Scalar S	Globally bounded
[20]	Bounded domain	8 7 , + ∞	Scalar S	Locally bounded
[32]	Bounded domain	9 8 , + ∞	Scalar S	Globally bounded
[28]	Bounded domain	7 6 , + ∞	Matrix S	Globally bounded

From the aforementioned results, the question of identifying an optimal condition on m > 1 ensuring global boundedness in the 3D chemotaxis-Stokes system (1.3) remains a challenging problem even for the scalar sensitivity S .

We would like to mention several recent results related to the current topic but with different boundary conditions for the chemical signal, in particular, Robin boundary conditions [21,36] or Dirichlet boundary conditions [23,25,37]. The related analytical studies of chemotaxis(-fluid) models also provide an engine of growth source for the development of partial differential equations of the parabolic type [3,8,9,24,35].

Main results. It is the main purpose of the present work to develop an analytical approach to further advance the analysis of system (1.3) in the larger porous medium cell diffusion range m ≥ 65 63 and with general sensitivity S . Precisely, we shall consider the following chemotaxis-Stokes system

(1.5) n t + u ⋅ ∇ n = ∇ ⋅ ( n m − 1 ∇ n ) − ∇ ⋅ ( n S ( x , n , c ) ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0

in a bounded domain Ω ⊂ R 3 with smooth boundary, under the boundary conditions

(1.6) ( n m − 1 ∇ n − n S ( x , n , c ) ∇ c ) ⋅ ν = 0 , ∂ c ∂ ν = 0 , and u = 0 , x ∈ ∂ Ω , t > 0 ;

and along with the initial conditions

(1.7) n ( x , 0 ) = n 0 ( x ) , c ( x , 0 ) = c 0 ( x ) and u ( x , 0 ) = u 0 ( x ) , x ∈ Ω .

Throughout this article, the chemotactic sensitivity S and the potential function ϕ are assumed to satisfy

(1.8) S ∈ C 2 ( Ω ¯ × [ 0 , ∞ ) 2 ; R 3 × 3 ) and ∣ S ( x , n , c ) ∣ ≤ C S , ϕ ∈ W 2 , ∞ ( Ω ) .

As for the initial data, we shall suppose that

(1.9) n 0 ∈ C ω ( Ω ¯ ) for some ω > 0 , n 0 ≥ 0 and n 0 ≢ 0 in Ω ¯ , c 0 ∈ W 1 , ∞ ( Ω ) , c 0 ≥ 0 and c 0 ≢ 0 in Ω ¯ , u 0 ∈ D ( A β ) for some β ∈ 3 4 , 1 .

A consequence of the degeneracy of the porous medium cell diffusion is that we do not expect to have classical solutions of system (1.5) when the initial data take on the value n = 0 in an open subset of Ω . Therefore, we first introduce the definition of global weak solutions to systems (1.5)–(1.7).

Definition 1.1

By a global weak solution to systems (1.5)–(1.7) in Ω × ( 0 , ∞ ) , we mean a quadruple of functions ( n , c , u , P ) such that

(1.10) n ∈ L loc 1 ( Ω ¯ × [ 0 , ∞ ) ) , c ∈ L loc ∞ ( Ω ¯ × [ 0 , ∞ ) ) ∩ L loc 1 ( [ 0 , ∞ ) ; W 1 , 1 ( Ω ) ) , u ∈ L loc 1 ( [ 0 , ∞ ) ; W 1 , 1 ( Ω ) ) ,

with n ≥ 0 and c ≥ 0 in Ω × ( 0 , ∞ ) , that

(1.11) n m , n ∣ ∇ c ∣ , and n ∣ u ∣ belong to L loc 1 ( Ω ¯ × [ 0 , ∞ ) ) ,

that ∇ ⋅ u = 0 in the distributional sense in Ω × ( 0 , ∞ ) , and that

∫ 0 ∞ ∫ Ω n ψ 1 t + ∫ Ω n 0 ψ 1 ( ⋅ , 0 ) = 1 m ∫ 0 ∞ ∫ Ω n m Δ ψ 1 − ∫ 0 ∞ ∫ Ω n ( S ( x , n , c ) ∇ c ) ⋅ ∇ ψ 1 − ∫ 0 ∞ ∫ Ω n u ⋅ ∇ ψ 1 , ∫ 0 ∞ ∫ Ω c ψ 2 t + ∫ Ω c 0 ψ 2 ( ⋅ , 0 ) = ∫ 0 ∞ ∫ Ω ∇ c ⋅ ∇ ψ 2 + ∫ 0 ∞ ∫ Ω n c ψ 2 − ∫ 0 ∞ ∫ Ω c u ⋅ ∇ ψ 2 , ∫ 0 ∞ ∫ Ω u ⋅ ψ 3 t + ∫ Ω u 0 ⋅ ψ 3 ( ⋅ , 0 ) = ∫ 0 ∞ ∫ Ω ∇ u ⋅ ∇ ψ 3 − ∫ 0 ∞ ∫ Ω n ∇ ϕ ⋅ ψ 3

for all ψ 1 ∈ C 0 ∞ ( Ω ¯ × [ 0 , ∞ ) ) with ∂ ψ 1 ∂ ν = 0 , ψ 2 ∈ C 0 ∞ ( Ω ¯ × [ 0 , ∞ ) ) , and ψ 3 ∈ C 0 ∞ ( Ω × [ 0 , ∞ ) ; R 3 ) satisfying ∇ ⋅ ψ 3 ≡ 0 in Ω × ( 0 , ∞ ) .

In the context of the above assumptions, our main result asserts the global existence of a bounded solution to systems (1.5)–(1.7) in the following sense.

Theorem 1.1

Let Ω ⊂ R 3 be a bounded domain with smooth boundary ∂ Ω and suppose that (1.8)–(1.9) hold. Then for any m ≥ 65 63 , systems (1.5)–(1.7) admit at least one global weak solution ( n , c , u , P ) , which is uniformly bounded in the sense that

‖ n ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ c ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) + ‖ A β u ( ⋅ , t ) ‖ L 2 ( Ω ) ≤ C for all t > 0

with β ∈ 3 4 , 1 given by (1.9), where C is a positive constant depending only on Ω , m , C S , ϕ , and the initial data.

Remark 1.1

Theorem 1.1 extended the precedent results which asserted global solvability within the larger range m > 7 6 for general sensitivity [28] or m > 9 8 for scalar sensitivity [31]. Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.

Outline of our approach. Before going into details, let us briefly outline the main steps of our analysis. As we have mentioned previously, the quasi-energy structure (1.4) is the core of the scalar sensitivity S . For the general sensitivity S , the core of our argument lies in a new observation on a refined quasi-energy structure of the form

(1.12) d d t ∫ Ω n ε ln n ε + ∫ Ω ∣ ∇ c ε ∣ 2 + 2 C ∫ Ω ∣ u ε ∣ 2 + 1 C ∫ Ω ∣ ∇ n ε ∣ 2 n ε 2 − m + ∫ Ω ∣ D 2 c ε ∣ 2 + ∫ Ω n ε ∣ ∇ c ε ∣ 2 + ∫ Ω ∣ ∇ u ε ∣ 2 ≤ C

for 1 < m ≤ 2 (see (3.12)). As a starting point, the corresponding a priori estimates on

∫ t t + 1 ∫ Ω n ε ∣ ∇ c ε ∣ 2

obtained from (1.12) (Lemma 3.1) will ensure the key L m -boundedness of the approximate solution component n ε (Lemma 3.2), which together with some new boundedness and dissipation estimates under some presupposed bounds on n ε (Lemmas 3.4 and 3.5) can be used as a new starting point for an induction argument on the further higher regular estimates of the form

∫ Ω n ε k ( m − 1 ) + 1 ( ⋅ , t ) ≤ C for all t > 0

for k = 1 , 2 , … (Lemma 3.6 and Corollary 3.2). Then we will be able to obtain the W 1 , ∞ -bounds of u ε by using the energy techniques and the smoothing effect of Stokes semigroup (Lemma 3.8) and of c ε by an essentially standard argument on the Neumann heat semigroup (Lemma 3.9). These estimates together with some time regularity properties of n ε (Lemmas 3.11 and 3.12) and uniform Hölder regularity properties of c ε , ∇ c ε , and u ε (Lemma 3.13) thereby will provide appropriate compactness properties, which will allow us to construct global bounded weak solutions to systems (1.5)–(1.7) via a suitable approximation procedure (Lemma 3.14).

Organization of this article. The rest of this article is organized as follows. In Section 2, we first construct a nondegenerate approximate system for (1.5)–(1.7) and show its global solvability. We will also present some basic estimates for the approximate solutions in this section. Then in Section 3, we will establish the a priori estimates for the approximate solutions by a series of comparatively subtle reasonings and show the global existence of weak solutions to systems (1.5)–(1.7) by a limit procedure. Finally in Section 4, we collect miscellaneous remarks, including the large time behavior of the obtained solutions and several generalizations of Theorem 1.1, which are still new in the existing literature.

2 Approximation by nondegenerate problems

Our main goal is to construct solutions of (1.5) as limits of solutions to appropriately regularized problems. To achieve this, following natural regularization procedures we will approximate the diffusion coefficient n m − 1 in (1.5) by using the shifted function ( n + ε ) m − 1 for ε ∈ ( 0 , 1 ) . On the other hand, it will be convenient to deal with homogeneous Neumann boundary conditions for both n and c rather than with the nonlinear no-flux relation in (1.6). In order to achieve this, we introduce families ( ρ ε ) ε ∈ ( 0 , 1 ) and ( χ ε ) ε ∈ ( 0 , 1 ) of functions

ρ ε ∈ C 0 ∞ ( Ω ) , 0 ≤ ρ ε ≤ 1 in Ω , and ρ ε → 1 in Ω as ε → 0

and

χ ε ∈ C 0 ∞ ( [ 0 , ∞ ) ) , 0 ≤ χ ε ≤ 1 in [ 0 , ∞ ) , and χ ε → 1 in [ 0 , ∞ ) as ε → 0 ,

and define smooth approximations S ε of S by letting

(2.1) S ε ( x , n , c ) ≔ ρ ε ( x ) χ ε ( n ) S ( x , n , c ) for all x ∈ Ω ¯ , n ≥ 0 , c ≥ 0

for all ε ∈ ( 0 , 1 ) . These choices guarantee that each of the approximate variants of systems (1.5)–(1.7) given by

(2.2) n ε t + u ε ⋅ ∇ n ε = ∇ ⋅ ( ( n ε + ε ) m − 1 ∇ n ε ) − ∇ ⋅ ( n ε S ε ( x , n ε , c ε ) ∇ c ε ) , x ∈ Ω , t > 0 , c ε t + u ε ⋅ ∇ c ε = Δ c ε − n ε c ε , x ∈ Ω , t > 0 , u ε t + ∇ P ε = Δ u ε + n ε ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u ε = 0 , x ∈ Ω , t > 0 , ∂ n ε ∂ ν = 0 , ∂ c ε ∂ ν = 0 , u ε = 0 , x ∈ ∂ Ω , t > 0 , n ε ( x , 0 ) = n 0 ( x ) , c ε ( x , 0 ) = c 0 ( x ) , u ε ( x , 0 ) = u 0 ( x ) , x ∈ Ω

possesses globally defined classical solutions:

Lemma 2.1

Let m > 1 . Suppose that (1.8)–(1.9) hold. Then for each ε ∈ ( 0 , 1 ) , there exists a global classical solution ( n ε , c ε , u ε , P ε ) to system (2.2) such that

n ε ∈ C 0 ( Ω ¯ × [ 0 , + ∞ ) ; R ) ∩ C 2 , 1 ( Ω ¯ × ( 0 , + ∞ ) ; R ) , c ε ∈ C 0 ( Ω ¯ × [ 0 , + ∞ ) ; R ) ∩ C 2 , 1 ( Ω ¯ × ( 0 , + ∞ ) ; R ) , u ε ∈ C 0 ( Ω ¯ × [ 0 , + ∞ ) ; R 3 ) ∩ C 2 , 1 ( Ω ¯ × ( 0 , + ∞ ) ; R 3 ) , P ε ∈ C 1 , 0 ( Ω ¯ × ( 0 , + ∞ ) ; R )

and that n ε and c ε are nonnegative in Ω ¯ × ( 0 , + ∞ ) . This solution is unique, up to addition of constants to P ε .

Proof

By means of standard point arguments, one can readily show the existence of a local-in-time smooth solution ( n ε , c ε , u ε , P ε ) , the nonnegativity in its first two components by the maximum principle, and the extension up to a maximal time T max , ε ∈ ( 0 , ∞ ] which in the case T max , ε < ∞ has the property that

(2.3) limsup t ↗ T max , ε ( ∥ n ε ( ⋅ , t ) ∥ C 2 ( Ω ¯ ) + ∥ c ε ( ⋅ , t ) ∥ C 2 ( Ω ¯ ) + ∥ u ε ( ⋅ , t ) ∥ C 2 ( Ω ¯ ) ) = ∞ .

