This paper was devoted to establishing some regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term. We focused on considering the role of the damping term in regularity criteria and the global existence brought by these criteria.
Citation: Jianlong Wu. Regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term[J]. AIMS Mathematics, 2024, 9(6): 16250-16259. doi: 10.3934/math.2024786
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This paper was devoted to establishing some regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term. We focused on considering the role of the damping term in regularity criteria and the global existence brought by these criteria.
In this paper, we consider the following Cauchy problem of Navier-Stokes equations with the damping term:
ut+(u⋅∇)u+∇π+Λ2αu+|u|β−1u=0, (t,x)∈R+×R3, | (1.1) |
divu=0, (t,x)∈R+×R3, | (1.2) |
u(x,0)=u0, x∈R3, | (1.3) |
where u=u(x,t)∈R3, π=π(x,t)∈R represent the unknown velocity field and the pressure respectively. α≥0, β≥1 are real parameters. Λ:=(−Δ)12 is defined in terms of Fourier transform by
^Λf(ξ)=|ξ|ˆf(ξ). |
Damping originates from the dissipation of energy by resistance, which describes many physical phenomena such as porous media flow, resistance or frictional effects, and some dissipation mechanisms (see [1] and references cited therein). When α=1, Cai and Jiu first proved that there exists a weak solution of (1.1)–(1.3) if β>1. Furthermore, if β≥72, the global existence of the strong solution was established. Later, this result was improved by Zhang, Wu and Lu in [2], where the lower bound of β decreased to 3. Zhou[3] proved the lower bound 3 is critical in some sense. For the general case, it is proved that when 34≤α<1, β≥2α+54α−2 or 1≤α<54, β≥1+104α+1, the global existence of the solution was established in [4]. For the asymptotic behavior, one can refer to [5,6,7] for details.
For the generalized Navier-Stokes equations (our system without damping term) when α=1, there are many regularity criteria to the system (1.1)–(1.3). The classical Prodi-Serrin's-type criteria was given in [8,9,10], where it was proved that if a weak solution u∈Lp(0,T;Lq(R3)) with 2p+3q=1, q≥3, then the solution is regular and unique. Beirão da Veiga [11] established the analogous result: ∇u∈Lp(0,T;Lq(R3)) with 2p+3q=2, q≥32. For the general case, in [12], Jiang and Zhu proved that if Λθu∈Lp(0,T;Lq(R3)) with 2αp+3q≤2α−1+θ, θ∈[1−α,1], q>32α−1+θ, then the solution remains smooth on [0, T]. One can refer to [11,13,14] for more classical regularity criteria. For the large time behavior, Jiu and Yu proved the algebraic decay of the solution under specific conditions (see [15]).
Our paper devotes to considering the role of damping terms in regularity criteria for the system (1.1)–(1.3). We will explain the role of damping term in the following two questions:
(1) When does the dissipative term work better than the damping term?
(2) How does the damping term work?
For the first question, if α≥54, the generalized Navier-Stokes equations (our system without damping term) exists a global strong solution u∈L∞(0,T;H1(R3))∩L2(0,T;H1+α(R3)). Consequently, we only consider the case when 12<α<54.
For the second question, we utilize two structures brought by the damping term: ‖|u|β−12∇u‖2L2 (Theorems 1.1 and 1.2, when 1<α<54) and 1β+1ddt‖u‖β+1Lβ+1 (Theorems 1.3 and 1.4, when 12<α<1). Actually, ‖|u|β−12∇u‖2L2 works better than 1β+1ddt‖u‖β+1Lβ+1, because ‖|u|β−12∇u‖2L2 is a first-order estimate resulting from the damping term while 1β+1ddt‖u‖β+1Lβ+1 is a zero-order estimate resulting from the damping term. However, because of the technical limitation, we still use 1β+1ddt‖u‖β+1Lβ+1 when 12<α<1. Consequently, when 12<α<1, how to utilize ‖|u|β−12∇u‖2L2 may be an insteresting question.
We give our main theorems as follows.
