We investigated impulsive consensus in second-order nonlinear multi-agent systems (MASs) under Denial-of-Service (DoS) attacks. We consided scenarios where the communication network is subjected to DoS attacks, disrupting communication links and causing changes in the communication topology. An event-triggered impulsive control(ETIC) approach is proposed to flexibly address these issues. Additionally, an upper bound on the DoS attack period is introduced. Finally, a numerical example is given to verify the validity of the major results.
Citation: Qiushi Wang, Hongwei Ren, Zhiping Peng. Event-triggered impulsive control for second-order nonlinear multi-agent systems under DoS attacks[J]. AIMS Mathematics, 2024, 9(6): 13998-14011. doi: 10.3934/math.2024680
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We investigated impulsive consensus in second-order nonlinear multi-agent systems (MASs) under Denial-of-Service (DoS) attacks. We consided scenarios where the communication network is subjected to DoS attacks, disrupting communication links and causing changes in the communication topology. An event-triggered impulsive control(ETIC) approach is proposed to flexibly address these issues. Additionally, an upper bound on the DoS attack period is introduced. Finally, a numerical example is given to verify the validity of the major results.
In this paper, we consider the two-dimensional viscous, compressible and heat conducting magnetohydrodynamic equations in the Eulerian coordinates (see [1])
{ρt+div(ρu)=0,(ρu)t+div(ρu⊗u)+∇P=μ△u+(μ+λ)∇(divu)+H⋅∇H−12∇|H|2,cv((ρθ)t+div(ρuθ))+Pdivu=κΔθ+λ(divu)2+ν|curlH|2+2μ|D(u)|2,Ht+u⋅∇H−H⋅∇u+Hdivu=νΔH,divH=0. | (1.1) |
Here x=(x1,x2)∈Ω is the spatial coordinate, Ω is a bounded smooth domain in R2, t≥0 is the time, and the unknown functions ρ=ρ(x,t), θ=θ(x,t), u=(u1,u2)(x,t) and H=(H1,H2)(x,t) denote, respectively, the fluid density, absolute temperature, velocity and magnetic field. In addition, the pressure P is given by
P(ρ)=Rθρ,(R>0), |
where R is a generic gas constant. The deformation tensor D(u) is defined by
D(u)=12(∇u+(∇u)tr). |
The shear viscosity μ and the bulk one λ satisfy the hypotheses as follows
μ>0,μ+λ≥0. |
Positive constants cv, κ and ν represent, respectively, the heat capacity, heat conductivity and magnetic diffusivity coefficient.
The initial condition and boundary conditions for Eq (1.1) are given as follows
(ρ,θ,u,H)(x,t=0)=(ρ0,θ0,u0,H0), | (1.2) |
∂θ∂n=0,u=0,H⋅n=0,curlH=0,on∂Ω, | (1.3) |
where n denotes the unit outward normal vector of ∂Ω.
Remark 1.1. The boundary condition imposed on H (1.3) is physical and means that the container is perfectly conducting, see [1,2,3,4].
In the absence of electromagnetic effect, namely, in the case of H≡0, the MHD system reduces to the Navier-Stokes equations. Due to the strong coupling and interplay interaction between the fluid motion and the magnetic field, it is rather complicated to investigate the well-posedness and dynamical behaviors of MHD system. There are a huge amount of literature on the existence and large time behavior of solutions to the Navier-Stokes system and MHD one due to the physical importance, complexity, rich phenomena and mathematical challenges, see [1,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and the reference therein. However, many physically important and mathematically fundamental problems are still open due to the lack of smoothing mechanism and the strong nonlinearity. When the initial density contain vacuum states, the local large strong solutions to Cauchy problem of 3D full MHD equations and 2D isentropic MHD system have been obtained, respectively, by Fan-Yu [5] and Lü-Huang [6]. For the global well-posedness of strong solutions, Li-Xu-Zhang [7] and Lü-Shi-Xu [8] established the global existence and uniqueness of strong solutions to the 3D and 2D MHD equations, respectively, provided the smooth initial data are of small total energy. In particular, the initial density can have compact support in [7,8]. Furthermore, Hu-Wang [9,10] and Fan-Yu [11] proved the global existence of renormalized solutions to the compressible MHD equations for general large initial data. However, it is an outstanding challenging open problem to establish the global well-posedness for general large strong solutions with vacuum.
