Global dynamics of a delayed diffusive virus infection model with cell-mediated immunity and cell-to-cell transmission

1.   Introduction

We consider the compressible isentropic Navier-Stokes equations with degenerate viscosities in $\mathbb{R}^3$, which gives the conservation laws of mass and momentum of fluids. This model comes from the Boltzmann equations through the Chapman-Enskog expansion to the second order, and the viscosities depend on the density $\rho\geq 0$ by the laws of Boyle and Gay-Lussac for ideal gas. This system can be written as

where $x\in \mathbb{R}^3$ is the spatial coordinate; $t\ge 0$ is the time; $\rho$ is the density of the fluid; $u = (u^{(1)},u^{(2)}, u^{(3)})^\top\in \mathbb{R}^3$ is the velocity of the fluid; $P$ is the pressure, and for the polytropic fluid

where $A$ is a positive constant, $\gamma$ is the adiabatic index; $\mathbb{T}$ is the stress tensor given by

where $\mathbb{I}_3$ is the $3\times 3$ unit matrix, $\mu(\rho)$ is the shear viscosity, $\lambda(\rho)$ is the second viscosity, and

where the constants $\alpha$ and $\beta$ satisfy

Here, the initial data are given by

and the far field behavior is given by

The aim of this paper is to prove a blow-up criterion for the regular solution to the Cauchy problem (1) with (5)-(6).

Throughout the paper, we adopt the following simplified notations for the standard homogeneous and inhomogeneous Sobolev space:

A detailed study of homogeneous Sobolev space can be found in [5].

The compressible isentropic Navier-Stokes system is a well-known mathematical model, which has attracted great attention from the researchers, and some significant processes have been made in the well-posedness for this system.

When $(\mu,\lambda)$ are both constants, with the assumption that there is no vacuum, the local existence of the classical solutions to system (1) follows from a standard Banach fixed point argument. For the existence results with vacuum and general data, the main breakthrough is due to Lions [16]. He established the global existence of weak solutions in $\mathbb{R}^3$, periodic domains or bounded domains, under the homogenous Dirichlet boundary conditions and the restriction $\gamma >9/5$. Later, the restriction on $\gamma$ was improved to $\gamma>3/2$ by Feireisl-Novotný-Petzeltová [4]. Recently, Cho-Choe-Kim [2] introduced the following initial layer compatibility condition

for some $g\in L^2$ to deal with the vacuum. They proved the local existence of the strong solutions in $\mathbb{R}^3$ or bounded domains with homogenous Dirichlet boundary conditions. Moreover, Huang-Li-Xin proved the global existence of the classical solutions to the Cauchy problem of the isentropic system with small energy and vacuum in [8].

When $(\mu, \lambda)$ depend on density in the following form

where $\delta_1>0$, $\delta_2\ge0$, $\alpha>0$ and $\beta$ are all real constants, system (1) has received a lot of attention. However, except for the $1$D problems, there are few results on the strong solutions for the multi-dimensional problems, since the possible degeneracy of the Lam$\acute{\text{e}}$ operator caused by initial vacuum. This degeneracy gives rise to some difficulties in the regularity estimates because of the less regularizing effect of the viscosity on the solutions. Recently, Li-Pan-Zhu [11] have obtained the local existence of the classical solutions to system (1) in $2$D space under

and (6), where the vacuum cannot appear in any local point. They [12] also prove the same existence result in $3$D space under

with initial vacuum appearing in some open set or the far field, the constant $\bar{\rho}\ge0$ and

We also refer readers to [3], [6], [10], [13], [18], [26] and references therein for other interesting progress for this compressible degenerate system, corresponding radiation hydrodynamic equations and magnetohydrodynamic equations.

It should be noted that one should not always expect the global existence of solutions with better regularities or general initial data because of Xin's results [23] and Rozanova's results [20]. It was proved that there is no global smooth solutions to (1), if the initial density has nontrivial compact support ($1$D) or the solutions are highly decreasing at infinity($d$D, $d\ge1$). These motivate us to find the blow-up mechanisms and singularity structures of the solutions.

For constant viscosity, Beale-Kato-Majda [1] first proved that the maximum norm of the vorticity controls the blow-up of the smooth solutions to $3$D incompressible Euler equations

where $T^*$ is the maximum existence time. Later, for the same problem, Ponce [19] proved that the maximum norm of the deformation tensor controls the blow-up of the smooth solutions

where the deformation tensor $D(u) = \frac12\big(\nabla u+(\nabla u)^{\top}\big)$. Huang-Li-Xin [7] proved that the criterion (12) holds for the strong solutions to the system (1). Sun-Wang-Zhang [22] proved that the upper bound of the density controls the blowup of the strong solution to the system (1). There are some other interesting results about infinite time blowup and finite time blowup results on the nonlinear wave equation with different initial energy levels, refer to [14], [15], [24] and references therein for detailed study.