It follows from (2.1) that for each fixed ε ∈ ( 0 , 1 ) , the function S ε ( x , n ε , c ε ) vanishes for all sufficiently large n ε . Then we can apply the well-known L p estimation techniques and methods from higher order regularity theories for scalar parabolic equations and the Stokes system to deduce that for each T > 0 there exists C ( ε , T ) > 0 such that

∥ n ε ( ⋅ , t ) ∥ C 2 ( Ω ¯ ) + ∥ c ε ( ⋅ , t ) ∥ C 2 ( Ω ¯ ) + ∥ u ε ( ⋅ , t ) ∥ C 2 ( Ω ¯ ) ≤ C ( ε , T ) for all t ∈ ( τ ε , T ˜ max , ε )

with τ ε ≔ 1 2 min { T , T max , ε } and T ˜ max , ε ≔ min { T , T max , ε } . This implies that (2.3) cannot hold unless T max , ε = + ∞ .□

In the rest of this article, we will denote ( n ε , c ε , u ε , P ε ) by the approximate solutions of system (2.2) in Ω × ( 0 , + ∞ ) . The positive constants C , C 1 , C 2 , … are independent of ε .

Some basic properties of these approximate solutions can be summarized as follows.

Lemma 2.2

For each ε ∈ ( 0 , 1 ) , we have

(2.4) ‖ n ε ( ⋅ , t ) ‖ L 1 ( Ω ) = ‖ n 0 ‖ L 1 ( Ω ) for all t > 0

and

(2.5) ‖ c ε ( ⋅ , t ) ‖ L ∞ ( Ω ) ≤ ‖ c 0 ‖ L ∞ ( Ω ) for all t > 0 .

Proof

The first identity directly results from an integration of the first equation in (2.2) over Ω , whereas the second inequality is a consequence of the parabolic maximum principle applied to the second equation in (2.2) due to n ε ≥ 0 .□

Let us also include in this preliminary section the following elementary estimates.

Lemma 2.3

(Lemma 3.4 in [33]) Let a > 0 , T > 0 and y ∈ C 0 ( [ 0 , T ) ) ∩ C 1 ( 0 , T ) be such that

y ′ ( t ) + a y ( t ) ≤ g ( t ) for all t ∈ ( 0 , T ) ,

where g ( t ) ≥ 0 and g ∈ L loc 1 ( R ) has the property that

1 τ ∫ ( t − τ ) + t g ( s ) d s ≤ b for all t ∈ ( 0 , T )

with some τ > 0 and b > 0 . Then

y ( t ) ≤ y ( 0 ) + b τ 1 − e a τ for all t ∈ ( 0 , T ) .

Lemma 2.4

(Lemma 4.2 in [17]) Assume that Ω is bounded and let w ∈ C 2 ( Ω ¯ ) satisfy ∂ w ∂ ν = 0 on ∂ Ω . Then we have

∂ ∣ ∇ w ∣ 2 ∂ ν ≤ 2 κ ∣ ∇ w ∣ 2 on ∂ Ω ,

where κ = κ ( Ω ) > 0 is upper bounded for the curvatures of ∂ Ω .

Lemma 2.5

(Proposition 4.22(ii), Theorem 4.24(i) in [10]) Let r ∈ ( 0 , ∞ ) and Ω ⊂ R 3 be a bounded domain with smooth boundary. Then the following properties hold:

W r + 1 2 , 2 ( Ω ) ↪ W r , 2 ( ∂ Ω ) ↪ L 2 ( ∂ Ω ) ,

where the latter is a compact embedding.

Lemma 2.6

(Lemma 3.8 in [28]) Suppose that Ω ⊂ R 3 is a bounded domain with smooth boundary, that q ≥ 1 , and that

(2.6) r ∈ [ 2 q + 2 , 4 q + 1 ] .

Then there exists C > 0 such that for any φ ∈ C 2 ( Ω ¯ ) fulfilling φ ⋅ ∂ φ ∂ ν = 0 on ∂ Ω we have

‖ ∇ φ ‖ L r ( Ω ) ≤ C ∥ ∣ ∇ φ ∣ q − 1 D 2 φ ∥ L 2 ( Ω ) 2 r − 6 ( 2 q − 1 ) r ‖ φ ‖ L ∞ ( Ω ) 6 q − r ( 2 q − 1 ) r + C ‖ φ ‖ L ∞ ( Ω ) .

Corollary 2.1

Let q ≥ 1 and r > 1 satisfy (2.6). Then there exists a positive constant C depending only on Ω and ‖ c 0 ‖ L ∞ ( Ω ) such that

‖ ∇ c ε ‖ L r ( Ω ) ≤ C ‖ ∣ ∇ c ε ∣ q − 1 D 2 c ε ‖ L 2 ( Ω ) 2 r − 6 ( 2 q − 1 ) r + C for all t > 0 .

Proof

A direct application of Lemma 2.6 to c ε and (2.5) gives the desired result.□

Lemma 2.7

(Lemma 2.10 in [28]) Let N = 3 and p ∈ [ 1 , ∞ ) and s ∈ [ 1 , ∞ ] be such that

s < 3 p 3 − p if p ≤ 3 , s ≤ ∞ if p > 3 .

Then for all K > 0 there exists C = C ( p , s , K , u 0 ) such that if for some ε , we have

‖ n ε ( ⋅ , t ) ‖ L p ( Ω ) ≤ K for all t > 0 ,

then

‖ ∇ u ε ( ⋅ , t ) ‖ L s ( Ω ) ≤ C for all t > 0 .

3 A priori estimates and global existence

The purpose of this section is to provide some fundamental a priori estimates for the approximate solutions. An analysis of the quasi-energy functional in (1.12) constitutes our main ingredient in an enhancement of bounds for n ε from the evident mass conservation, which will furthermore serve as a starting point for our induction argument.

3.1 A basic spatio-temporal estimate

As a useful preparation for the regularity of ( n ε , c ε , u ε ) , we first establish the following basic spatio-temporal estimate. Its derivation is the only place in this work where we need to require 1 < m ≤ 2 .

Lemma 3.1

Let 1 < m ≤ 2 . Then there exists a positive constant C depending only on Ω , m , C S , ϕ , and the initial data such that

(3.1) ∫ Ω ( ∣ ∇ c ε ( ⋅ , t ) ∣ 2 + ∣ u ε ( ⋅ , t ) ∣ 2 ) ≤ C for all t > 0 ,

(3.2) ∫ ( t − 1 ) + t ∫ Ω ( ∣ D 2 c ε ∣ 2 + n ε ∣ ∇ c ε ∣ 2 + ∣ ∇ u ε ∣ 2 ) ≤ C for all t > 0 .

Proof

We begin with applying the standard testing procedures and subtle estimates to gain a differential inequality. First, testing the first equation in (2.2) by ( 1 + ln n ε ) and using the integration by parts over Ω , we see from (1.8) that

(3.3) d d t ∫ Ω n ε ln n ε + ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 ≤ d d t ∫ Ω n ε ln n ε + ∫ Ω ( n ε + ε ) m − 1 n ε ∣ ∇ n ε ∣ 2 = ∫ Ω ∇ n ε ⋅ ( S ε ( x , n ε , c ε ) ⋅ ∇ c ε ) ≤ C S ∫ Ω ∣ ∇ n ε ∣ ∣ ∇ c ε ∣ .

Second, we apply the gradient operator ∇ to the both sides of the second equation in (2.2) and take the inner product with ∇ c ε to obtain

(3.4) 1 2 ( ∣ ∇ c ε ∣ 2 ) t + ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) = ∇ c ε ⋅ ∇ Δ c ε − ∇ c ε ⋅ ∇ ( n ε c ε ) ,

which together with the pointwise identities ∇ c ε ⋅ ∇ Δ c ε = 1 2 Δ ∣ ∇ c ε ∣ 2 − ∣ D 2 c ε ∣ 2 and ∇ c ε ⋅ ∇ ( n ε c ε ) = c ε ∇ n ε ⋅ ∇ c ε + n ε ∣ ∇ c ε ∣ 2 yields that

(3.5) d d t ∫ Ω ∣ ∇ c ε ∣ 2 + 2 ∫ Ω ∣ D 2 c ε ∣ 2 + ∫ Ω n ε ∣ ∇ c ε ∣ 2 = ∫ Ω Δ ∣ ∇ c ε ∣ 2 − 2 ∫ Ω ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) − 2 ∫ Ω c ε ∇ n ε ⋅ ∇ c ε = ∫ ∂ Ω ∂ ∣ ∇ c ε ∣ 2 ∂ ν − 2 ∫ Ω ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) − 2 ∫ Ω c ε ∇ n ε ⋅ ∇ c ε .

For the first term on the right-hand side of (3.5), Lemmas 2.4 and 2.5 entail that

∫ ∂ Ω ∂ ∣ ∇ c ε ∣ 2 ∂ ν ≤ C 1 ∫ ∂ Ω ∣ ∇ c ε ∣ 2 ≤ C 2 ‖ ∇ c ε ‖ W r + 1 2 , 2 ( Ω ) 2

for any fixed r ∈ 1 4 , 1 2 . It then follows from the Gagliardo-Nirenberg inequality and (2.5) that

(3.6) ∫ ∂ Ω ∂ ∣ ∇ c ε ∣ 2 ∂ ν ≤ C 3 ‖ D 2 c ε ‖ L 2 ( Ω ) 4 r ‖ c ε ‖ L ∞ ( Ω ) 2 − 4 r + C 3 ‖ c ε ‖ L ∞ ( Ω ) 2 ≤ C 3 ‖ D 2 c ε ‖ L 2 ( Ω ) 4 r ‖ c 0 ‖ L ∞ ( Ω ) 2 − 4 r + C 3 ‖ c 0 ‖ L ∞ ( Ω ) 2 ≤ 1 2 ∫ Ω ∣ D 2 c ε ∣ 2 + C 4 .

For the second term on the right-hand side of (3.5), since the integration by parts in conjunction with u ε = 0 on ∂ Ω and ∇ ⋅ u ε = 0 in Ω ensures that

− 2 ∫ Ω ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) = − 2 ∫ Ω ∇ c ε ⋅ ( ∇ u ε ⋅ ∇ c ε ) − 2 ∑ i , j = 1 3 ∫ Ω ∂ i c ε ∂ i j c ε u ε j = − 2 ∫ Ω ∇ c ε ⋅ ( ∇ u ε ⋅ ∇ c ε ) − ∫ Ω ∇ ∣ ∇ c ε ∣ 2 ⋅ u ε = − 2 ∫ Ω ∇ c ε ⋅ ( ∇ u ε ⋅ ∇ c ε ) − ∫ ∂ Ω ∣ ∇ c ε ∣ 2 u ε ⋅ ν + ∫ Ω ∣ ∇ c ε ∣ 2 ∇ ⋅ u ε = − 2 ∫ Ω ∇ c ε ⋅ ( ∇ u ε ⋅ ∇ c ε ) ,

we may use Hölder’s inequality and Lemma 2.1 to see that

(3.7) − 2 ∫ Ω ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) ≤ 2 ∥ ∣ ∇ c ε ∣ 2 ∥ L 2 ( Ω ) ‖ ∇ u ε ‖ L 2 ( Ω ) = 2 ‖ ∇ c ε ‖ L 4 ( Ω ) 2 ‖ ∇ u ε ‖ L 2 ( Ω ) ≤ C 5 ( ‖ D 2 c ε ‖ L 2 ( Ω ) + 1 ) ‖ ∇ u ε ‖ L 2 ( Ω ) ≤ 1 2 ∫ Ω ∣ D 2 c ε ∣ 2 + C 6 ∥ ∇ u ε ‖ L 2 ( Ω ) 2 + C 6 .

As for the third term on the right-hand side of (3.5), the boundedness (2.5) warrants that

(3.8) − 2 ∫ Ω c ε ∇ n ε ⋅ ∇ c ε ≤ 2 ‖ c ε ‖ L ∞ ( Ω ) ∫ Ω ∣ ∇ n ε ∣ ∣ ∇ c ε ∣ ≤ 2 ‖ c 0 ‖ L ∞ ( Ω ) ∫ Ω ∣ ∇ n ε ∣ ∣ ∇ c ε ∣ .

Substituting (3.6), (3.7), and (3.8) into (3.5), we obtain

(3.9) d d t ∫ Ω ∣ ∇ c ε ∣ 2 + ∫ Ω ∣ D 2 c ε ∣ 2 + 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 ≤ C 7 ∫ Ω ∣ ∇ n ε ∣ ∣ ∇ c ε ∣ + ‖ ∇ u ε ‖ L 2 ( Ω ) 2 + 1 .

Third, testing the third equation in (2.2) by u ε and integrating by parts, we obtain

1 2 d d t ∫ Ω ∣ u ε ∣ 2 + ∫ Ω ∣ ∇ u ε ∣ 2 = ∫ Ω n ε ∇ ϕ ⋅ u ε .