Theorem 1.1. When 1<α<54, β<1+104α+1, assume that the initial data u0(x)∈H1(R3) with divu0=0, and u(x,t) is a local strong solution of the system (1.1)–(1.3). If u(x,t)∈Lp(0,T;Lq(R3)) with
2αp+3q≤max{2(α−1)3−β,2α−1},min{9−3β2(α−1),32α−1}<q≤∞, | (1.4) |
then, for any T>0, the system (1.1)–(1.3) has a global strong solution satisfying
u∈L∞(0,T;H1(R3))∩L2(0,T;H1+α(R3))∩Lβ+1(0,T;Lβ+1(R3)). |
Remark 1.1. In Theorem 1.1, we roughly combine the regularity criteria brought by the dissipative term and the damping term. In fact, we can verify that if 1<α<54, 2+12α−1<β<1+104α+1, then 2(α−1)3−β>2α−1. Consequently, (1.4) becomes
2αp+3q≤2(α−1)3−β,9−3β2(α−1)<q≤∞, | (1.5) |
which means that damping the term works better than the dissipative term.
Theorem 1.2. When 1<α<54, 5−2α<β<1+104α+1, assume that the initial data u0(x)∈H1(R3) with divu0=0, and u(x,t) is a local strong solution of the system (1.1)–(1.3). If Λαu(x,t)∈Lp(0,T;Lq(R3)) with
(3−β)αp(2α−5+β)+3q≤α+32,31+α≤q<∞, |
then, for any T>0, the system (1.1)–(1.3) has a global strong solution satisfying
u∈L∞(0,T;H1(R3))∩L2(0,T;H1+α(R3))∩Lβ+1(0,T;Lβ+1(R3)). |
Remark 1.2. In Theorems 1.1 and 1.2, we consider the regularity criteria when β<1+104α+1, because the global existence was established in [4] when β≥1+104α+1. If β≥1+104α+1, the regularity criteria in Theorem 1.1 is satisfied naturally, so we recover the result in [4] when 1<α<54.
Theorem 1.3. When 12<α<1, β<min{2α+54α−2,3α+2α}, assume that the initial data u0(x)∈H1(R3)∩Lβ+1(R3) with divu0=0, and u(x,t) is a local strong solution of the system (1.1)–(1.3). If u(x,t)∈Lp(0,T;Lq(R3)) with
6α−(2α−1)(β+1)p+3q≤2α−1,32α−1<q≤6α2α−1, | (1.6) |
then, for any T>0, the system (1.1)–(1.3) has a global strong solution satisfying
u∈L∞(0,T;H1(R3))∩L2(0,T;Hα+1(R3))∩L∞(0,T;Lβ+1(R3)),ut∈L2(0,T;L2(R3)). |
Remark 1.3. If β≥2α+54α−2, the regularity criteria in Theorem 1.3 is satisfied naturally, so we recover the result in [4] when 34≤α<1.