When the viscosities depend on density in the form of (4), S. Zhu [25] introduced the regular solutions, which can be defined as

Definition 1.1. [25] Let $T> 0$ be a finite constant, $(\rho,u)$ is called a regular solution to the Cauchy problem (1) with (5)-(6) on $[0,T]\times \mathbb{R}^3$ if $(\rho,u)$ satisfies

The local existence of the regular solutions has been obtained by Zhu [25].

Theorem 1.2. [25] Let $1< \gamma \leq 2$ or $\gamma = 3$. If the initial data $( \rho_0, u_0)$ satisfies the regularity conditions

then there exist a small time $T_*$ and a unique regular solution $(\rho, u)$ to the Cauchy problem (1) with (5)-(6). Moreover, we also have $\rho(t,x)\in C([0,T_*]\times \mathbb{R}^3)$ and

Based on Theorem 1.2, we establish the blow-up criterion for the regular solution in terms of $\nabla\log\rho$ and the deformation tensor $D(u)$, which is similar to the Beale-Kato-Majda criterion for the ideal incompressible Euler equations and the compressible Navier-Stokes equations.

Theorem 1.3. Let $(\rho, u)$ be a regular solution obtained in Theorem 1.2. Then if $\overline{T}< +\infty$ is the maximal existence time, one has both

and

The rest of the paper can be organized as follows. In Section 2, we will give the proof for the criterion (14). Section 3 is an appendix which will present some important lemmas which are frequently used in our proof, and also give the detail derivation for the desired system used in our following proof.

2.   Blow-up criterion

In this section, we give the proof for Theorem 1.3. We use a contradiction argument to prove $(14)$, let $( \rho, u)$ be the unique regular solution to the Cauchy problem (1) with (5)-(6) and the maximal existence time $\overline{T}$. We assume that $\overline{T}<+\infty$ and

for some constant $0<C_0<\infty$. If we prove that under assumption (16), $\overline{T}$ is actually not the maximal existence time for the regular solution, there will be a contradiction, thus (14) holds.

Notice that, one can also prove (15) by contradiction argument. Assume that

for some constant $0<C'_0<\infty$. Combing (17) with the mass equation, we know that

which implies that under assumption (17), we have (16). Thus, if we prove that (14) holds, then (15) holds immediately.

In the rest part of this section, based on the assumption (16), we will prove that $\bar{T}$ is not the maximal existence time for the regular solution.

From the definition of the regular solution, we know for

$(\phi, \psi, u)$ satisfies

where $L$ is the so-called Lam$\acute{ \text{e} }$ operator given by

and terms $\big( Q(u),\theta\big)$ are given by

See our appendix for the detailed process of the reformulation.

For $(19)_2$, we have the equivalent form

Here $A_l = (a^{(l)}_{ij})_{3\times 3}$ ($i,j,l = 1,2,3$) are symmetric with $a^{(l)}_{ij} = u^{(l)}$ when $i = j$; and $a^{(l)}_{ij} = 0$, otherwise. $B = (\nabla u)^{\top}$, so (22) is a positive symmetric hyperbolic system. By direct computation, one knows

combing this with $(16)$, one has $\psi\in L^{\infty}([0,T]; L^6)$.

Under (16) and (19), we first show that the density $\rho$ is uniformly bounded.

Lemma 2.1. Let $( \rho, u)$ be the unique regular solution to the Cauchy problem (1) with (5)-(6) on $[0,\overline{T})\times \mathbb{R}^3$ satisfying (16). Then

where $C>0$ depends on $C_0$, constant $q\in [2,+\infty]$ and $\overline{T}$.

Proof. First, it is obvious that $\phi$ can be represented by

where $W\in C^1([0,T]\times[0,T]\times \mathbb{R}^3)$ is the solution to the initial value problem

Then it is clear that

Similarly,

Next, multiplying $(19)_1$ by $2\phi$ and integrating over $\mathbb{R}^3$, we get

from (16), (27) and the Gronwall's inequality, we immediately obtain

Combing (25)-(28) together, one has

We complete the proof of this lemma.

Before go further, notice that

where we have used $(16)$ and Lemma 2.1. Next, we give the basic energy estimates on $u$.

Lemma 2.2. Let $( \rho, u)$ be the unique regular solution to the Cauchy problem (1) with (5)-(6) on $[0,\overline{T})\times \mathbb{R}^3$ satisfying (16). Then

where $C$ only depends on $C_0$ and $\overline{T}$.