Since Sobolev’s embedding W 0 1 , 2 ( Ω ) ↪ L 4 ( Ω ) and Young’s inequality entail that

∫ Ω n ε ∇ ϕ ⋅ u ε ≤ ‖ ∇ ϕ ‖ L ∞ ( Ω ) ‖ n ε ‖ L 4 3 ( Ω ) ‖ u ε ‖ L 4 ( Ω ) ≤ C 8 ‖ ∇ ϕ ‖ L ∞ ( Ω ) ‖ n ε ‖ L 4 3 ( Ω ) ‖ ∇ u ε ‖ L 2 ( Ω ) ≤ 1 2 ‖ ∇ u ε ‖ L 2 ( Ω ) 2 + 1 2 C 8 ‖ ∇ ϕ ‖ L ∞ ( Ω ) 2 ‖ n ε ‖ L 4 3 ( Ω ) 2 ,

we then have

(3.10) d d t ∫ Ω ∣ u ε ∣ 2 + ∫ Ω ∣ ∇ u ε ∣ 2 ≤ C 2 ‖ n ε ‖ L 4 3 ( Ω ) 2 .

In view of the Gagliardo-Nirenberg inequality and the mass conservation (2.4), we see that

(3.11) C 2 ‖ n ε ‖ L 4 3 ( Ω ) 2 ≤ C 9 ‖ ∇ n ε m 2 ‖ L 2 ( Ω ) 3 3 m − 1 ‖ n ε m 2 ‖ L 2 m ( Ω ) 9 m − 4 3 m 2 − m + C 9 ‖ n ε m 2 ‖ L 2 m ( Ω ) 4 m = C 9 ‖ ∇ n ε m 2 ‖ L 2 ( Ω ) 3 3 m − 1 ‖ n ε ‖ L 1 ( Ω ) 9 m − 4 2 ( 3 m − 1 ) + C 9 ‖ n ε ‖ L 1 ( Ω ) 2 ≤ C 10 ‖ ∇ n ε m 2 ‖ L 2 ( Ω ) 3 3 m − 1 + C 10 ≤ 1 8 C 7 ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 + C 11

due to 3 3 m − 1 < 2 , and thus that

d d t ∫ Ω ∣ u ε ∣ 2 + ∫ Ω ∣ ∇ u ε ∣ 2 ≤ 1 8 C 7 ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 + C 11 ,

which combined with (3.3) and (3.9) yields that

d d t ∫ Ω n ε ln n ε + ∫ Ω ∣ ∇ c ε ∣ 2 + 2 C 7 ∫ Ω ∣ u ε ∣ 2 + 3 4 ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 + ∫ Ω ∣ D 2 c ε ∣ 2 + 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 + C 7 ∫ Ω ∣ ∇ u ε ∣ 2 ≤ C 12 ∫ Ω ∣ ∇ n ε ∣ ∣ ∇ c ε ∣ + C 12 ≤ 1 4 ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 + C 13 ∫ Ω n ε 2 − m ∣ ∇ c ε ∣ 2 + C 13 for all t > 0 .

Since 1 < m ≤ 2 , we can further use Hölder’s inequality and Corollary 2.1 to deduce that

C 13 ∫ Ω n ε 2 − m ∣ ∇ c ε ∣ 2 ≤ C 13 ∥ n ε ∣ ∇ c ε ∣ 2 ∥ L 1 ( Ω ) 2 − m ∥ ∣ ∇ c ε ∣ 2 ( m − 1 ) ∥ L 1 m − 1 ( Ω ) ≤ C 13 ∥ n ε ∣ ∇ c ε ∣ 2 ∥ L 1 ( Ω ) 2 − m ‖ ∇ c ε ‖ L 4 ( Ω ) 2 ( m − 1 ) ∣ Ω ∣ m − 1 2 ≤ C 14 ∫ Ω n ε ∣ ∇ c ε ∣ 2 2 − m ( ‖ D 2 c ε ‖ L 2 ( Ω ) m − 1 + 1 ) ≤ 3 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 + 1 2 ∫ Ω ∣ D 2 c ε ∣ 2 + C 15

and then obtain a differential inequality:

(3.12) d d t ∫ Ω n ε ln n ε + ∫ Ω ∣ ∇ c ε ∣ 2 + 2 C 7 ∫ Ω ∣ u ε ∣ 2 + 1 2 ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 + 1 2 ∫ Ω ∣ D 2 c ε ∣ 2 + 1 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 + C 7 ∫ Ω ∣ ∇ u ε ∣ 2 ≤ C 16 .

We now establish an ordinary differential inequality (ODI) for the energy-type functional

y ( t ) ≔ ∫ Ω n ε ln n ε + ∫ Ω ∣ ∇ c ε ∣ 2 + 2 C 7 ∫ Ω ∣ u ε ∣ 2 .

For this purpose, we first use the pointwise inequality x ln x ≤ 3 e x 4 3 for all x > 0 and (3.11) to obtain

∫ Ω n ε ln n ε ≤ 3 e ‖ n ε ‖ L 4 3 ( Ω ) 4 3 ≤ C 17 ‖ n ε ‖ L 4 3 ( Ω ) 2 + C 17 ≤ C 18 ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 + C 18 .

Next, it follows from Corollary 2.1 and Young’s inequality that

∫ Ω ∣ ∇ c ε ∣ 2 ≤ ‖ ∇ c ε ‖ L 4 ( Ω ) 2 ∣ Ω ∣ 1 2 ≤ ∫ Ω ∣ D 2 c ε ∣ 2 + C 19 .

Moreover, Poincaré’s inequality entails that

∫ Ω ∣ u ε ∣ 2 ≤ C 20 ∫ Ω ∣ ∇ u ε ∣ 2 .

The above three inequalities in combination with (3.12) yield that

(3.13) y ′ ( t ) + 1 C 21 y ( t ) + 1 C 21 h ( t ) ≤ C 22 for all t > 0 ,

where

h ( t ) ≔ ∫ Ω n ε m − 2 ∣ ∇ n ε ∣ 2 + ∫ Ω ∣ D 2 c ε ∣ 2 + ∫ Ω n ε ∣ ∇ c ε ∣ 2 + ∫ Ω ∣ ∇ u ε ∣ 2 .

By a comparison argument, the ODI (3.13) in particular entails that

y ( t ) ≤ C 23 ≔ max { y ( 0 ) , C 21 C 22 } for all t > 0 ,

which together with the pointwise inequality − x ln x ≤ 1 e for all x > 0 ensures that

∫ Ω ∣ ∇ c ε ∣ 2 + 2 C 7 ∫ Ω ∣ u ε ∣ 2 = y ( t ) − ∫ Ω n ε ln n ε ≤ C 23 + ∣ Ω ∣ e for all t > 0

and that

1 C 21 ∫ ( t − 1 ) + t h ( s ) d s ≤ C 22 − y ( t ) + y ( ( t − 1 ) + ) − 1 C 21 ∫ ( t − 1 ) + t y ( s ) d s ≤ C 22 + ∣ Ω ∣ e + C 23 + ∣ Ω ∣ C 21 e .

Here in the last inequality we also used the fact

− y ( t ) ≤ − ∫ Ω n ε ln n ε ≤ ∣ Ω ∣ e for all t > 0 .

This completes the proof of Lemma 3.1.□

3.2 L m -boundedness of n ε

The bound on ∫ ( t − 1 ) + t ∫ Ω n ε ∣ ∇ c ε ∣ 2 obtained in Lemma 3.1 will be a cornerstone of subsequent a priori estimates on the approximate solutions and of our induction argument.

Lemma 3.2

Let m > 1 . Then there exists a positive constant C depending only on Ω , m , C S , ϕ , and the initial data such that

(3.14) ∫ Ω ( n ε m ( ⋅ , t ) + ∣ ∇ c ε ∣ 2 ( ⋅ , t ) + ∣ u ε ∣ 2 ( ⋅ , t ) ) ≤ C and ∫ ( t − 1 ) + t ∫ Ω ( n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + ∣ ∇ u ε ∣ 2 ) ≤ C

for all t > 0 .

Proof

We begin with the case 1 < m ≤ 2 . Since S ε ( x , n ε , c ε ) vanishes whenever x ∈ ∂ Ω according to (2.1), we multiply the first equation in (2.2) by n ε m − 1 and integrate by parts over Ω to obtain

1 m d d t ∫ Ω n ε m + ( m − 1 ) ∫ Ω n ε m − 2 ( n ε + ε ) m − 1 ∣ ∇ n ε ∣ 2 = ( m − 1 ) ∫ Ω n ε m − 1 ∇ n ε ⋅ ( S ( x , n ε , c ε ) ∇ c ε ) .

Then using (1.8) and Young’s inequality, we have

1 m d d t ∫ Ω n ε m + ( m − 1 ) ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 ≤ ( m − 1 ) C S ∫ Ω n ε m − 1 ∣ ∇ n ε ∣ ∣ ∇ c ε ∣ ≤ m − 1 2 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + m − 1 2 C S 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 ,

which implies that

(3.15) d d t ∫ Ω n ε m + m ( m − 1 ) 2 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 ≤ C 1 ∫ Ω n ε ∣ ∇ c ε ∣ 2 .

In view of the Gagliardo-Nirenberg inequality, (2.4), and Young’s inequality, we see that

‖ n ε ‖ L m ( Ω ) m = n ε 2 m − 1 2 L 2 m 2 m − 1 ( Ω ) 2 m 2 m − 1 ≤ C 2 ∇ n ε 2 m − 1 2 L 2 ( Ω ) 3 ( m − 1 ) 3 m − 2 n ε 2 m − 1 2 L 2 2 m − 1 ( Ω ) 5 m − 3 ( 3 m − 2 ) ( 2 m − 1 ) + n ε 2 m − 1 2 L 2 2 m − 1 ( Ω ) 2 m 2 m − 1 = C 2 ∇ n ε 2 m − 1 2 L 2 ( Ω ) 3 ( m − 1 ) 3 m − 2 ‖ n ε ‖ L 1 ( Ω ) 5 m − 3 2 ( 3 m − 2 ) + ‖ n ε ‖ L 1 ( Ω ) m ≤ C 3 ∇ n ε 2 m − 1 2 L 2 ( Ω ) 3 ( m − 1 ) 3 m − 2 + C 3 ≤ m ( m − 1 ) 4 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + C 4

and thus that

d d t ∫ Ω n ε m + ∫ Ω n ε m + m ( m − 1 ) 4 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 ≤ C 1 ∫ Ω n ε ∣ ∇ c ε ∣ 2 + C 4 for all t > 0 .

By setting

y 1 ( t ) ≔ ∫ Ω n ε m and g 1 ( t ) ≔ C 1 ∫ Ω n ε ∣ ∇ c ε ∣ 2 + C 4 ,

we obtain

y 1 ′ ( t ) + y 1 ( t ) ≤ g 1 ( t ) for all t > 0 .

Thanks to (3.2), we have

∫ ( t − 1 ) + t g 1 ( s ) d s ≤ C 5 for all t > 0 ,

which together with Lemma 2.3 entails that

∫ Ω n ε m ( ⋅ , t ) = y 1 ( t ) ≤ C 6

holds for all t > 0 , which together with Lemma 3.1 gives (3.14) for 1 < m ≤ 2 .

We now turn to the case m > 2 . Following (3.5)–(3.7), (3.10), (3.15), and a variant of (3.11), we obtain

(3.16) d d t ∫ Ω n ε m + ∫ Ω ∣ ∇ c ε ∣ 2 + C 7 ∫ Ω ∣ u ε ∣ 2 + m ( m − 1 ) 4 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + ∫ Ω ∣ D 2 c ε ∣ 2 + C 7 ∫ Ω ∣ ∇ u ε ∣ 2 ≤ C 8 ∫ Ω n ε ∣ ∇ c ε ∣ 2 − 2 ∫ Ω c ε ∇ n ε ⋅ ∇ c ε + C 8 .

For the first term on the right-hand side of (3.16), we use Hölder’s inequality and Corollary 2.1 to obtain

C 8 ∫ Ω n ε ∣ ∇ c ε ∣ 2 ≤ C 8 ‖ n ε ‖ L 2 ( Ω ) ‖ ∇ c ε ‖ L 4 ( Ω ) 2 ≤ C 9 ‖ n ε ‖ L 2 ( Ω ) ( ‖ D 2 c ε ‖ L 2 ( Ω ) + 1 ) .