Theorem 1.4. When 12<α<1, β<min{2α+54α−2,3α+2α}, assume that the initial data u0(x)∈H1(R3)∩Lβ+1(R3) with divu0=0, and u(x,t) is a local strong solution of the system (1.1)–(1.3). If Λαu(x,t)∈Lp(0,T;Lq(R3)) with
6α−(2α−1)(β+1)p+3q≤3α−1,33α−1<q≤6α3α−1, |
then, for any T>0, the system (1.1) has a global strong solution satisfying
u∈L∞(0,T;H1(R3))∩L2(0,T;Hα+1(R3))∩L∞(0,T;Lβ+1(R3)),ut∈L2(0,T;L2(R3)). |
Proof of the Theorem 1.1. Multiplying (1.1) by −△u, after integration by parts and taking the divergence-free property into account, we have
12ddt‖∇u‖2L2+‖Λ1+αu‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2=∫R3(u⋅∇)u⋅Δudx. |
For ∫R3(u⋅∇)u⋅Δudx, we have
∫R3(u⋅∇)u⋅Δudx≤C‖|u|β−12∇u‖L2‖|u|3−β2Δu‖L2≤12‖|u|β−12∇u‖2L2+C‖u‖3−βLq‖Δu‖2L2qq−3+β≤12‖|u|β−12∇u‖2L2+‖u‖3−βLq‖∇u‖2(1−θ1)L2‖Λ1+αu‖2θ1L2≤12‖|u|β−12∇u‖2L2+12‖Λ1+αu‖2L2+C‖u‖3−β1−θ1Lq‖∇u‖2L2≤12‖|u|β−12∇u‖2L2+12‖Λ1+αu‖2L2+C‖u‖2qα(3−β)2(α−1)−9+3βLq‖∇u‖2L2, |
where
12−3−β2q=13+(12−α3)θ1+1−θ12, |
with θ1=2q+9−3β2αq. The conditions in Theorem 1.1 imply θ1∈[1α,1). By direct calculation, we have
3−β1−θ1=2qα(3−β)2(α−1)q−9+3β. |
Combining the above estimates, we obtain
12ddt‖∇u‖2L2+‖Λ1+αu‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2≤12‖|u|β−12∇u‖2L2+12‖Λ1+αu‖2L2+C‖u‖3−β1−θ1Lq‖∇u‖2L2. |
A standard Gronwall's inequality shows that
‖∇u‖2L2+∫t0(‖Λα+1u‖2L2+‖|u|β−12∇u‖2L2+‖∇|u|β+12‖2L2)(s)ds≤C( t,‖u0‖H1). |
This completes the proof of the Theorem 1.1.
Proof of the Theorem 1.2. Multiplying (1.1) by −△u, after integration by parts and taking the divergence-free property into account, we have
12ddt‖∇u‖2L2+‖Λ1+αu‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2=∫R3(u⋅∇)u⋅Δudx. |
For ∫R3(u⋅∇)u⋅Δudx, we have
∫R3(u⋅∇)u⋅Δu≤C‖|u|β−12∇u‖L2‖|u|3−β2Δu‖L2≤12‖|u|β−12∇u‖2L2+C‖u‖3−βL3‖Δu‖2L6β≤12‖|u|β−12∇u‖2L2+C‖u‖(3−β)(1−θ2)L2‖Λαu‖(3−β)θ2Lq‖∇u‖2(1−θ3)L2‖Λ1+αu‖2θ3L2≤12‖|u|β−12∇u‖2L2+12‖Λ1+αu‖2L2+C‖Λαu‖(3−β)θ21−θ3Lq‖∇u‖2L2≤12‖|u|β−12∇u‖2L2+12‖Λ1+αu‖2L2+C‖Λαu‖2(3−β)αq[(2α+3)q−6][2α−5+β]Lq‖∇u‖2L2, |
where
{13=θ2(1q−α3)+1−θ22,β6=13+θ3(12−α3)+1−θ32, |
with θ2=q(2α+3)q−6, θ3=5−β2α. The conditions in Theorem 1.2 imply θ2∈(0,1], θ3∈(1α,1). By direct calculation, we have
(3−β)θ21−θ3=2(3−β)αq[(2α+3)q−6][2α−5+β]. |
Combining the above estimates, we obtain
12ddt‖∇u‖2L2+‖Λ1+αu‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2≤12‖|u|β−12∇u‖2L2+12‖Λ1+αu‖2L2+C‖Λαu‖2(3−β)αq[(2α+3)q−6][2α−5+β]Lq‖∇u‖2L2. |
A standard Gronwall's inequality shows that
‖∇u‖2L2+∫t0(‖Λα+1u‖2L2+‖|u|β−12∇u‖2L2+‖∇|u|β+12‖2L2)(s)ds≤C( t,‖u0‖H1). |
This completes the proof of the Theorem 1.2.