Therefore, it is important to study the mechanism of blow-up and structure of possible singularities of strong (or smooth) solutions to the compressible MHD system (1.1). The pioneering work can be traced to Serrin's criterion [12] on the Leray-Hopf weak solutions to the 3D incompressible Navier-Stokes equations, that is
limt→T∗‖u‖Ls(0,t;Lr)=∞,for3r+2s=1,3<r≤∞, | (1.4) |
where T∗ is the finite blow up time. Later, He-Xin [13] established the same Serrin's criterion (1.4) for strong solutions to the incompressible MHD equations.
First of all, we recall several known blow-up criteria for the compressible Navier-Stokes equations. In the isentropic case, Huang-Li-Xin [14] established the Serrin type criterion as follows
limt→T∗(‖u‖Ls(0,t;Lr)+‖divu‖L1(0,t;L∞))=∞,for3r+2s=1,3<r≤∞. | (1.5) |
For the full compressible Navier-Stokes equations, Fan-Jiang-Ou [15] obtained that
limt→T∗(‖θ‖L∞(0,t;L∞)+‖∇u‖L1(0,t;L∞))=∞, | (1.6) |
under the condition
7μ>λ. | (1.7) |
Later, the restriction (1.7) was removed in Huang-Li-Xin [16]. Recently, Wang [17] established a blow-up criterion for the initial boundary value problem (IBVP) on a smooth bounded domain in R2, namely,
limt→T∗‖divu‖L1(0,t;L∞)=∞. | (1.8) |
Then, let's return to the compressible MHD system (1.1). Under the three-dimensional isentropic condition, Xu-Zhang [18] founded the same criterion (1.5) as [14]. For the three-dimensional full compressible MHD system, the criterion (1.6) is also established by Lu-Du-Yao [19] under the condition
μ>4λ. | (1.9) |
Soon, the restriction (1.9) was removed by Chen-Liu [20]. Later, for the Cauchy problem and the IBVP of three-dimensional full compressible MHD system, Huang-Li [21] proved that
limt→T∗(‖u‖Ls(0,t;Lr)+‖ρ‖L∞(0,t;L∞))=∞,for3r+2s≤1,3<r≤∞. | (1.10) |
Recently, Fan-Li-Nakamura [22] extended the results of [17] to the MHD system and established a blow-up criterion which depend only on H and divu as follows
limt→T∗(‖H‖L∞(0,t;L∞)+‖divu‖L1(0,t;L∞))=∞. | (1.11) |
In fact, if H≡0 in (1.11), the criterion (1.11) becomes (1.8).
The purpose of this paper is to loosen and weaken the regularity of H required in the blow-up criterion (1.11) for strong solutions of the IBVP (1.1)–(1.3).
In this paper, we denote
∫⋅dx≜∫Ω⋅dx. |
Furthermore, for s≥0 and 1≤r≤∞, we define the standard Lebesgue and Sobolev spaces as follows
{Lr=Lr(Ω),Ws,r=Ws,r(Ω),Hs=Ws,2,Ws,r0={f∈Ws,r|f=0on∂Ω},Hs0=Ws,20. |
To present our results, we first recall the local existence theorem of the strong solution. Fan-Yu [5] attained the local existence and uniqueness of strong solution with full compressible MHD system in R3. In fact, when Ω is a bounded domain in R2, the method applied in [5,23] can also be used to the case here. The corresponding result can be expressed as follows.