Since Gagliardo-Nirenberg inequality and the mass conservation (2.4) entail that

(3.17) ‖ n ε ‖ L 2 ( Ω ) = n ε 2 m − 1 2 L 4 2 m − 1 ( Ω ) 2 2 m − 1 ≤ C 10 ∇ n ε 2 m − 1 2 L 2 ( Ω ) 3 2 ( 3 m − 2 ) n ε 2 m − 1 2 L 2 2 m − 1 ( Ω ) 6 m − 5 2 ( 3 m − 2 ) ( 2 m − 1 ) + C 10 n ε 2 m − 1 2 L 2 2 m − 1 ( Ω ) 2 2 m − 1 = C 10 ∇ n ε 2 m − 1 2 L 2 ( Ω ) 3 2 ( 3 m − 2 ) ‖ n ε ‖ L 1 ( Ω ) 6 m − 5 4 ( 3 m − 2 ) + C 10 ‖ n ε ‖ L 1 ( Ω ) ≤ C 11 ∇ n ε 2 m − 1 2 L 2 ( Ω ) 3 2 ( 3 m − 2 ) + C 11 ,

we see from 3 2 ( 3 m − 2 ) < 1 due to m > 2 and Young’s inequality that

(3.18) C 8 ∫ Ω n ε ∣ ∇ c ε ∣ 2 ≤ C 9 C 11 ∇ n ε 2 m − 1 2 L 2 θ ( Ω ) 3 2 ( 3 m − 2 ) + 1 ( ‖ D 2 c ε ‖ L 2 ( Ω ) + 1 ) ≤ m ( m − 1 ) 16 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + 1 8 ∫ Ω ∣ D 2 c ε ∣ 2 + C 12 .

For the second term on the right-hand side of (3.16), since the integration by parts and Hölder’s inequality ensure that

− 2 ∫ Ω c ε ∇ n ε ⋅ ∇ c ε = 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 + 2 ∫ Ω n ε c ε Δ c ε ≤ 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 + 2 ‖ c ε ‖ L ∞ ( Ω ) ‖ n ε ‖ L 2 ( Ω ) ‖ Δ c ε ‖ L 2 ( Ω ) ,

we can use (2.5), (3.17), and (3.18) to deduce that

(3.19) − 2 ∫ Ω c ε ∇ n ε ⋅ ∇ c ε ≤ m ( m − 1 ) 16 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + 1 8 ∫ Ω ∣ D 2 c ε ∣ 2 + C 13 ∫ Ω n ε 2 + 1 8 ∫ Ω ∣ D 2 c ε ∣ 2 + C 13 ≤ m ( m − 1 ) 8 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + 1 4 ∫ Ω ∣ D 2 c ε ∣ 2 + C 14 .

We now substitute (3.18) and (3.19) into (3.16) to conclude that

d d t ∫ Ω n ε m + ∫ Ω ∣ ∇ c ε ∣ 2 + C 7 ∫ Ω ∣ u ε ∣ 2 + m ( m − 1 ) 16 ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 + 5 8 ∫ Ω ∣ D 2 c ε ∣ 2 + C 7 ∫ Ω ∣ ∇ u ε ∣ 2 ≤ C 15 .

Then a similar ODI argument as in the proof of Lemma 3.1 and of the case of 1 < m ≤ 2 gives (3.14).□

Corollary 3.1

Let m > 1 . Then there exist s > 3 2 and a positive constant C depending only on Ω , m , s , C S , ϕ , and the initial data such that

(3.20) ‖ ∇ u ε ( ⋅ , t ) ‖ L s ( Ω ) ≤ C for all t > 0 .

Proof

It is a direct consequence of Lemmas 2.7 and 3.2.□

3.3 L p -boundedness of n ε

In this subsection, we will establish a coupled estimate of the form ∫ Ω n ε p + ∫ Ω ∣ ∇ c ε ∣ 2 q for all p > 1 and q > 1 . We begin with two standard differential inequalities.

Lemma 3.3

Let m > 1 , p > 1 , and q > 1 . Then there exists a positive constant C depending on Ω , m , p , q , C S , ϕ , and the initial data such that

(3.21) d d t ∫ Ω n ε p + 3 p ( p − 1 ) 4 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 ≤ C ∫ Ω n ε p − m + 1 ∣ ∇ c ε ∣ 2

and

(3.22) 1 2 q d d t ∫ Ω ∣ ∇ c ε ∣ 2 q + q − 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 2 ) ∣ ∇ ∣ ∇ c ε ∣ 2 ∣ 2 + ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 ≤ − ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( n ε c ε ) + C .

Proof

Similar to the proof of (3.15), we have

1 p d d t ∫ Ω n ε p + ( p − 1 ) ∫ Ω n ε p − 2 ( n ε + ε ) m − 1 ∣ ∇ n ε ∣ 2 ≤ ( p − 1 ) C S ∫ Ω n ε p − 1 ∣ ∇ n ε ∣ ∣ ∇ c ε ∣ ≤ p − 1 4 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + ( p − 1 ) C S 2 ∫ Ω n ε p − m + 1 ∣ ∇ c ε ∣ 2 ,

which implies that (3.21) holds.

On the other hand, we can test equation (3.4) by ∣ ∇ c ε ∣ 2 ( q − 1 ) and use the pointwise identity ∇ c ε ⋅ ∇ Δ c ε = 1 2 Δ ∣ ∇ c ε ∣ 2 − ∣ D 2 c ε ∣ 2 to obtain

(3.23) 1 2 q d d t ∫ Ω ∣ ∇ c ε ∣ 2 q + q − 1 2 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 2 ) ∣ ∇ ∣ ∇ c ε ∣ 2 ∣ 2 + ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 = 1 2 ∫ ∂ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∂ ∣ ∇ c ε ∣ 2 ∂ ν − ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( n ε c ε ) − ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) .

For the first term on the right-hand side of (3.23), Lemmas 2.4 and 2.5 entail that

1 2 ∫ ∂ Ω ∣ ∇ c ε ∣ 2 q − 2 ∂ ∣ ∇ c ε ∣ 2 ∂ ν ≤ κ 2 ∫ ∂ Ω ∣ ∇ c ε ∣ 2 q ≤ C 2 ‖ ∣ ∇ c ε ∣ q ‖ W 3 4 , 2 ( Ω ) 2 .

It then follows from the Gagliardo-Nirenberg inequality and (3.14) that

C 2 ‖ ∣ ∇ c ε ∣ q ‖ W 3 4 , 2 ( Ω ) ≤ C 3 ‖ ∇ ∣ ∇ c ε ∣ q ‖ L 2 ( Ω ) 3 ( 2 q − 1 ) 2 ( 3 q − 1 ) ‖ ∣ ∇ c ε ∣ q ‖ L 2 q ( Ω ) 1 2 ( 3 q − 1 ) + ‖ ∣ ∇ c ε ∣ q ‖ L 2 q ( Ω ) = C 3 ‖ ∇ ∣ ∇ c ε ∣ q ‖ L 2 ( Ω ) 3 ( 2 q − 1 ) 2 ( 3 q − 1 ) ‖ ∇ c ε ‖ L 2 ( Ω ) 1 2 q ( 3 q − 1 ) + ‖ ∇ c ε ‖ L 2 ( Ω ) q ≤ C 4 ‖ ∇ ∣ ∇ c ε ∣ q ‖ L 2 ( Ω ) 3 ( 2 q − 1 ) 2 ( 3 q − 1 ) + C 4

and thus that

(3.24) 1 2 ∫ ∂ Ω ∣ ∇ c ε ∣ 2 q ∂ ∣ ∇ c ε ∣ 2 ∂ ν ≤ q − 1 8 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 2 ) ∣ ∇ ∣ ∇ c ε ∣ 2 ∣ 2 + C 5

by Young’s inequality. For the third term on the right-hand side of (3.23), we use the integration by parts twice, the divergence free condition on u ε and Hölder’s inequality to obtain

− ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) = − ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ u ε ⋅ ∇ c ε ≤ ∫ Ω ∣ ∇ c ε ∣ 2 q ∣ ∇ u ε ∣ ≤ ∥ ∣ ∇ c ε ∣ 2 q ∣ ∥ L s s − 1 ( Ω ) ‖ ∇ u ε ‖ L s ( Ω ) ,

where s > 3 2 is taken from Corollary 3.1. Noting that Gagliardo-Nirenberg inequality and (3.14) ensures that

∥ ∣ ∇ c ε ∣ 2 q ∥ L s s − 1 ( Ω ) = ‖ ∣ ∇ c ε ∣ q ‖ L 2 s s − 1 ( Ω ) 2 ≤ C 6 ‖ ∇ ∣ ∇ c ε ∣ q ‖ L 2 ( Ω ) 6 ( ( q − 1 ) s + 1 ) ( 3 q − 1 ) s ‖ ∣ ∇ c ε ∣ q ‖ L 2 q ( Ω ) 2 ( 2 s − 3 ) ( 3 q − 1 ) s + C 6 ‖ ∣ ∇ c ε ∣ q ‖ L 2 q ( Ω ) 2 = C 6 ‖ ∇ ∣ ∇ c ε ∣ q ‖ L 2 ( Ω ) 6 ( ( q − 1 ) s + 1 ) ( 3 q − 1 ) s ‖ ∇ c ε ‖ L 2 ( Ω ) 2 q ( 2 s − 3 ) ( 3 q − 1 ) s + C 6 ‖ ∇ c ε ‖ L 2 ( Ω ) 2 q ≤ C 7 ‖ ∇ ∣ ∇ c ε ∣ q ‖ L 2 ( Ω ) 6 ( ( q − 1 ) s + 1 ) ( 3 q − 1 ) s + C 7 ,

we can deduce from Young’s inequality and Corollary 3.1 that

(3.25) − ∫ Ω ∣ ∇ c ε ∣ 2 q ∇ c ε ⋅ ∇ ( u ε ⋅ ∇ c ε ) ≤ C 7 ‖ ∇ ∣ ∇ c ε ∣ q ‖ L 2 ( Ω ) 6 ( ( q − 1 ) s + 1 ) ( 3 q − 1 ) s + ‖ ∇ u ε ‖ L s ( Ω ) ≤ q − 1 8 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 2 ) ∣ ∇ ∣ ∇ c ε ∣ 2 ∣ 2 + C 8 .

Substituting (3.24) and (3.25) into (3.23), we complete the proof of (3.22).□

We now derive the boundedness of n ε and ∇ c ε under some presupposed bounds on n ε .

Lemma 3.4

Let m > 1 . Suppose that ι ≥ 1 satisfies 2 ( ι + 3 ) > 1 m − 1 and that

(3.26) ∫ Ω n ε ι ( m − 1 ) + 1 ≤ K for all t > 0

holds for some K > 0 . Then for all p > m + ι ( m − 1 ) and q > max { ι ( m − 1 ) , 1 } satisfying

(3.27) 3 q − ( 2 ι + 3 ) ( m − 1 ) + 1 < 3 p < 2 q + 2 ( ι + 3 ) ( m − 1 ) q + 3 ( m − 1 ) ,

there exists a positive constant C depending on Ω , m , ι , p , q , C S , ϕ , and the initial data such that

∫ Ω n ε p + ∫ Ω ∣ ∇ c ε ∣ 2 q ≤ C for all t > 0 .

Proof

It follows from Lemma 3.3 that

(3.28) d d t ∫ Ω n ε p + 1 2 q ∫ Ω ∣ ∇ c ε ∣ 2 q + 3 p ( p − 1 ) 4 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + q − 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 2 ) ∣ ∇ ∣ ∇ c ε ∣ 2 ∣ 2 + ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 ≤ C 1 ∫ Ω n ε p − m + 1 ∣ ∇ c ε ∣ 2 − ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( n ε c ε ) + C 1 .

To estimate the first term on the right-hand side of (3.28), we first use Hölder’s inequality to see

(3.29) C 1 ∫ Ω n ε p − m + 1 ∣ ∇ c ε ∣ 2 ≤ C 1 ‖ n ε p − m + 1 ‖ L q + 1 q ( Ω ) ∥ ∣ ∇ c ε ∣ 2 ∥ L q + 1 ( Ω ) .

Then for the first factor on the right-hand side of (3.29), Gagliardo-Nirenberg inequality and the assumption (3.26) ensure that

(3.30) ‖ n ε p − m + 1 ‖ L q + 1 q ( Ω ) = n ε p + m − 1 2 L 2 ( p − m + 1 ) ( q + 1 ) ( p + m − 1 ) q ( Ω ) 2 ( p − m + 1 ) p + m − 1 ≤ C 2 ∇ n ε p + m − 1 2 L 2 ( Ω ) 2 ( p − m + 1 ) p + m − 1 θ 1 n ε p + m − 1 2 L 2 ( ι ( m − 1 ) + 1 ) p + m − 1 ( Ω ) 2 ( p − m + 1 ) p + m − 1 ( 1 − θ 1 ) + C 2 n ε p + m − 1 2 L 2 ( ι ( m − 1 ) + 1 ) p + m − 1 ( Ω ) 2 ( p − m + 1 ) p + m − 1 = C 2 ∇ n ε p + m − 1 2 L 2 ( Ω ) 2 ( p − m + 1 ) p + m − 1 θ 1 ‖ n ε ‖ L ι ( m − 1 ) + 1 ( Ω ) ( p − m + 1 ) ( 1 − θ 1 ) + C 2 ‖ n ε ‖ L ι ( m − 1 ) + 1 ( Ω ) p − m + 1 ≤ C 3 ∇ n ε p + m − 1 2 L 2 ( Ω ) 2 ( p − m + 1 ) p + m − 1 θ 1 + C 3 ,

where

θ 1 ≔ 3 ( p + m − 1 ) ( ( p − m ) ( q + 1 ) − ι ( m − 1 ) q + 1 ) ( 3 ( p + m − 1 ) − ι ( m − 1 ) − 1 ) ( p − m + 1 ) ( q + 1 ) ∈ ( 0 , 1 )

due to p > m + ι ( m − 1 ) and q > 1 , while for the second factor on the right-hand side of (3.29), we see from Corollary 2.1 that

(3.31) C 1 ‖ ∣ ∇ c ε ∣ 2 ‖ L q + 1 ( Ω ) = C 1 ‖ ∇ c ε ‖ L 2 ( q + 1 ) ( Ω ) 2 ≤ C 4 ∥ ∣ ∇ c ε ∣ q − 1 D 2 c ε ∥ L 2 ( Ω ) 2 q + 1 + 1 .