Proof. Multiplying $(19)_3$ by $2u$ and integrating over $\mathbb{R}^3$, we have

The right-hand side terms can be estimated as follows.

where we have used (16), (23) and the facts

Thus (30) and (31) yield

By the Gronwall's inequality, (16) and (33), we have

This completes the proof of this lemma.

The next lemma provides the key estimates on $\nabla \phi$ and $\nabla u$.

Lemma 2.3. Let $( \rho, u)$ be the unique regular solution to the Cauchy problem (1) with (5)-(6) on $[0,\overline{T})\times \mathbb{R}^3$ satisfying (16). Then

where $C$ only depends on $C_0$ and $\overline{T}$.

Proof. Multiplying $(19)_3$ by $-Lu-\theta \nabla \phi^2$ and integrating over $\mathbb{R}^3$, we have

where we have used the fact that

First, from the standard elliptic estimate shown in Lemma 3.3, we have

Second, we estimate the right-hand side of (35) term by term. According to

Hölder's inequality, Young's inequality, (16), (23), (29), Lemma 2.2, Lemma 3.1 and $(19)_1$, one can obtain that

where $\epsilon> 0$ is a sufficiently small constant. Thus (35)-(38) imply

Third, applying $\nabla$ to $(19)_1$ and multiplying by $(\nabla \phi)^{\top}$, we have

Integrating (40) over $\mathbb{R}^3$, we get

Adding (41) to (39), from the Gronwall's inequality and (16), we immediately obtain

that is

which implies

Finally, due to $u_t = -Lu-u\cdot \nabla u-2\theta\phi\nabla \phi+\psi\cdot Q(u)$, we deduce that

Thus we complete the proof of this lemma.

Next, we proceed to improve the regularity of $\phi$, $\psi$ and $u$. First, we start with the estimates on the velocity.

Lemma 2.4. Let $( \rho, u)$ be the unique regular solution to the Cauchy problem (1) with (5)-(6) on $[0,\overline{T})\times \mathbb{R}^3$ satisfying (16). Then

where $C$ only depends on $C_0$ and $\overline{T}$.

Proof. From the standard elliptic estimate shown in Lemma 3.3 and

one has

where we have used Sobolev inequalities, (16), (23), (29) and Lemmas 2.1-2.3. Then we immediately obtain that

Next, differentiating $(19)_3$ with respect to $t$, it reads

Multiplying (46) by $u_t$ and integrating over $\mathbb{R}^3$, one has

Similarly, based on (16), (23), $(29)$ and Lemmas 2.1-2.3, we estimate the right-hand side of (47) term by term as follows.

where we also used Hölder's inequality, Young's inequality and

It is clear from (47)-(50) and (45) that

Integrating (52) over $(\tau,t)$ $(\tau \in( 0,t))$, we have

From the momentum equations $(19)_3$, we obtain

which, together with the definition of regular solution, gives

Letting $\tau \rightarrow 0$ in (53), applying the Gronwall's inequality, (16) and Lemma 2.3, we arrive at

This completes the proof of this lemma.

The following lemma gives bounds of $\nabla \phi$ and $\nabla^2 u$.

Lemma 2.5. Let $( \rho, u)$ be the unique regular solution to the Cauchy problem (1) with (5)-(6) on $[0,\overline{T})\times \mathbb{R}^3$ satisfying (16). Then

where $C$ only depends on $C_0$ and $\overline{T}$.

Proof. First, taking $q = 6$ in Lemma 2.1, combing with $(29)$ we have

Second, one has

where we have used Lemmas 2.3-2.4 and $(29)$.

Third, according to

and the standard elliptic estimate shown in Lemma 3.3, one has

where we have used (16), (23), (29), Lemmas 2.1-2.4 and

Thus, (60) implies that

Combing (62) with Lemma 2.4, one has

The proof of this lemma is completed.

Lemma 2.5 implies that

for any $t\in [0, \overline{T})$ with $C>0$ a finite number. Noting that (19) is essentially a parabolic-hyperbolic system, it is then standard to derive other higher order estimates for the regularity of the regular solutions. We will show this fact in the following lemma.

Lemma 2.6. Let $( \rho, u)$ be the unique regular solution to the Cauchy problem (1) with (5)-(6) on $[0,\overline{T})\times \mathbb{R}^3$ satisfying (16). Then

where $C$ only depends on $C_0$ and $\overline{T}$.

