Research article

Regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term

  • Received: 17 January 2024 Revised: 08 April 2024 Accepted: 08 April 2024 Published: 08 May 2024
  • MSC : 35B65, 76W05, 35Q35

  • 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|β1u=0,   (t,x)R+×R3, (1.1)
    divu=0,   (t,x)R+×R3, (1.2)
    u(x,0)=u0,   xR3, (1.3)

    where u=u(x,t)R3, π=π(x,t)R represent the unknown velocity field and the pressure respectively. α0, β1 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 β72, 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 34α<1, β2α+54α2 or 1α<54, β1+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 uLp(0,T;Lq(R3)) with 2p+3q=1, q3, then the solution is regular and unique. Beirão da Veiga [11] established the analogous result: uLp(0,T;Lq(R3)) with 2p+3q=2, q32. For the general case, in [12], Jiang and Zhu proved that if ΛθuLp(0,T;Lq(R3)) with 2αp+3q2α1+θ, θ[1α,1], q>32α1+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 α54, the generalized Navier-Stokes equations (our system without damping term) exists a global strong solution uL(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|β12u2L2 (Theorems 1.1 and 1.2, when 1<α<54) and 1β+1ddtuβ+1Lβ+1 (Theorems 1.3 and 1.4, when 12<α<1). Actually, |u|β12u2L2 works better than 1β+1ddtuβ+1Lβ+1, because |u|β12u2L2 is a first-order estimate resulting from the damping term while 1β+1ddtuβ+1Lβ+1 is a zero-order estimate resulting from the damping term. However, because of the technical limitation, we still use 1β+1ddtuβ+1Lβ+1 when 12<α<1. Consequently, when 12<α<1, how to utilize |u|β12u2L2 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+3qmax{2(α1)3β,2α1},min{93β2(α1),32α1}<q, (1.4)

    then, for any T>0, the system (1.1)–(1.3) has a global strong solution satisfying

    uL(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+12α1<β<1+104α+1, then 2(α1)3β>2α1. Consequently, (1.4) becomes

    2αp+3q2(α1)3β,93β2(α1)<q, (1.5)

    which means that damping the term works better than the dissipative term.

    Theorem 1.2. When 1<α<54, 52α<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

    (3β)αp(2α5+β)+3qα+32,31+αq<,

    then, for any T>0, the system (1.1)–(1.3) has a global strong solution satisfying

    uL(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 β1+104α+1. If β1+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+54α2,3α+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(2α1)(β+1)p+3q2α1,32α1<q6α2α1, (1.6)

    then, for any T>0, the system (1.1)–(1.3) has a global strong solution satisfying

    uL(0,T;H1(R3))L2(0,T;Hα+1(R3))L(0,T;Lβ+1(R3)),utL2(0,T;L2(R3)).

    Remark 1.3. If β2α+54α2, the regularity criteria in Theorem 1.3 is satisfied naturally, so we recover the result in [4] when 34α<1.

    Theorem 1.4. When 12<α<1, β<min{2α+54α2,3α+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(2α1)(β+1)p+3q3α1,33α1<q6α3α1,

    then, for any T>0, the system (1.1) has a global strong solution satisfying

    uL(0,T;H1(R3))L2(0,T;Hα+1(R3))L(0,T;Lβ+1(R3)),utL2(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

    12ddtu2L2+Λ1+αu2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L2=R3(u)uΔudx.

    For R3(u)uΔudx, we have

    R3(u)uΔudxC|u|β12uL2|u|3β2ΔuL212|u|β12u2L2+Cu3βLqΔu2L2qq3+β12|u|β12u2L2+u3βLqu2(1θ1)L2Λ1+αu2θ1L212|u|β12u2L2+12Λ1+αu2L2+Cu3β1θ1Lqu2L212|u|β12u2L2+12Λ1+αu2L2+Cu2qα(3β)2(α1)9+3βLqu2L2,

    where

    123β2q=13+(12α3)θ1+1θ12,

    with θ1=2q+93β2αq. The conditions in Theorem 1.1 imply θ1[1α,1). By direct calculation, we have

    3β1θ1=2qα(3β)2(α1)q9+3β.

    Combining the above estimates, we obtain

    12ddtu2L2+Λ1+αu2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L212|u|β12u2L2+12Λ1+αu2L2+Cu3β1θ1Lqu2L2.

    A standard Gronwall's inequality shows that

    u2L2+t0(Λα+1u2L2+|u|β12u2L2+|u|β+122L2)(s)dsC( t,u0H1).

    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

    12ddtu2L2+Λ1+αu2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L2=R3(u)uΔudx.

    For R3(u)uΔudx, we have

    R3(u)uΔuC|u|β12uL2|u|3β2ΔuL212|u|β12u2L2+Cu3βL3Δu2L6β12|u|β12u2L2+Cu(3β)(1θ2)L2Λαu(3β)θ2Lqu2(1θ3)L2Λ1+αu2θ3L212|u|β12u2L2+12Λ1+αu2L2+CΛαu(3β)θ21θ3Lqu2L212|u|β12u2L2+12Λ1+αu2L2+CΛαu2(3β)αq[(2α+3)q6][2α5+β]Lqu2L2,

    where

    {13=θ2(1qα3)+1θ22,β6=13+θ3(12α3)+1θ32,

    with θ2=q(2α+3)q6, θ3=5β2α. The conditions in Theorem 1.2 imply θ2(0,1], θ3(1α,1). By direct calculation, we have

    (3β)θ21θ3=2(3β)αq[(2α+3)q6][2α5+β].

    Combining the above estimates, we obtain

    12ddtu2L2+Λ1+αu2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L212|u|β12u2L2+12Λ1+αu2L2+CΛαu2(3β)αq[(2α+3)q6][2α5+β]Lqu2L2.