Thus by inserting (3.30) and (3.31) into (3.29), we deduce that

C 1 ∫ Ω n ε p − m + 1 ∣ ∇ c ε ∣ 2 ≤ C 3 C 4 ‖ ∇ n ε p + m − 1 2 ‖ L 2 ( Ω ) 2 ( p − m + 1 ) p + m − 1 θ 1 + 1 ∥ ∣ ∇ c ε ∣ q − 1 D 2 c ε ∥ L 2 ( Ω ) 2 q + 1 + 1 ≤ 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 5 ‖ ∇ n ε p + m − 1 2 ‖ L 2 ( Ω ) 2 ( p − m + 1 ) ( q + 1 ) ( p + m − 1 ) q θ 1 + C 5 .

Since

( p − m + 1 ) ( q + 1 ) ( p + m − 1 ) q θ 1 = 3 ( ( p − m ) ( q + 1 ) − ι ( m − 1 ) q + 1 ) ( 3 ( p + m − 1 ) − ι ( m − 1 ) − 1 ) q < 1

due to 3 p < 2 q + 2 ( ι + 3 ) ( m − 1 ) q + 3 ( m − 1 ) , Young’s inequality entails that

(3.32) C 1 ∫ Ω n ε p − m + 1 ∣ ∇ c ε ∣ 2 ≤ p ( p − 1 ) 4 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 6 .

To estimate the second term on the right-hand side of (3.28), we use the integration by parts to obtain

− ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( n ε c ε ) = 2 ( q − 1 ) ∫ Ω n ε c ε ∣ ∇ c ε ∣ 2 ( q − 2 ) ( D 2 c ε ⋅ ∇ c ε ) ⋅ ∇ c ε + ∫ Ω n ε c ε ∣ ∇ c ε ∣ 2 ( q − 1 ) Δ c ε .

It then follows from ∣ Δ c ε ∣ ≤ 3 ∣ D 2 c ε ∣ , Lemma 2.2, Hölder’s inequality, and Corollary 2.1 that

− ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( n ε c ε ) ≤ ( 2 ( q − 1 ) + 3 ) ‖ c ε ‖ L ∞ ( Ω ) ∫ Ω n ε ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ ≤ 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 7 ∫ Ω n ε 2 ∣ ∇ c ε ∣ 2 ( q − 1 ) ≤ 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 7 ‖ n ε ‖ L q + 1 ( Ω ) 2 ‖ ∇ c ε ‖ L 2 ( q + 1 ) ( Ω ) 2 ( q − 1 ) ≤ 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 8 ‖ n ε ‖ L q + 1 ( Ω ) 2 ‖ ∇ c ε ∣ q − 1 D 2 c ε ‖ L 2 ( Ω ) 2 ( q − 1 ) q + 1 + 1 .

Since Gagliardo-Nirenberg inequality and the assumption (3.40) entail that

(3.33) ‖ n ε ‖ L q + 1 ( Ω ) 2 = ‖ n ε p + m − 1 2 ‖ L 2 ( q + 1 ) p + m − 1 ( Ω ) 4 p + m − 1 ≤ C 9 ‖ ∇ n ε p + m − 1 2 ‖ L 2 ( Ω ) 4 θ 2 p + m − 1 ‖ n ε p + m − 1 2 ‖ L 2 ( ι ( m − 1 ) + 1 ) p + m − 1 ( Ω ) 4 ( 1 − θ 2 ) p + m − 1 + C 9 ‖ n ε p + m − 1 2 ‖ L 2 ( ι ( m − 1 ) + 1 ) p + m − 1 ( Ω ) 4 p + m − 1 = C 9 ‖ ∇ n ε p + m − 1 2 ‖ L 2 ( Ω ) 4 θ 2 p + m − 1 ‖ n ε ‖ L ι ( m − 1 ) + 1 ( Ω ) 2 ( 1 − θ 2 ) + C 9 ‖ n ε ‖ L ι ( m − 1 ) + 1 ( Ω ) 2 ≤ C 10 ‖ ∇ n ε p + m − 1 2 ‖ L 2 ( Ω ) 4 θ 2 p + m − 1 + C 10 ,

where

θ 2 ≔ 3 ( p + m − 1 ) ( q − ι ( m − 1 ) ) ( 3 ( p + m − 1 ) − ι ( m − 1 ) − 1 ) ( q + 1 ) ∈ ( 0 , 1 )

due to q > ι ( m − 1 ) and 3 q − ( 2 ι + 3 ) ( m − 1 ) + 1 < 3 p . Noting that

2 θ 2 p + m − 1 + q − 1 q + 1 = 6 ( q − ι ( m − 1 ) ) ( 3 ( p + m − 1 ) − ι ( m − 1 ) − 1 ) ( q + 1 ) + q − 1 q + 1 < 2 q + 1 + q − 1 q + 1 = 1

again due to 3 q − ( 2 ι + 3 ) ( m − 1 ) + 1 < 3 p , we see from Young’s inequality that

(3.34) − ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( n ε c ε ) ≤ 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 8 C 10 ‖ ∇ n ε p + m − 1 2 ‖ L 2 ( Ω ) 4 θ 2 p + m − 1 + 1 ‖ ∇ c ε ∣ q − 1 D 2 c ε ‖ L 2 ( Ω ) 2 ( q − 1 ) q + 1 + 1 ≤ p ( p − 1 ) 4 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + 1 2 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 11 .

Substituting (3.32) and (3.34) into (3.28), we obtain

d d t ∫ Ω n ε p + 1 2 q ∫ Ω ∣ ∇ c ε ∣ 2 q + p ( p − 1 ) 4 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + q − 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 2 ) ∣ ∇ ∣ ∇ c ε ∣ 2 ∣ 2 + 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 ≤ C 12 .

Then similar to (3.33) and (3.31), we can deduce that

(3.35) ∫ Ω n ε p ≤ p ( p − 1 ) 4 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + C 13 and 1 2 q ∫ Ω ∣ ∇ c ε ∣ 2 q ≤ 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 14

and thus that

d d t ∫ Ω n ε p + 1 2 q ∫ Ω ∣ ∇ c ε ∣ 2 q + ∫ Ω n ε p + 1 2 q ∫ Ω ∣ ∇ c ε ∣ 2 q ≤ C 15 ,

which together with Lemma 2.3 yields the desired result.□

We will establish some dissipation estimates under some presupposed bounds on some derivatives of n ε . In particular, the bound for ∫ t t + 1 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q is one of the main novelties of this article.

Lemma 3.5

Let 1 < m ≤ 2 . Suppose that 1 < p ≤ 3 − m and q = p + m − 1 , and that

(3.36) ∫ ( t − 1 ) + t ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 ≤ K for all t > 0

holds for some K > 0 . Then there exists a positive constant C depending on Ω , m , p , q , C S , ϕ , and the initial data such that

(3.37) ∫ ( t − 1 ) + t ∫ Ω n ε ∣ ∇ c ε ∣ 2 q ≤ C for all t > 0 .

Proof

Since q = p + m − 1 , we can take a similar calculation as (3.31) to obtain

∫ Ω ∣ ∇ c ε ∣ 2 q ( p + m − 1 ) − 2 p + m − 2 = ∫ Ω ∣ ∇ c ε ∣ 2 ( q + 1 ) ≤ C 1 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + 1 .

Then Young’s inequality entails

∫ Ω c ε ∣ ∇ c ε ∣ 2 q − 1 ∣ ∇ n ε ∣ ≤ ‖ c ε ‖ L ∞ ( Ω ) ∫ Ω n ε p + m − 3 2 ∣ ∇ n ε ∣ ( n ε ∣ ∇ c ε ∣ 2 q ) 3 − ( p + m ) 2 ∣ ∇ c ε ∣ q ( p + m − 1 ) − 1 ≤ 1 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q + 1 2 C 1 ∫ Ω ∣ ∇ c ε ∣ 2 q ( p + m − 1 ) − 2 p + m − 2 + C 2 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 ≤ 1 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q + 1 2 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 2 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + 1 2

due to p + m − 3 ≤ 0 , which implies that

(3.38) − ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ c ε ⋅ ∇ ( n ε c ε ) = − ∫ Ω n ε ∣ ∇ c ε ∣ 2 q − ∫ Ω c ε ∣ ∇ c ε ∣ 2 ( q − 1 ) ∇ n ε ⋅ ∇ c ε ≤ − ∫ Ω n ε ∣ ∇ c ε ∣ 2 q + ∫ Ω c ε ∣ ∇ c ε ∣ 2 q − 1 ∣ ∇ n ε ∣ ≤ − 1 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q + 1 2 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + C 2 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + 1 2 .

Thus substituting (3.38) into (3.22) in Lemma 3.3, we deduce that

1 2 q d d t ∫ Ω ∣ ∇ c ε ∣ 2 q + q − 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 2 ) ∣ ∇ ∣ ∇ c ε ∣ 2 ∣ 2 + 1 2 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + 1 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q ≤ C 2 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + 1 2 .

By an estimate of the form (3.35), we conclude that

1 2 q d d t ∫ Ω ∣ ∇ c ε ∣ 2 q + 1 2 q ∫ Ω ∣ ∇ c ε ∣ 2 q + 1 4 ∫ Ω ∣ ∇ c ε ∣ 2 ( q − 1 ) ∣ D 2 c ε ∣ 2 + 1 2 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q ≤ C 2 ∫ Ω n ε p + m − 3 ∣ ∇ n ε ∣ 2 + C 3 ,

which together with Lemma 2.3 and the assumption (3.36) yields the desired (3.37).□

The following lemma is the foundation of our induction argument.

Lemma 3.6

Let 65 63 ≤ m < 3 2 . For all k ∈ Z + satisfying 1 ≤ k < 2 − m m − 1 , if

(3.39) ∫ Ω n ε k ( m − 1 ) + 1 ( ⋅ , t ) ≤ K and ∫ ( t − 1 ) + t ∫ Ω n ε ( k + 1 ) ( m − 1 ) − 1 ∣ ∇ n ε ∣ 2 ≤ K

hold for some K > 0 and all t > 0 , then there exists a positive constant C depending on Ω , m , k , C S , ϕ , K and the initial data such that

∫ Ω n ε ( k + 1 ) ( m − 1 ) + 1 ( ⋅ , t ) ≤ C and ∫ ( t − 1 ) + t ∫ Ω n ε ( k + 2 ) ( m − 1 ) − 1 ∣ ∇ n ε ∣ 2 ≤ C for all t > 0 .

Proof

For simplicity, we set p k ≔ k ( m − 1 ) + 1 . Assume that

(3.40) ∫ Ω n ε p k ( ⋅ , t ) ≤ K and ∫ ( t − 1 ) + t ∫ Ω ∣ ∇ n ε p k + 1 2 ∣ 2 ≤ K for all t > 0

are valid for some 1 ≤ k < 2 − m m − 1 and K > 0 . Then Lemma 3.5 with p ≔ p k ∈ ( 1 , 3 − m ) entails that

(3.41) ∫ ( t − 1 ) + t ∫ Ω n ε ∣ ∇ c ε ∣ 2 q k ≤ C 1 ,

where q k ≔ p k + 1 . We need to show that

(3.42) ∫ Ω n ε p k + 1 ( ⋅ , t ) ≤ C 2 and ∫ ( t − 1 ) + t ∫ Ω ∣ ∇ n ε p k + 2 2 ∣ 2 ≤ C 2 for all t > 0

hold for some C 2 > 0 .

For this purpose, we first invoke Lemma 3.3 with p ≔ p k + 1 to see that

(3.43) d d t ∫ Ω n ε p k + 1 + C 3 ∫ Ω ∣ ∇ n ε p k + 2 2 ∣ 2 ≤ C 4 ∫ Ω n ε p k ∣ ∇ c ε ∣ 2 .