Theorem 1.1. (Local existence theorem) For q>2, assume that the initial data (ρ0,θ0,u0,H0) satisfies
{0≤ρ0∈W1,q,0≤θ0∈H2,u0∈H10∩H2,H0∈H2,divH0=0,∂θ0∂n|∂Ω=0,u0|∂Ω=0,H0⋅n|∂Ω=curlH0|∂Ω=0, | (1.12) |
and the compatibility conditions as follows
−μ△u0−(μ+λ)∇divu0+R∇(ρ0θ0)−H0⋅∇H0+12∇|H0|2=ρ1/20g1, | (1.13) |
−κ△θ0−2μ|D(u0)|2−λ(divu0)2−ν(curlH0)2=ρ1/20g2, | (1.14) |
for some g1,g2∈L2. Then there exists a time T0>0 such that the IBVP (1.1)–(1.3) has a unique strong solution (ρ,θ,u,H) on Ω×(0,T0] satisfying that
{0≤ρ∈C([0,T0];W1,q),ρt∈C([0,T0];Lq),(u,θ,H)∈C([0,T0];H2)∩L2(0,T0;W2,q),θ≥0,(ut,θt,Ht)∈L2(0,T0;H1),(√ρut,√ρθt,Ht)∈L∞(0,T0;L2). | (1.15) |
Then, our main result is stated as follows.
Theorem 1.2. Under the assumption of Theorem 1.1, suppose (ρ,θ,u,H) is the strong solution of the IBVP (1.1)–(1.3) obtained in Theorem 1.1. If T∗<∞ is the maximum existence time of the strong solution, then
limt→T∗(‖H‖L∞(0,t;Lb)+‖divu‖L1(0,t;L∞))=∞, | (1.16) |
for any b>2.
Remark 1.2. Compared to the blow-up criterion (1.11) attained in [22], Theorem 1.2 demonstrates some new message about the blow-up mechanism of the MHD system (1.1)–(1.3). Particularly, beside the same regularity on ‖divu‖L1(0,t;L∞) as (1.11) in [22], our result (1.16) improves the regularity on ‖H‖L∞(0,t;L∞) by relaxing it to ‖H‖L∞(0,t;Lb) for any b>2.
The rest of the paper is arranged as follows. We state several basic facts and key inequalities which are helpful for later analysis in Section 2. Sections 3 is devoted to a priori estimate which is required to prove Theorem 1.2, while we give its proof in Section 4.
In this section, we will recall several important inequalities and well-known facts. First of all, Gagliardo-Nirenberg inequality (see [27]) is described as follows.
Lemma 2.1. (Gagliardo-Nirenberg) For q∈(1,∞),r∈(2,∞) and s∈[2,∞), there exists some generic constant C>0 which may depend only on q,r and s such that for f∈C∞0(Ω), we have
‖f‖sLs(Ω)≤C‖f‖2L2(Ω)‖∇f‖s−2L2(Ω), | (2.1) |
‖g‖L∞(Ω)≤C‖g‖q(r−2)/(2r+q(r−2))Lq(Ω)‖∇g‖2r/(2r+q(r−2))Lr(Ω). | (2.2) |
Then, we give several regularity results for the following Lamé system with Dirichlet boundary condition (see [24])
{LU≜μΔU+(μ+λ)∇divU=F,x∈Ω,U=0,x∈∂Ω. | (2.3) |
We assume that U∈H10 is a weak solution of the Lamé system, due to the uniqueness of weak solution, it could be denoted by U=L−1F.
Lemma 2.2. Let r∈(1,∞), then there exists some generic constant C>0 depending only on μ,λ,r and Ω such that
● If F∈Lr, then
‖U‖W2,r(Ω)≤C‖F‖Lr(Ω). | (2.4) |
● If F∈W−1,r (i.e., F=divf with f=(fij)2×2,fij∈Lr), then
‖U‖W1,r(Ω)≤C‖f‖Lr(Ω). | (2.5) |
Furthermore, for the endpoint case, if fij∈L2∩L∞, then ∇U∈BMO(Ω) and
‖∇U‖BMO(Ω)≤C‖f‖L∞(Ω)+C‖f‖L2(Ω). | (2.6) |
The following Lp-bound for elliptic systems, whose proof is similar to that of [28,Lemma 12], is a direct consequence of the combination of a well-known elliptic theory due to Agmon-Douglis-Nirenberg[29,30] with a standard scaling procedure.
Lemma 2.3. For k≥0 and p>1, there exists a constant C>0 depending only on k and p such that
‖∇k+2v‖Lp(Ω)≤C‖Δv‖Wk,p(Ω), | (2.7) |
for every v∈Wk+2,p(Ω) satisfying either
v⋅n=0,rotv=0,on ∂Ω, |
or
v=0,on ∂Ω. |
Finally, we give two critical Sobolev inequalities of logarithmic type, which are originally due to Brezis-Gallouet [31] and Brezis-Wainger [32].