    A standard Gronwall's inequality shows that

    u2L2+t0(Λα+1u2L2+|u|β12u2L2+|u|β+122L2)(s)dsC( t,u0H1).

    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

    12ddtu2L2+12ddtΛαu2L2+1β+1ddtuβ+1Lβ+1+Λ1+αu2L2+ut2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L2=R3(u)uΔudxR3(u)uutdxCu3L3+Cuu2L2+12ut2L2.

    For u3L3, we have

    Cu3L3Cuδ1(1θ4)LqΛ1+αuδ1θ4L2u(3δ1)(1θ5)Lβ+1Λ1+αu(3δ1)θ5L214Λ1+αu2L2+Cu2δ1(1θ4)2δ1θ4(3δ1)θ5Lqu2(3δ1)(1θ5)2δ1θ4(3δ1)θ5Lβ+114Λ1+αu2L2+Cu2δ1(1θ4)2δ1θ4(3δ1)θ5Lquβ+1Lβ+114Λ1+αu2L2+Cu[6α(2α1)(β+1)]q(2α1)q3Lquβ+1Lβ+1,

    where

    {13=13+θ4(121+α3)+1θ4q,13=13+θ5(121+α3)+1θ5β+1,2(3δ1)(1θ5)2δ1θ4(3δ1)θ5=β+1.

    By directly calculating, we have

    {θ4=6(2α1)q+6,θ5=6(2α1)(β+1)+6,δ1=[(2α1)q+6][6α(2α1)(β+1)]2(α+1)[(2α1)q+6]3[(2α1)(β+1)+6],2δ1(1θ4)2δ1θ4(3δ1)θ5=[6α(2α1)(β+1)]q(2α1)q3.

    The conditions in Theorem 1.3 imply θ4[11+α,1), θ5[11+α,1), δ1(0,3).

    For uu2L2, we have

    Cuu2L2Cuδ2Lqu2δ2Lβ+1uL21δ2q2δ2β+1Cuδ2Lqu2δ2Lβ+1u2(1θ6)Lβ+1Λ1+αu2θ6L214Λ1+αu2L2+Cuδ21θ6Lqu2δ21θ6Lβ+1u2Lβ+114Λ1+αu2L2+Cuδ21θ6Lquβ+1Lβ+1=14Λ1+αu2L2+Cu(3α+2αβ)qαq3Lquβ+1Lβ+114Λ1+αu2L2+C(u[6α(2α1)(β+1)]q(2α1)q3Lq+1)uβ+1Lβ+1,

    where

    {12δ22q2δ22(β+1)=13+θ6(121+α3)+1θ6β+1,2δ21θ6=β1.

    By direct calculation, we have

    {θ6=2q+93β2(α+1)q+33β,δ21θ6=21θ6+1β=(3α+2αβ)qαq3.

    The conditions in Theorem 1.3 imply θ6[11+α,1).

    Combining the above estimates, we obtain

    12ddtu2L2+12ddtΛαu2L2+1β+1ddtuβ+1Lβ+1+Λ1+αu2L2+ut2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L212Λ1+αu2L2+C(u[6α(2α1)(β+1)]q(2α1)q3Lq+1)uβ+1Lβ+1.

    A standard Gronwall's inequality shows that

    u2L2+uβ+1Lβ+1+Λαu2L2+t0(|u|β+122L2+|u|β12u2L2+Λ1+αu2L2+ut2L2)(τ)dτC( t,u0H1,u0Lβ+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

    12ddtu2L2+12ddtΛαu2L2+1β+1ddtuβ+1Lβ+1+Λ1+αu2L2+ut2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L2=R3(u)uΔudxR3(u)uutdxCu3L3+Cuu2L2+12ut2L2.

    For u3L3, we have

    Cu3L3CΛαuδ3(1θ7)LqΛ1+αuδ3θ7L2u(3δ3)(1θ5)Lβ+1Λ1+αu(3δ3)θ5L212Λ1+αu2L2+CΛαu2δ3(1θ7)2δ3θ7(3δ3)θ5Lqu2(3δ3)(1θ5)2δ3θ7(3δ3)θ5Lβ+112Λ1+αu2L2+CΛαu2δ3(1θ7)2δ3θ7(3δ3)θ5Lquβ+1Lβ+112Λ1+αu2L2+CΛαu[6α(2α1)(β+1)]q(3α1)q3Lquβ+1Lβ+1,

    where

    {13=1α3+θ7(1213)+1θ7q,13=13+θ5(121+α3)+1θ5β+1,2(3δ3)(1θ5)2δ3θ7(3δ3)θ5=β+1.

    By direct calculation, we have

    {θ7=62αq6q,θ5=6(2α1)(β+1)+6,δ3=(6q)[6α(2α1)(β+1)]2(α+1)(6q)(3αq)[(2α1)(β+1)+6].

    The conditions in Theorem 1.3 imply θ7[1α,1), θ5[11+α,1), δ3(0,3).

    We can estimate uu2L2 similarily.

    Combining the above estimates, we obtain

    12ddtu2L2+12ddtΛαu2L2+1β+1ddtuβ+1Lβ+1+Λ1+αu2L2+ut2L2+|u|β12u2L2+4(β1)(β+1)2|u|β+122L212Λ1+αu2L2+CΛαu[6α(2α1)(β+1)]q(3α1)q3Lquβ+1Lβ+1.

    A standard Gronwall's inequality shows that

    u2L2+uβ+1Lβ+1+Λαu2L2+t0(|u|β+122L2+|u|β12u2L2+Λ1+αu2L2+ut2L2)(τ)dτC( t,u0H1,u0Lβ+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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