Since Gagliardo-Nirenberg inequality and the assumption (3.40) ensure that

∫ Ω n ε p k q k − 1 q k − 1 = n ε p k + 2 2 L 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) ( Ω ) 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) ≤ C 5 ∇ n ε p k + 2 2 L 2 ( Ω ) 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) θ k n ε p k + 2 2 L 2 p k p k + 2 ( Ω ) 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) ( 1 − θ k ) + C 5 n ε p k + 2 2 L 2 p k p k + 2 ( Ω ) 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) = C 5 ∇ n ε p k + 2 2 L 2 ( Ω ) 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) θ k ‖ n ε ‖ L p k ( Ω ) p k q k − 1 q k − 1 ( 1 − θ k ) + C 5 ‖ n ε ‖ L p k ( Ω ) p k q k − 1 q k − 1 ≤ C 6 ∇ n ε p k + 2 2 L 2 ( Ω ) 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) θ k + C 6 ,

where

θ k ≔ 3 p k + 2 ( p k − 1 ) ( 3 p k + 2 − p k ) ( p k q k − 1 ) = 3 p k + 2 ( p k − 1 ) 2 p k + 3 ( p k q k − 1 ) ∈ ( 0 , 1 )

due to

2 p k + 3 ( p k q k − 1 ) − 3 p k + 2 ( p k − 1 ) = ( 2 p k 2 + 8 ( m − 1 ) p k − 3 p k + 6 ( m − 1 ) 2 − 6 ( m − 1 ) + 1 ) p k = ( 2 ( m − 1 ) k 2 + ( 8 m − 7 ) k + 6 m − 4 ) ( m − 1 ) p k > 0 ,

we can deduce from (3.43) and Young’s inequality that

d d t ∫ Ω n ε p k + 1 + C 3 ∫ Ω ∣ ∇ n ε p k + 2 2 ∣ 2 ≤ C 4 ∫ Ω n ε p k − 1 q k ( n ε ∣ ∇ c ε ∣ 2 q k ) 1 q k ≤ C 3 2 C 6 ∫ Ω n ε p k q k − 1 q k − 1 + C 7 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q k ≤ C 3 2 ∥ ∇ n ε p k + 2 2 ∥ L 2 ( Ω ) 2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) θ k + C 8 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q k + C 8 .

Noting that

2 ( p k q k − 1 ) p k + 2 ( q k − 1 ) θ k = 3 ( p k − 1 ) p k + 3 ( q k − 1 ) ≤ 2

after a serious of basic calculations,^[3] we use Young’s inequality again to obtain

d d t ∫ Ω n ε p k + 1 + C 3 2 ∫ Ω ∣ ∇ n ε p k + 2 2 ∣ 2 ≤ C 8 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q k + C 9 .

Then invoking the Gagliardo-Nirenberg inequality to find positive constants C 9 , C 10 , and C 11 fulfilling

∫ Ω n ε p k + 1 = n ε p k + 2 2 L 2 p k + 1 p k + 2 ( Ω ) 2 p k + 1 p k + 2 ≤ C 10 ∇ n ε p k + 2 2 L 2 ( Ω ) 6 ( p k + 1 − p k ) 3 p k + 2 − p k n ε p k + 2 2 L 2 p k p k + 2 ( Ω ) 2 p k ( 3 p k + 2 − p k + 1 ) p k + 2 ( 3 p k + 2 − p k ) + C 10 n ε p k + 2 2 L 2 p k p k + 2 ( Ω ) 2 p k + 1 p k + 2 = C 10 ∇ n ε p k + 2 2 L 2 ( Ω ) 6 ( p k + 1 − p k ) 3 p k + 2 − p k ‖ n ε ‖ L p k ( Ω ) p k ( 3 p k + 2 − p k + 1 ) 3 p k + 2 − p k + C 10 n ε p k + 2 2 L p k ( Ω ) p k + 1 ≤ C 11 ∇ n ε p k + 2 2 L 2 ( Ω ) 6 ( p k + 1 − p k ) 3 p k + 2 − p k + C 11 ≤ C 3 4 ∫ Ω ∣ ∇ n ε p k + 2 2 ∣ 2 + C 12 ,

we conclude that

d d t ∫ Ω n ε p k + 1 + ∫ Ω n ε p k + 1 + C 3 4 ∫ Ω ∣ ∇ n ε p k + 2 2 ∣ 2 ≤ C 8 ∫ Ω n ε ∣ ∇ c ε ∣ 2 q k + C 9 + C 12 ,

which together with (3.41) and Lemma 2.3 entails that the estimates (3.42) are valid. This completes the proof of Lemma 3.6.□

Corollary 3.2

Let 65 63 ≤ m < 3 2 . For all k ∈ Z + satisfying 1 ≤ k < 2 − m m − 1 , there exists a positive constant C depending on Ω , m , k , C S , ϕ , and the initial data such that

(3.44) ∫ Ω n ε ( k + 1 ) ( m − 1 ) + 1 ( ⋅ , t ) ≤ C for all t > 0 .

Proof

For any fixed m ∈ 65 63 , 3 2 , we will prove the conclusion by an induction on k . According to Lemma 3.2, we have

∫ Ω n ε m ≤ C 1 and ∫ ( t − 1 ) + t ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 ≤ C 1 ,

which implies the assumption (3.39) with k = 1 in Lemma 3.6 is valid and entails that

∫ Ω n ε 2 ( m − 1 ) + 1 ≤ C 2 and ∫ ( t − 1 ) + t ∫ Ω n ε 2 ( m − 1 ) + m − 2 ∣ ∇ n ε ∣ 2 ≤ C 2 .

This yields the desired conclusion (3.44) for k = 1 .

Assuming now that (3.44) is valid for some positive integer k satisfying 1 ≤ k < 2 − m m − 1 , that is,

∫ Ω n ε k ( m − 1 ) + 1 ≤ C 3 and ∫ ( t − 1 ) + t ∫ Ω n ε k ( m − 1 ) + m − 2 ∣ ∇ n ε ∣ 2 ≤ C 3 .

Invoking Lemma 3.6 again, we deduce that

∫ Ω n ε ( k + 1 ) ( m − 1 ) + 1 ≤ C 4 and ∫ ( t − 1 ) + t ∫ Ω n ε ( k + 1 ) ( m − 1 ) + m − 2 ∣ ∇ n ε ∣ 2 ≤ C 4 .

This completes the proof of Corollary 3.2.□

We now derive the boundedness of n ε and ∇ c ε without any presupposed bounds.

Lemma 3.7

Let m ≥ 65 63 . Then for all p > 1 and q > 1 , there exists a positive constant C depending only on Ω , m , p , q , C S , ϕ , and the initial data such that

(3.45) ∫ Ω n ε p + ∫ Ω ∣ ∇ c ε ∣ 2 q ≤ C for all t > 0 .

Proof

It follows from Lemma 3.2 that

(3.46) ∫ Ω n ε m ≤ C 1 and ∫ ( t − 1 ) + t ∫ Ω n ε 2 m − 3 ∣ ∇ n ε ∣ 2 ≤ C 1

for all m > 1 . In particular, when m > 9 8 , we first set ι ∗ ≔ 1 , which implies that 2 ( ι ∗ + 3 ) > 1 m − 1 . Then for any p > 2 m − 1 and q > max { m − 1 , 1 } satisfying

3 q − 5 m + 6 < 3 p < ( 8 m − 6 ) q + 3 m − 3 ,

Lemma 3.4 with ι = ι ∗ entails that

∫ Ω n ε p + ∫ Ω ∣ ∇ c ε ∣ 2 q ≤ C 2 for all t > 0 ,

which together with Hölder’s inequality yields (3.45) for all p > 1 and q > 1 .

On the other hand, when m ∈ 65 63 , 9 8 ⊂ 65 63 , 3 2 , Corollary 3.2 ensures that

∫ Ω n ε ( k + 1 ) ( m − 1 ) + 1 ( ⋅ , t ) ≤ C 3

for all t > 0 and k ∈ Z + satisfying k < 2 − m m − 1 and thus that

∫ Ω n ε ι ∗ ( m − 1 ) + 1 ( ⋅ , t ) ≤ C 4 for all t > 0 ,

where ι ∗ is the greatest integer less than 1 m − 1 . Noting that 2 ( ι ∗ + 3 ) > 1 m − 1 , we deduce from Lemma 3.4 with ι = ι ∗ that

∫ Ω n ε p + ∫ Ω ∣ ∇ c ε ∣ 2 q ≤ C 2 for all t > 0

for all p > m + ι ∗ ( m − 1 ) and q > max { ι ∗ ( m − 1 ) , 1 } satisfying

3 q − ( 2 ι ∗ + 3 ) ( m − 1 ) + 1 < 3 p < 2 q + 2 ( ι ∗ + 3 ) ( m − 1 ) q + 3 ( m − 1 ) .

A direct application of Hölder’s inequality yields (3.45) for all p > 1 and q > 1 . This completes the proof of Lemma 3.7.□

3.4 W 1 , ∞ -boundedness of u ε

Our next result is the W 1 , ∞ -boundedness of u ε derived from the regularity of n ε .

Lemma 3.8

Let m ≥ 65 63 . Then there exists a positive constant C depending only on Ω , m , C S , ϕ , and the initial data such that

(3.47) ‖ u ε ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) ≤ C for all t > 0 .

Proof

Combining Lemmas 3.7 with 2.7, we obtain the desired estimate.□

3.5 W 1 , ∞ -boundedness of c ε

With the boundedness of n ε and u ε at hand, a further boundedness of c ε can now be obtained by essentially straightforward argument of a standard regularization feature of the Neumann heat semigroup.

Lemma 3.9

Let m ≥ 65 63 . Then there exists a positive constant C depending only on Ω , m , C S , ϕ , and the initial data such that

‖ c ε ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) ≤ C for all t > 0 .

Proof

The L ∞ -boundedness of c ε follows from (2.5) and thus we only need to consider the boundedness of ∇ c ε . Applying ∇ to both sides of the representation formula

c ε ( ⋅ , t ) = e − t Δ c 0 − ∫ 0 t e − ( t − s ) Δ ( n ε c ε + u ε ⋅ ∇ c ε ) d s for all t > 0 ,

and recalling the known regularization properties of the homogeneous Neumann heat semigroup, we obtain that for any fixed q > 3 ,

‖ ∇ c ε ( ⋅ , t ) ‖ L ∞ ( Ω ) ≤ ‖ e − t Δ ∇ c 0 ‖ L ∞ ( Ω ) + ∫ 0 t ‖ ∇ e − ( t − s ) Δ ( n ε c ε + u ε ⋅ ∇ c ε ) ‖ L ∞ ( Ω ) d s ≤ ‖ ∇ c 0 ‖ L ∞ ( Ω ) + C 1 ∫ 0 t 1 + ( t − s ) − 1 2 − 3 2 q e − μ ( t − s ) ( ‖ n ε c ε ‖ L q ( Ω ) + ‖ u ε ⋅ ∇ c ε ‖ L q ( Ω ) ) d s

with some positive constant μ . Since

‖ n ε c ε ‖ L q ( Ω ) ≤ ‖ n ε ‖ L q ( Ω ) ‖ c ε ‖ L ∞ ( Ω ) ≤ ‖ n ε ‖ L q ( Ω ) ‖ c 0 ‖ L ∞ ( Ω ) ≤ C 2

by Lemma 3.7 and (2.5), and

‖ u ε ⋅ ∇ c ε ‖ L q ( Ω ) ≤ ‖ u ε ‖ L ∞ ( Ω ) ‖ ∇ c ε ‖ L q ( Ω ) ≤ C 3 ‖ ∇ c ε ‖ L ∞ ( Ω ) q − 2 q ‖ ∇ c ε ‖ L 2 ( Ω ) 2 q ≤ C 4 ‖ ∇ c ε ‖ L ∞ ( Ω ) q − 2 q

by (3.47), we further infer that

‖ ∇ c ε ( ⋅ , t ) ‖ L ∞ ( Ω ) ≤ ‖ ∇ c 0 ‖ L ∞ ( Ω ) + C 1 ∫ 0 t 1 + ( t − s ) − 1 2 − 3 2 q e − μ ( t − s ) C 2 + C 4 ‖ ∇ c ε ‖ L ∞ ( Ω ) q − 2 q d s ≤ C 5 + C 5 ∫ 0 t 1 + ( t − s ) − 1 2 − 3 2 q e − μ ( t − s ) d s + C 5 ∫ 0 t 1 + ( t − s ) − 1 2 − 3 2 q e − μ ( t − s ) ‖ ∇ c ε ‖ L ∞ ( Ω ) q − 2 q d s ≤ C 6 + C 5 ∫ 0 t 1 + ( t − s ) − 1 2 − 3 2 q e − μ ( t − s ) ‖ ∇ c ε ‖ L ∞ ( Ω ) q − 2 q d s

due to q > 3 . Then introducing the finite number

N ( T ) ≔ sup t ∈ ( 0 , T ) ‖ ∇ c ε ( ⋅ , t ) ‖ L ∞ ( Ω )

for each T > 0 , we have

‖ ∇ c ε ( ⋅ , t ) ‖ L ∞ ( Ω ) ≤ C 6 + C 5 N ( T ) q − 2 q ∫ 0 t 1 + ( t − s ) − 1 2 − 3 2 q e − μ ( t − s ) d s ≤ C 6 + C 5 N ( T ) q − 2 q ∫ 0 t 1 + s − 1 2 − 3 2 q e − μ s d s ≤ C 6 + C 7 N ( T ) q − 2 q for all t ∈ ( 0 , T ) ,

which in combination with Young’s inequality entails that

N ( T ) ≤ C 8 + 1 2 N ( T ) .