Lemma 2.4. Let Ω⊂R2 be a bounded Lipschitz domain and f∈W1,q with q>2, then it holds that
‖f‖L∞(Ω)≤C‖f‖BMO(Ω)ln(e+‖f‖W1,q(Ω))+C, | (2.8) |
with a constant C depending only on q.
Lemma 2.5. Let Ω⊂R2 be a smooth domain and f∈L2(s,t;H10∩W1,q) with q>2, then it holds that
‖f‖2L2(s,t;L∞)≤C‖f‖2L2(s,t;H1)ln(e+‖f‖L2(s,t;W1,q))+C, | (2.9) |
with a constant C depending only on q.
Let (ρ,θ,u,H) be the strong solution of the IBVP (1.1)–(1.3) obtained in Theorem 1.1. Assume that (1.16) is false, namely, there exists a constant M>0 such that
limt→T∗(‖H‖L∞(0,t;Lb)+‖divu‖L1(0,t;L∞))≤M<∞,for anyb>2. | (3.1) |
First of all, the upper bound of the density can be deduced from (1.1)1 and (3.1), see [14,Lemma 3.4].
Lemma 3.1. Under the assumptions of Theorem 1.2 and (3.1), it holds that for any t∈[0,T∗),
sup0≤s≤t‖ρ‖L1∩L∞≤C, | (3.2) |
where (and in what follows) C represents a generic positive constant depending only on μ,λ,cv,κ, ν, q, b, M, T∗ and the initial data.
Then, we give the following estimates, which are similar to the energy estimates.
Lemma 3.2. Under the assumptions of Theorem 1.2 and (3.1), it holds that for any t∈[0,T∗),
sup0≤s≤t(‖ρθ‖L1+‖ρ1/2u‖2L2+‖H‖2L2)+∫t0(‖∇u‖2L2+‖∇H‖2L2)ds≤C. | (3.3) |
Proof. First, using the standard maximum principle to (1.1)3 together with θ0≥0 (see [15,25]) gives
infΩ×[0,t]θ(x,t)≥0. | (3.4) |
Then, utilizing the standard energy estimates to (1.1) shows
sup0≤s≤t(‖ρθ‖L1+‖ρ1/2u‖2L2+‖H‖2L2)≤C. | (3.5) |
Next, adding (1.1)2 multiplied by u to (1.1)4 multiplied by H, and integrating the summation by parts, we have
12ddt(‖ρ1/2u‖2L2+‖H‖2L2)+μ‖∇u‖2L2+ν‖∇H‖2L2+(μ+λ)‖divu‖2L2≤C‖ρθ‖L1‖divu‖L∞, | (3.6) |
where one has used the following well-known fact
‖∇H‖L2≤C‖curlH‖L2, | (3.7) |
due to divH=0 and H⋅n|∂Ω=0.
Hence, the combination of (3.6) with (3.1), (3.4) and (3.5) yields (3.3). This completes the proof of Lemma 3.2.
The following lemma shows the estimates on the spatial gradients of both the velocity and the magnetic, which are crucial for obtaining the higher order estimates of the solution.
Lemma 3.3. Under the assumptions of Theorem 1.2 and (3.1), it holds that for any t∈[0,T∗),
sup0≤s≤t(‖ρ1/2θ‖2L2+‖∇u‖2L2+‖curlH‖2L2)+∫t0(‖ρ1/2˙u‖2L2+‖∇θ‖2L2+‖Ht‖2L2+‖ΔH‖2L2)ds≤C, | (3.8) |
where ˙f≜u⋅∇f+ft represents the material derivative of f.