Proof of the Theorem 1.3. Multiplying (1.1) by −△u, ut and adding the two equations, after integration by parts and taking the divergence-free property into account, we have
12ddt‖∇u‖2L2+12ddt‖Λαu‖2L2+1β+1ddt‖u‖β+1Lβ+1+‖Λ1+αu‖2L2+‖ut‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2=∫R3(u⋅∇)u⋅Δudx−∫R3(u⋅∇)u⋅utdx≤C‖∇u‖3L3+C‖u⋅∇u‖2L2+12‖ut‖2L2. |
For ‖∇u‖3L3, we have
C‖∇u‖3L3≤C‖u‖δ1(1−θ4)Lq‖Λ1+αu‖δ1θ4L2‖u‖(3−δ1)(1−θ5)Lβ+1‖Λ1+αu‖(3−δ1)θ5L2≤14‖Λ1+αu‖2L2+C‖u‖2δ1(1−θ4)2−δ1θ4−(3−δ1)θ5Lq‖u‖2(3−δ1)(1−θ5)2−δ1θ4−(3−δ1)θ5Lβ+1≤14‖Λ1+αu‖2L2+C‖u‖2δ1(1−θ4)2−δ1θ4−(3−δ1)θ5Lq‖u‖β+1Lβ+1≤14‖Λ1+αu‖2L2+C‖u‖[6α−(2α−1)(β+1)]q(2α−1)q−3Lq‖u‖β+1Lβ+1, |
where
{13=13+θ4(12−1+α3)+1−θ4q,13=13+θ5(12−1+α3)+1−θ5β+1,2(3−δ1)(1−θ5)2−δ1θ4−(3−δ1)θ5=β+1. |
By directly calculating, we have
{θ4=6(2α−1)q+6,θ5=6(2α−1)(β+1)+6,δ1=[(2α−1)q+6][6α−(2α−1)(β+1)]2(α+1)[(2α−1)q+6]−3[(2α−1)(β+1)+6],2δ1(1−θ4)2−δ1θ4−(3−δ1)θ5=[6α−(2α−1)(β+1)]q(2α−1)q−3. |
The conditions in Theorem 1.3 imply θ4∈[11+α,1), θ5∈[11+α,1), δ1∈(0,3).
For ‖u⋅∇u‖2L2, we have
C‖u⋅∇u‖2L2≤C‖u‖δ2Lq‖u‖2−δ2Lβ+1‖∇u‖L21−δ2q−2−δ2β+1≤C‖u‖δ2Lq‖u‖2−δ2Lβ+1‖u‖2(1−θ6)Lβ+1‖Λ1+αu‖2θ6L2≤14‖Λ1+αu‖2L2+C‖u‖δ21−θ6Lq‖u‖2−δ21−θ6Lβ+1‖u‖2Lβ+1≤14‖Λ1+αu‖2L2+C‖u‖δ21−θ6Lq‖u‖β+1Lβ+1=14‖Λ1+αu‖2L2+C‖u‖(3α+2−αβ)qαq−3Lq‖u‖β+1Lβ+1≤14‖Λ1+αu‖2L2+C(‖u‖[6α−(2α−1)(β+1)]q(2α−1)q−3Lq+1)‖u‖β+1Lβ+1, |
where
{12−δ22q−2−δ22(β+1)=13+θ6(12−1+α3)+1−θ6β+1,2−δ21−θ6=β−1. |
By direct calculation, we have
{θ6=2q+9−3β2(α+1)q+3−3β,δ21−θ6=21−θ6+1−β=(3α+2−αβ)qαq−3. |
The conditions in Theorem 1.3 imply θ6∈[11+α,1).
Combining the above estimates, we obtain
12ddt‖∇u‖2L2+12ddt‖Λαu‖2L2+1β+1ddt‖u‖β+1Lβ+1+‖Λ1+αu‖2L2+‖ut‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2≤12‖Λ1+αu‖2L2+C(‖u‖[6α−(2α−1)(β+1)]q(2α−1)q−3Lq+1)‖u‖β+1Lβ+1. |
A standard Gronwall's inequality shows that
‖∇u‖2L2+‖u‖β+1Lβ+1+‖Λαu‖2L2+∫t0(‖∇|u|β+12‖2L2+‖|u|β−12∇u‖2L2+‖Λ1+αu‖2L2+‖ut‖2L2)(τ)dτ≤C( t,‖u0‖H1,‖u0‖Lβ+1). |
This completes the proof of the Theorem 1.3.