Proof. From $(19)_3$ and Lemma 3.3, we have

where we have used Young's inequality, Lemma 2.5, (16), (23), (29) and (61). Thus (65) offers that

Next, applying $\partial_i$ ($i = 1,2,3$) to $(19)_2$ with respect to $x$, we obtain

Multiplying (67) by $2(\partial_i\psi)^{\top}$, integrating over $\mathbb{R}^3$, and then summing over $i$, noting that $A_l$ ($l = 1,2,3$) are symmetric, it is not difficult to show that

where $\text{div}A = \sum\limits_{l = 1}^3\partial_{l}A_l$. When $|\zeta| = 1$, choosing $r = 2,\ a = 3$, $b = 6$ in (83), we have

Thus

Combining (70) with (66) and Lemma 3.1, we have

On the other hand, let $\nabla \phi = G = (G^{(1)},G^{(2)}, G^{(3)})^\top$. Applying $\nabla^2$ to $(19)_1$, we have

similarly to the previous step, we multiply (72) by $2\nabla G$ and integrate it over ${\mathbb {R}}^3$ to derive

where we have used the Young's inequality, (29) and (66). This estimate, together with (71), gives that

Then the Gronwall's inequality, (42), (64) and (74) imply

Combing (75) with (66) and Lemma 2.4, one has

Finally, using the following relations

according to Hölder's inequality, (16), (29), Lemmas 2.1-2.5, one has

Thus

and according to (42), one has

The proof of this lemma is completed.

Now we know from Lemmas 2.1-2.6 that, if the regular solution $(\rho, u)(x,t)$ exists up to the time ${\overline{T}}>0$, with the maximal time ${\overline{T}}<+\infty$ such that the assumption (16) holds, then

satisfies the conditions imposed on the initial data $(13)$. If we solve the system (1) with the initial time ${\overline{T}}$, then Theorem 1.1 ensures that $(\rho, u)(x,t)$ extends beyond ${\overline{T}}$ as the unique regular solution. This contradicts to the fact that ${\overline{T}}$ is the maximal existence time. We thus complete the proof of Theorem 1.3.

3.   Appendix

3.1.   Some important lemmas

In this subsection, we present some important lemmas which are frequently used in our previous proof. The first one is the well-known Gagliardo-Nirenberg inequality, which can be found in [9].

Lemma 3.1. [9] Let $r\in (1,+\infty)$ and $\ h\in W^{1,p}(\mathbb{R}^3) \cap L^r(\mathbb{R}^3)$. Then the following inequality holds for some constant $C(c,p,r)$

where

If $p< 3$, then $q\in [r,\frac{3p}{3-p}]$ when $r<\frac{3p}{3-p}$; and $q\in [\frac{3p}{3-p},r]$ when $r\geq\frac{3p}{3-p}$. If $p = 3$, then $q\in [r,+\infty)$. If $p> 3$, then $q\in [r,+\infty]$.

Some common versions of this inequality can be written as

which have be used frequently in our previous proof.

The second one can be found in Majda [17], and we omit its proof.

Lemma 3.2. [17] Let positive constants $r$, $a$ and $b$ satisfy the relation

and $1\leq a,\ b, \ r\leq +\infty$. $\forall s\geq 1$, if $f, g \in W^{s,a} (\mathbb{R}^3)\cap W^{s,b}(\mathbb{R}^3)$, then we have

where $C_s> 0$ is a constant depending only on $s$, and $\nabla^s f$ means that the set of all elements of $\partial^{\zeta} f$ with $|\zeta| = s$.

The third one is on the regularity estimates for Lam$\acute{ \text{e}}$ operator. For the elliptic problem

one has

Lemma 3.3. [21] If $u\in D^{k,q}$ with $1< q< +\infty$ is a weak solution to the problem (84), then

where k is an integer and the constant $C>0$ depend on $\alpha, \beta$ and $q$. Moreover, if $u\in D^{k,q}$ is a weak solution to the following problem

then (85) holds and if $f = \mathit{\text{div}} g$, we also have

The proof can be obtained via the classical estimates from harmonic analysis, which can be found in [21] or [22]. We omit it here.

3.2.   Reformulation of the system (1)

Now we show that, via introducing new variables

the system (1) can be rewritten as

Proof. First, from the momentum equation, one has

where $P = A\rho^{\gamma}$, divide both side by $\rho$, one has

Denote

we have

Second, for $\psi = \nabla\log\rho$, one has

Third, for $\phi = \rho^{\frac{\gamma-1}{2}}$, one has

Combing (90)-(92) together, we complete the proof of the transformation.

Acknowledgments

The author sincerely appreciates Dr. Shengguo Zhu for his very helpful suggestions and discussions on the problem solved in this paper. The research of Y. Cao was supported in part by China Scholarship Council 201806230126 and National Natural Science Foundation of China under Grants 11571232.

Conflict of Interest: The authors declare that they have no conflict of interest.