This implies the desired result by the arbitrariness of T .□

3.6 L ∞ -boundedness of n ε

Invoking a Moser-type iteration developed by [19] in conjunction with standard parabolic regularity arguments, we can now establish the L ∞ -boundedness of n ε .

Lemma 3.10

Let m ≥ 65 63 . Then there exists a positive constant C depending only on Ω , m , C S , ϕ , and the initial data such that

‖ n ε ( ⋅ , t ) ‖ L ∞ ( Ω ) ≤ C for all t > 0 .

Proof

We rewrite equation ( 2.2 ) 1 as the form

n ε t = ∇ ⋅ ( ( n ε + ε ) m − 1 ∇ n ε ) − ∇ ⋅ ( n ε S ε ( x , n ε , c ε ) ∇ c ε + u ε n ε ) ≔ ∇ ⋅ ( ( n ε + ε ) m − 1 ∇ n ε ) − ∇ ⋅ g ε ( x , t ) .

Thanks to Lemmas 3.7, 3.8, and 3.9, we see that

‖ g ε ( ⋅ , t ) ‖ L 6 ( Ω ) ≤ C S ‖ n ε ‖ L 6 ( Ω ) ‖ ∇ c ε ‖ L ∞ ( Ω ) + ‖ u ε ‖ L ∞ ( Ω ) ‖ n ε ‖ L 6 ( Ω ) ≤ C 1 for all t > 0 .

It then follows from Lemma A.1 in Tao and Winkler [19] that our desired L ∞ -boundedness holds. This completes the proof of Lemma 3.10.□

3.7 Regularity properties of time derivatives

In order to provide some compactness and equicontinuity properties of n ε , we further derive two statements on time regularity of n ε in a straightforward manner.

Lemma 3.11

Let m ≥ 65 63 . Then there exists a positive constant C depending only on Ω , m , C S , ϕ , and the initial data such that

(3.48) ‖ n ε t ( ⋅ , t ) ‖ ( W 0 2 , 2 ( Ω ) ) ∗ ≤ C for all t > 0

and in particular that

(3.49) ‖ n ε ( ⋅ , t ) − n ε ( ⋅ , s ) ‖ ( W 0 2 , 2 ( Ω ) ) ∗ ≤ C ∣ t − s ∣ for all t > 0 , s > 0 .

Proof

For any fixed ψ ∈ W 0 2 , 2 ( Ω ) , we multiply the first equation in (2.2) by ψ and integrate by parts to obtain

∫ Ω n ε t ψ = ∫ Ω n ε u ε ⋅ ∇ ψ + 1 m ∫ Ω ( n ε + ε ) m Δ ψ + ∫ Ω n ε ∇ ψ ⋅ ( S ε ( x , n ε , c ε ) ∇ c ε ) ,

which together with Hölder’s inequality, and Lemmas 3.8, 3.9, and 3.10 yields that

(3.50) ∫ Ω n ε t ψ ≤ ‖ n ε ‖ L ∞ ( Ω ) ‖ u ε ‖ L ∞ ( Ω ) ∣ Ω ∣ 1 2 ‖ ∇ ψ ‖ L 2 ( Ω ) + 1 m ‖ n ε + 1 ‖ L ∞ ( Ω ) m ∣ Ω ∣ 1 2 ‖ Δ ψ ‖ L 2 ( Ω ) + C S ‖ n ε ‖ L ∞ ( Ω ) ‖ ∇ c ε ‖ L 2 ( Ω ) ‖ ∇ ψ ‖ L 2 ( Ω ) ≤ C 1 ‖ ψ ‖ W 0 2 , 2 ( Ω ) .

This implies (3.48). Moreover, integrating (3.50) from t to s yields (3.49). This completes the proof of Lemma 3.11.□

Lemma 3.12

Let m ≥ 65 63 . Then for each T > 0 and γ > 1 , there exists a positive constant C depending only on T , γ , Ω , m , C S , ϕ , and the initial data such that

∫ 0 T ‖ ∂ t ( n ε + ε ) γ ‖ ( W 0 2 , 2 ( Ω ) ) ∗ d t ≤ C , ∫ 0 T ∫ Ω ∣ ∇ ( n ε + ε ) γ ∣ 2 ≤ C .

Proof

For any fixed p > 1 , multiplying the first equation in (2.2) by ( n ε + ε ) p − 1 and integrating by parts, we first obtain

1 p d d t ∫ Ω ( n ε + ε ) p + 4 ( p − 1 ) ( p + m − 1 ) 2 ∫ Ω ∣ ∇ ( n ε + ε ) p + m − 1 2 ∣ 2 = ( p − 1 ) ∫ Ω ( n ε + ε ) p − 2 n ε ∇ n ε ⋅ ( S ε ( x , n ε , c ε ) ∇ c ε ) ≤ 2 ( p − 1 ) ( p + m − 1 ) 2 ∫ Ω ∣ ∇ ( n ε + ε ) p + m − 1 2 ∣ 2 + C 1 ∫ Ω ( n ε + ε ) p − m + 1 ∣ ∇ c ε ∣ 2 ,

which together with Lemmas 3.8, 3.9, and 3.10 implies

1 p d d t ∫ Ω ( n ε + ε ) p + 2 ( p − 1 ) ( p + m − 1 ) 2 ∫ Ω ∣ ∇ ( n ε + ε ) p + m − 1 2 ∣ 2 ≤ C 1 ‖ n ε + ε ‖ L ∞ ( Ω ) p − m + 1 ‖ ∇ c ε ‖ L 2 ( Ω ) 2 ≤ C 2 .

Then we integrate from 0 to T to obtain

(3.51) ∫ 0 T ∫ Ω ∣ ∇ ( n ε + ε ) p + m − 1 2 ∣ 2 ≤ C 4 .

For any fixed ψ ∈ W 0 2 , 2 ( Ω ) , we now test the first equation in (2.2) by ( n ε + ε ) γ − 1 ψ and integrate by parts to obtain

1 γ ∫ Ω ∂ t ( n ε + ε ) γ ψ = ∫ Ω ( n ε + ε ) γ − 1 ψ ∇ ⋅ ( ( n ε + ε ) m − 1 ∇ n ε − n ε S ε ( x , n ε , c ε ) ∇ c ε − u ε n ε ) = − ( γ − 1 ) ∫ Ω ( n ε + ε ) γ + m − 3 ∣ ∇ n ε ∣ 2 ψ − ∫ Ω ( n ε + ε ) γ + m − 2 ∇ n ε ⋅ ∇ ψ + ( γ − 1 ) ∫ Ω ( n ε + ε ) γ − 2 n ε ∇ n ε ⋅ ( S ε ( x , n ε , c ε ) ∇ c ε ) ψ + ∫ Ω ( n ε + ε ) γ − 1 n ε ( S ε ( x , n ε , c ε ) ∇ c ε ) ⋅ ∇ ψ + 1 γ ∫ Ω ( n ε + ε ) γ u ε ⋅ ∇ ψ .

It follows from Lemmas 3.8, 3.9, and 3.10 that

1 γ ∫ Ω ∂ t ( n ε + ε ) γ ψ ≤ C 1 ∇ ( n ε + ε ) γ + m − 1 2 L 2 ( Ω ) 2 ‖ ψ ‖ L ∞ ( Ω ) + ∥ ∇ ( n ε + ε ) γ + m − 1 ∥ L 2 ( Ω ) ‖ ∇ ψ ‖ L 2 ( Ω ) + ∥ ∇ ( n ε + ε ) γ ∥ L 2 ( Ω ) ‖ ∇ c ε ‖ L 2 ( Ω ) ‖ ψ ‖ L ∞ ( Ω ) + ‖ n ε + ε ‖ L ∞ ( Ω ) γ ‖ ∇ c ε ‖ L 2 ( Ω ) ‖ ∇ ψ ‖ L 2 ( Ω ) + ‖ n ε + ε ‖ L ∞ ( Ω ) γ ‖ u ε ‖ L 2 ( Ω ) ‖ ∇ ψ ‖ L 2 ( Ω ) ) ≤ C 2 ∇ ( n ε + ε ) γ + m − 1 2 L 2 ( Ω ) 2 + ∥ ∇ ( n ε + ε ) γ + m − 1 ∥ L 2 ( Ω ) + ∥ ∇ ( n ε + ε ) γ ∥ L 2 ( Ω ) + 1 ‖ ψ ‖ W 0 2 , 2 ( Ω ) ≤ C 3 ∇ ( n ε + ε ) γ + m − 1 2 L 2 ( Ω ) 2 + ∥ ∇ ( n ε + ε ) γ + m − 1 ∥ L 2 ( Ω ) 2 + ∥ ∇ ( n ε + ε ) γ ∥ L 2 ( Ω ) 2 + 1 ‖ ψ ‖ W 0 2 , 2 ( Ω )

for all t ∈ ( 0 , T ) . This implies that

∫ 0 T ‖ ∂ t ( n ε + ε ) γ ‖ ( W 0 2 , 2 ( Ω ) ) ∗ d t ≤ C 3 ∫ 0 T ∇ ( n ε + ε ) γ + m − 1 2 L 2 ( Ω ) 2 + ∥ ∇ ( n ε + ε ) γ + m − 1 ∥ L 2 ( Ω ) 2 + ∥ ∇ ( n ε + ε ) γ ∥ L 2 ( Ω ) 2 + 1 d t .

We only need to take p = γ , 2 γ + m − 1 , and 2 γ − m + 1 in (3.51) to conclude boundedness of the right-hand side. This completes the proof of Lemma 3.12.□

3.8 Uniform Hölder regularity properties

As one further class of a priori estimates, we finally state a straightforward consequence of Lemmas 3.8, 3.9, and 3.10 for uniform Hölder regularity properties of c ε , ∇ c ε , and u ε .

Lemma 3.13

Let m ≥ 65 63 . Then there exist α ∈ ( 0 , 1 ) and a positive constant C depending only on Ω , m , C S , ϕ , and the initial data such that

(3.52) ‖ c ε ‖ C α , α 2 ( Ω ¯ × [ t , t + 1 ] ) ≤ C and ‖ u ε ‖ C α , α 2 ( Ω ¯ × [ t , t + 1 ] ) ≤ C for all t > 0 .

Also for each τ > 0 , there exists a positive constant C ( τ ) such that

(3.53) ‖ ∇ c ε ‖ C α , α 2 ( Ω ¯ × [ t , t + 1 ] ) ≤ C ( τ ) for all t > τ .

Proof

According to the embeddings W 1 , q ( Ω ) ↪ C 1 − 3 q ( Ω ¯ ) for q > 3 and D ( A β ) ↪ C 2 β − 3 2 ( Ω ¯ ) for β ∈ ( 3 4 , 1 ) , we obtain the spatio-temporal Hölder continuity (3.52) from Lemmas 3.8 and 3.9.

For (3.53), we rewrite equation ( 2.2 ) 2 as

c ε t − Δ c ε = − u ε ⋅ ∇ c ε − n ε c ε ≔ f ε ( x , t ) .

Noting that f ε ( x , t ) is uniformly bounded in L ∞ ( Ω × ( 0 , ∞ ) ) from Lemmas 3.8, 3.9, and 3.10, we can use the standard parabolic regularity theory [13] to show (3.53).□

3.9 Global existence: Proof of Theorem 1.1

To show the global existence of weak solutions to systems (1.5)–(1.7), we now make use of the above a priori estimates and extract suitable subsequences in a standard manner.

Lemma 3.14

Let m ≥ 65 63 and ( n ε , c ε , u ε , P ε ) be solutions to system (2.2) in Ω × ( 0 , + ∞ ) . Then there exist a subsequence { ε j } j = 1 ∞ and a triple of functions ( n , c , u ) such that ε = ε j ↘ 0 and

(3.54) n ε ⇀ * n in L ∞ ( Ω × ( 0 , ∞ ) ) ,

(3.55) n ε → n a.e. in Ω × ( 0 , ∞ ) ,

(3.56) n ε → n in C loc 0 ( [ 0 , ∞ ) ; ( W 0 2 , 2 ( Ω ) ) ∗ ) ,

(3.57) c ε ⇀ * c in L ∞ ( ( 0 , ∞ ) ; W 1 , ∞ ( Ω ) ) ,

(3.58) c ε → c in C loc 0 ( Ω ¯ × [ 0 , ∞ ) ) ,

(3.59) ∇ c ε → ∇ c in L loc 2 ( Ω ¯ × ( 0 , ∞ ) ) ,

(3.60) u ε ⇀ * u in L ∞ ( Ω × ( 0 , ∞ ) ) ,

(3.61) u ε → u in C loc 0 ( Ω ¯ × [ 0 , ∞ ) ) ,

(3.62) ∇ u ε → ∇ u in L loc 2 ( Ω ¯ × ( 0 , ∞ ) )

as j → ∞ .