Proof. Above all, multiplying the equation (1.1)3 by θ and integrating by parts yield
cv2ddt‖ρ1/2θ‖2L2+κ‖∇θ‖2L2≤ν∫θ|curlH|2dx+C∫θ|∇u|2dx+C‖ρ1/2θ‖2L2‖divu‖L∞. | (3.9) |
Firstly, integration by parts together with (3.1) and Gagliardo-Nirenberg inequality implies that
ν∫θ|curlH|2dx≤C‖∇θ‖L2‖H‖Lb‖∇H‖L˜b+C‖θ‖L˜b‖H‖Lb‖∇2H‖L2≤C‖∇θ‖L2‖∇H‖L˜b+C‖∇2H‖L2(‖∇θ‖L2+1)≤ε‖∇θ‖2L2+C‖∇2H‖2L2+C(‖∇H‖2L2+1), | (3.10) |
where ˜b≜2bb−2>2 satisfies 1/b+1/˜b=1/2, and in the second inequality where one has applied the estimate as follows
‖θ‖Lr≤C(‖∇θ‖L2+1),for anyr≥1. | (3.11) |
Indeed, denote the average of θ by ˉθ=1|Ω|∫θdx, it follows from (3.2) and (3.3) that
ˉθ∫ρdx≤∫ρθdx+∫ρ|θ−ˉθ|dx≤C+C‖∇θ‖L2, | (3.12) |
which together with Poincaré inequality yields
‖θ‖L2≤C(1+‖∇θ‖L2). | (3.13) |
And consequently, (3.11) holds.
Secondly, according to [17,21,33], Multiplying equations (1.1)2 by uθ and integrating by parts yield
μ∫θ|∇u|2dx+(μ+λ)∫θ|divu|2dx=−∫ρ˙u⋅θudx−μ∫u⋅∇θ⋅∇udx−(μ+λ)∫divuu⋅∇θdx−∫∇P⋅θudx+∫H⋅∇H⋅θudx−12∫∇|H|2⋅θudx≜6∑i=1Ii. | (3.14) |
Using the same arguments in [17,33], we have
4∑i=1Ii≤η‖ρ1/2˙u‖2L2+ε‖∇θ‖2L2+C‖ρ1/2θ‖2L2‖divu‖L∞+C(‖ρ1/2θ‖2L2+‖∇u‖2L2)‖u‖2L∞. | (3.15) |
Besides, according to (3.1) and (3.11) yields
6∑i=5Ii≤C‖θ‖L˜b‖H‖Lb‖∇H‖L2‖u‖L∞≤ε‖∇θ‖2L2+C‖∇H‖2L2‖u‖2L∞+C. | (3.16) |
Substituting (3.10), (3.15) and (3.16) into (3.9), and choosing ε suitably small, we have
cvddt‖ρ1/2θ‖2L2+κ‖∇θ‖2L2≤2η‖ρ1/2˙u‖2L2+C1‖ΔH‖2L2+C(‖ρ1/2θ‖2L2+‖∇u‖2L2+‖∇H‖2L2+1)(‖divu‖L∞+‖u‖2L∞+1), | (3.17) |
where one has applied the key fact as follows
‖∇2H‖L2≤C‖ΔH‖L2. | (3.18) |
Furthermore, it follows from (3.1) and (1.1)4 that
‖Ht‖2L2+ν2‖ΔH‖2L2+νddt‖curlH‖2L2≤C‖∇u‖L2‖∇u‖L2˜b‖H‖Lb‖H‖L2˜b+C‖∇H‖2L2‖u‖2L∞≤C‖∇u‖L2‖∇u‖L2˜b(‖∇H‖L2+1)+C‖∇H‖2L2‖u‖2L∞. | (3.19) |
In order to estimate ‖∇u‖L2˜b, according to [24,26], we divide u into v and w. More precisely, let
u=v+w,andv=L−1∇P, | (3.20) |
then we get
Lw=ρ˙u−H⋅∇H+12∇|H|2. | (3.21) |
And hence, Lemma 2.2 implies that for any r>1,