Proof of the Theorem 1.4. Multiplying (1.1) by −△u, ut and adding the two equations, after integration by parts and taking the divergence-free property into account, we have
12ddt‖∇u‖2L2+12ddt‖Λαu‖2L2+1β+1ddt‖u‖β+1Lβ+1+‖Λ1+αu‖2L2+‖ut‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2=∫R3(u⋅∇)u⋅Δudx−∫R3(u⋅∇)u⋅utdx≤C‖∇u‖3L3+C‖u⋅∇u‖2L2+12‖ut‖2L2. |
For ‖∇u‖3L3, we have
C‖∇u‖3L3≤C‖Λαu‖δ3(1−θ7)Lq‖Λ1+αu‖δ3θ7L2‖u‖(3−δ3)(1−θ5)Lβ+1‖Λ1+αu‖(3−δ3)θ5L2≤12‖Λ1+αu‖2L2+C‖Λαu‖2δ3(1−θ7)2−δ3θ7−(3−δ3)θ5Lq‖u‖2(3−δ3)(1−θ5)2−δ3θ7−(3−δ3)θ5Lβ+1≤12‖Λ1+αu‖2L2+C‖Λαu‖2δ3(1−θ7)2−δ3θ7−(3−δ3)θ5Lq‖u‖β+1Lβ+1≤12‖Λ1+αu‖2L2+C‖Λαu‖[6α−(2α−1)(β+1)]q(3α−1)q−3Lq‖u‖β+1Lβ+1, |
where
{13=1−α3+θ7(12−13)+1−θ7q,13=13+θ5(12−1+α3)+1−θ5β+1,2(3−δ3)(1−θ5)2−δ3θ7−(3−δ3)θ5=β+1. |
By direct calculation, we have
{θ7=6−2αq6−q,θ5=6(2α−1)(β+1)+6,δ3=(6−q)[6α−(2α−1)(β+1)]2(α+1)(6−q)−(3−αq)[(2α−1)(β+1)+6]. |
The conditions in Theorem 1.3 imply θ7∈[1−α,1), θ5∈[11+α,1), δ3∈(0,3).
We can estimate ‖u⋅∇u‖2L2 similarily.
Combining the above estimates, we obtain
12ddt‖∇u‖2L2+12ddt‖Λαu‖2L2+1β+1ddt‖u‖β+1Lβ+1+‖Λ1+αu‖2L2+‖ut‖2L2+‖|u|β−12∇u‖2L2+4(β−1)(β+1)2‖∇|u|β+12‖2L2≤12‖Λ1+αu‖2L2+C‖Λαu‖[6α−(2α−1)(β+1)]q(3α−1)q−3Lq‖u‖β+1Lβ+1. |
A standard Gronwall's inequality shows that
‖∇u‖2L2+‖u‖β+1Lβ+1+‖Λαu‖2L2+∫t0(‖∇|u|β+12‖2L2+‖|u|β−12∇u‖2L2+‖Λ1+αu‖2L2+‖ut‖2L2)(τ)dτ≤C( t,‖u0‖H1,‖u0‖Lβ+1). |
This completes the proof of the Theorem 1.4.
In this paper, we have established some regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term. First, we consider the case where the dissipative term is superior to the damping term, which corresponds to when the damping term works. Second, in Remark 1.1, we show that the damping term works better than the dissipative term. Furthermore, we have presented that the damping term has different effects in different cases, which shows the balance and the interaction between the dissipative term and the damping term as well as the role of the damping term in regularity criteria. In fact, considering how the damping term works and the interaction between the dissipative term and the damping term is the main idea of this paper.
The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.
This work was partially supported by the National Natural Science Foundation of China (Grant No.12071439) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LY19A010016).
The author declares no conflict of interest.
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