Proof

First, in view of Lemmas 3.8, 3.9, and 3.10, we see that (3.54), (3.57), and (3.60) hold after extracting a subsequence { ε j } j ∈ N ⊂ ( 0 , 1 ) with ε j ↘ 0 as j → ∞ .

For any given T > 0 and γ > 1 , we have shown that ∂ t ( n ε + ε ) γ ∈ L 1 ( ( 0 , T ) ; ( W 0 2 , 2 ( Ω ) ) ∗ ) and ( n ε + ε ) γ ∈ L 2 ( ( 0 , T ) ; W 1 , 2 ( Ω ) ) in Lemma 3.12. Noting that the embedding W 1 , 2 ( Ω ) ↪ L 2 ( Ω ) is compact, and that the embedding L 2 ( Ω ) ↪ ( W 0 2 , 2 ( Ω ) ) ∗ is continuous, we can use the Aubin-Lions lemma to obtain a strong precompactness of n ε γ in L 2 ( Ω × ( 0 , T ) ) , which together with Egorov’s theorem implies (3.55).

Next, as the embedding L ∞ ( Ω ) ↪ ( W 0 2 , 2 ( Ω ) ) ∗ is compact, Lemma 3.11 together with the Arzelá-Ascoli theorem entails (3.56).

Also, by invoking Lemma 3.13 and using the Arzelá-Ascoli theorem gain, we can find a further sequence { ε j } j ∈ N ⊂ ( 0 , 1 ) with ε j ↘ 0 as j → ∞ such that (3.58) and (3.61) hold from a standard extraction procedure.

Finally, since the embeddings W 1 , q ( Ω ) ↪ W 1 , 2 ( Ω ) for q > 3 and D ( A β ) ↪ W 1 , 2 ( Ω ) for β ∈ ( 3 4 , 1 ) , we can show (3.59) and (3.62) by Lemmas 3.8, 3.9, and 3.13.□

Proof of Theorem 1.1

The verification of the weak solution property of ( n , c , u ) obtained in Lemma 3.14 is straightforward: whereas the nonnegativity of n and c and the integrability requirements in (1.10) and (1.11) are immediate from Lemma 3.14, the integral identities in Definition 1.1 can be derived by standard arguments from the corresponding weak formulations in the approximate system (2.2) upon letting ε = ε j ↘ 0 and using Lemma 3.14.□

4 Miscellaneous remarks

In this section, we will collect some miscellaneous remarks as by-products of Theorem 1.1 and its proof. Since the proofs of the following theorems are parallel to those of the corresponding conclusions that we mentioned, we omit the details here.

2D chemotaxis-Navier-Stokes system. Following the proof of Theorem 1.1, we can establish the global existence of weak solutions to the 2D system (1.5) coupled with Navier-Stokes flow

(4.1) n t + u ⋅ ∇ n = ∇ ⋅ ( n m − 1 ∇ n ) − ∇ ⋅ ( n S ( x , n , c ) ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ( u ⋅ ∇ ) u + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0

under the same initial-boundary value conditions (1.6) and (1.7). Precisely, we have

Theorem 4.1

Let Ω ⊂ R 2 be a bounded domain with smooth boundary ∂ Ω and suppose that (1.8)–(1.9) with R 3 replaced by R 2 and β ∈ 1 2 , 1 hold. Then for any m > 1 , systems (4.1)–(1.6)–(1.7) admit at least one global weak solution ( n , c , u , P ) , which is uniformly bounded in the sense that

‖ n ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ c ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) + ‖ A β u ( ⋅ , t ) ‖ L 2 ( Ω ) ≤ C for all t > 0 ,

where C is a positive constant depending only on Ω , m , C S , ϕ , and the initial data.

Remark 4.1

Theorem 4.1 extends the previous result in Tao and Winkler [18] to the matrix-valued chemotactic sensitivity and in Winkler [31] to the weak solutions. On the other hand, Theorems 1.1 and 4.1 provide some progress also in the fluid-free subcase of (4.1) obtained on letting u ≡ 0 . Indeed, even for the correspondingly gained 2D chemotaxis system with matrix-valued sensitivity the literature so far only contains very few results. For instance, global classical solutions are known to exist for m = 1 with small values of ‖ c 0 ‖ L ∞ in Li et al. [14] and global bounded weak solutions for large initial data can be found in Cao-Ishida [1] for m > 1 in a bounded planar convex domain.

General nonlinear diffusion. It is easy to confirm that Theorems 1.1 and 4.1 are valid when the porous medium cell diffusion n m − 1 is replaced by a generalized nonlinear diffusion D ( n ) satisfying

D ∈ C loc θ ( [ 0 , ∞ ) ) ∩ C 2 ( ( 0 , ∞ ) ) for some θ > 0 and D ( n ) ≥ k D n m − 1 for all n > 0

with m ≥ 65 63 and k D > 0 , which includes both degenerate and nondegenerate diffusion at n = 0 .

Large time behavior. Once we have established the global existence of weak solutions to systems (1.5)–(1.7), we may follow the corresponding studies on qualitative behavior in the related chemotaxis-fluid systems with signal absorption (e.g., in [27,28,32]) to show that the global weak solutions constructed in Theorem 1.1 stabilize toward the unique spatially constant steady state.

Theorem 4.2

Suppose that all assumptions in Theorem 1.1 hold. Then the global weak solution ( n , c , u , P ) constructed in Theorem 1.1 satisfies

‖ n ( ⋅ , t ) − n ¯ 0 ‖ L p ( Ω ) + ‖ c ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) + ‖ A β u ( ⋅ , t ) ‖ L 2 ( Ω ) → 0 a s t → + ∞

with β ∈ 3 4 , 1 given by (1.9) for any p > 1 , where n ¯ 0 ≔ 1 ∣ Ω ∣ ∫ Ω n 0 .

Funding information: Z. Xiang was supported by the NNSF of China (no. 11971093), the Applied Fundamental Research Program of Sichuan Province (no. 2020YJ0264), and the Fundamental Research Funds for the Central Universities (no. ZYGX2019J096).
Conflict of interest: The authors state no conflict of interest.

References

[1] X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899–1913. 10.1088/0951-7715/27/8/1899Search in Google Scholar

[2] X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. 55 (2016), 55–107. 10.1007/s00526-016-1027-2Search in Google Scholar

[3] J. A. Carrillo and K. Lin, Sharp conditions on global existence and blow-up in a degenerate two-species and cross-attraction system, Adv. Nonlinear Anal. 11 (2022), 1–39. 10.1515/anona-2020-0189Search in Google Scholar

[4] M. Chae, K. Kang, and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differ. Equ. 39 (2014), 1205–1235. 10.1080/03605302.2013.852224Search in Google Scholar

[5] M. Di Francesco, A. Lorz, and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. 28 (2010), 1437–1453. 10.3934/dcds.2010.28.1437Search in Google Scholar

[6] R. Duan, A. Lorz, and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differ. Equ. 35 (2010), 1635–1673. 10.1080/03605302.2010.497199Search in Google Scholar

[7] R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Notices 2014 (2014), 1833–1852. 10.1093/imrn/rns270Search in Google Scholar

[8] M. Fila and J. Lankeit, Lack of smoothing for bounded solutions of a semilinear parabolic equation, Adv. Nonlinear Anal. 9 (2020), 1437–1452. 10.1515/anona-2020-0059Search in Google Scholar

[9] T. Ghoul, V. Nguyen and H. Zaag, Construction of type I blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear Anal. 9 (2020), 388–412. 10.1515/anona-2020-0006Search in Google Scholar

[10] D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. 10.4171/042Search in Google Scholar

[11] P. He, Y. Wang, and L. Zhao, A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity, Appl. Math. Lett. 90 (2019), 23–29. 10.1016/j.aml.2018.09.019Search in Google Scholar

[12] C. Jin, Global bounded solution in three-dimensional chemotaxis-Stokes model with arbitrary porous medium slow diffusion, 2021, arXiv:2101.11235v1. Search in Google Scholar

[13] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence R.I., 1968. 10.1090/mmono/023Search in Google Scholar

[14] T. Li, A. Suen, C. Xue, and M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci. 25 (2015), 721–746. 10.1142/S0218202515500177Search in Google Scholar

[15] J. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. I. H. Poincaré – AN. 28 (2011), 643–652. 10.1016/j.anihpc.2011.04.005Search in Google Scholar

[16] A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci. 20 (2010), 987–1004. 10.1142/S0218202510004507Search in Google Scholar

[17] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. I. H. Poincaré - AN 31 (2014), 851–875. 10.1016/j.anihpc.2013.07.007Search in Google Scholar

[18] Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. 32 (2012), 1901–1914. 10.3934/dcds.2012.32.1901Search in Google Scholar

[19] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ. 252 (2012), 692–715. 10.1016/j.jde.2011.08.019Search in Google Scholar

[20] Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. I. H. Poincaré - AN 30 (2013), 157–178. 10.1016/j.anihpc.2012.07.002Search in Google Scholar

[21] Y. Tian and Z. Xiang, Global solutions to a 3D chemotaxis-stokes system with nonlinear cell diffusion and Robin signal boundary condition, J. Differ. Equ. 269 (2020), 2012–2056. 10.1016/j.jde.2020.01.031Search in Google Scholar

[22] I. Tuval, L. Cisneros, C. Dombrowski, C. w. Wolgemuth, J. O. Kessler, and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. National Acad. Sci. 102 (2005), 2277–2282. 10.1073/pnas.0406724102Search in Google Scholar PubMed PubMed Central

[23] Y. Wang, M. Winkler, and Z. Xiang, Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary, Comm. Partial Differ. Equ. 46 (2021), 1058–1091. 10.1080/03605302.2020.1870236Search in Google Scholar

[24] Y. Wang, M. Winkler, and Z. Xiang, Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation, Adv. Nonlinear Anal. 10 (2021), 707–731. 10.1515/anona-2020-0158Search in Google Scholar

[25] Y. Wang, M. Winkler and Z. Xiang, Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal, Anal. Appl. 20 (2022), 141–170. 10.1142/S0219530521500275Search in Google Scholar

[26] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differ. Equ. 37 (2012), 319–351. 10.1080/03605302.2011.591865Search in Google Scholar

[27] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal. 211 (2014), 455–487. 10.1007/s00205-013-0678-9Search in Google Scholar

[28] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828. 10.1007/s00526-015-0922-2Search in Google Scholar

[29] M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. I. H. Poincaré - AN 33 (2016), 1329–1352. 10.1016/j.anihpc.2015.05.002Search in Google Scholar

[30] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Tran Amer. Math. Soc. 369 (2017), 3067–3125. 10.1090/tran/6733Search in Google Scholar

[31] M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151. 10.1016/j.jde.2018.01.027Search in Google Scholar

[32] M. Winkler, Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components, J. Evolution Equ. 18 (2018), 1267–1289. 10.1007/s00028-018-0440-8Search in Google Scholar

[33] M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal. 276 (2019), 1339–1401. 10.1016/j.jfa.2018.12.009Search in Google Scholar

[34] M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems? Int. Math. Res. Notices 2021 (2021), 8106–8152. 10.1093/imrn/rnz056Search in Google Scholar

[35] M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1, Adv. Nonlinear Anal. 9 (2020), 526–566. 10.1515/anona-2020-0013Search in Google Scholar

[36] C. Wu and Z. Xiang, Asymptotic dynamics on a chemotaxis-Navier-Stokes system with nonlinear diffusion and inhomogeneous boundary conditions, Math. Models Methods Appl. Sci. 30 (2020), 1325–1374. 10.1142/S0218202520500244Search in Google Scholar

[37] C. Wu and Z. Xiang, Saturation of the signal on the boundary: Global weak solvability in a chemotaxis-Stokes system with porous-media type diffusion, J. Differ. Equ. 315 (2022), 122–158. 10.1016/j.jde.2022.01.033Search in Google Scholar

[38] C. Xue, Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling, J. Math. Biol. 70 (2015), 1–44. 10.1007/s00285-013-0748-5Search in Google Scholar PubMed

[39] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math. 70 (2009), 133–167. 10.1137/070711505Search in Google Scholar PubMed PubMed Central

Received: 2021-08-20

Revised: 2021-12-29

Accepted: 2022-01-05

Published Online: 2022-08-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2022-0228

Keywords for this article

chemotaxis-Stokes system; porous medium cell diffusion; general sensitivity; global boundedness

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