‖∇v‖Lr≤C‖θρ‖Lr, | (3.22) |
and
‖∇2w‖Lr≤C‖ρ˙u‖Lr+C‖|H||∇H|‖Lr. | (3.23) |
Consequently, it follows from Gagliardo-Nirenberg inequality, (3.2), (3.11), (3.20), (3.22) and (3.23) that for any s≥2,
‖∇u‖Ls≤C‖∇v‖Ls+C‖∇w‖Ls≤C‖ρθ‖Ls+C‖∇w‖L2+C‖∇w‖2/sL2‖∇2w‖1−2/sL2≤C‖ρθ‖Ls+C‖∇w‖L2+C‖∇w‖2/sL2(‖ρ˙u‖L2+‖|H||∇H|‖L2)1−2/s≤η‖ρ1/2˙u‖L2+C‖ρθ‖Ls+C‖∇w‖L2+C‖|H||∇H|‖L2≤η‖ρ1/2˙u‖L2+C‖∇u‖L2+C‖∇θ‖L2+C‖∇H‖L2+C‖ΔH‖L2+C. | (3.24) |
Putting (3.24) into (3.19) and utilizing Young inequality lead to
‖Ht‖2L2+ν22‖ΔH‖2L2+νddt‖curlH‖2L2≤ε‖∇θ‖2L2+η‖ρ1/2˙u‖2L2+C(‖∇u‖2L2+‖u‖2L∞+1)(‖∇H‖2L2+1). | (3.25) |
Adding (3.25) multiplied by 2ν−2(C1+1) to (3.17) and choosing ε suitably small, we have
κ2‖∇θ‖2L2+2ν−2(C1+1)‖Ht‖2L2+‖ΔH‖2L2+ddt(cv‖ρ1/2θ‖2L2+2ν−1(C1+1)‖curlH‖2L2)≤C(‖ρ1/2θ‖2L2+‖∇u‖2L2+‖∇H‖2L2+1)(‖∇u‖2L2+‖u‖2L∞+‖divu‖L∞+1)+Cη‖ρ1/2˙u‖2L2. | (3.26) |
Then, multiplying (1.1)2 by ut and integrating by parts, we get
12ddt(μ‖∇u‖2L2+(μ+λ)‖divu‖2L2)+‖ρ1/2˙u‖2L2≤η‖ρ1/2˙u‖2L2+C‖∇u‖2L2‖u‖2L∞+ddt(∫Pdivudx+12∫|H|2divudx−∫H⋅∇u⋅Hdx)−∫Ptdivudx−∫H⋅Htdivudx+∫Ht⋅∇u⋅Hdx+∫H⋅∇u⋅Htdx. | (3.27) |
Notice that
∫Ptdivudx=∫Ptdivvdx+∫Ptdivwdx, | (3.28) |
integration by parts together with (3.20) leads to
∫Ptdivvdx=12ddt((μ+λ)‖divv‖2L2+μ‖∇v‖2L2). | (3.29) |
Moreover, define
E≜cvθ+12|u|2, |
according to (1.1) that E satisfies
(ρE)t+div(ρuE+Pu)=Δ(κθ+12μ|u|2)+μdiv(u⋅∇u)+λdiv(udivu)+H⋅∇H⋅u−12u⋅∇|H|2+ν|curlH|2. | (3.30) |
Motivated by [17,21], it can be deduced from (3.30) that
−∫Ptdivwdx=−Rcv(∫(ρE)tdivwdx−∫12(ρ|u|2)tdivwdx)=−Rcv{∫((cv+R)ρθu+12ρ|u|2u−κ∇θ−μ∇u⋅u−μu⋅∇u−λudivu)⋅∇divwdx−12∫ρ|u|2u⋅∇divwdx−∫ρ˙u⋅udivwdx−∫divHH⋅udivwdx−∫H⋅∇u⋅Hdivwdx−∫(H⋅u)H⋅∇divwdx+12∫divu|H|2divwdx+12∫|H|2u⋅∇divwdx−ν∫∇divw×curlH⋅Hdx−ν∫curl(curlH)⋅Hdivwdx}≤Cη‖ρ1/2˙u‖2L2+C‖∇θ‖2L2+C‖ΔH‖2L2+C(‖∇u‖2L2+‖u‖2L∞+1)(‖ρ1/2θ‖2L2+‖∇u‖2L2+‖∇H‖2L2+1). | (3.31) |
Additionally, combining (3.1) and (3.24) yields
∫Ht⋅∇u⋅Hdx+∫H⋅∇u⋅Htdx−∫H⋅Htdivudx≤C‖Ht‖2L2+C‖∇u‖2L˜b‖H‖2Lb≤Cη‖ρ1/2˙u‖2L2+C(‖Ht‖2L2+‖∇θ‖2L2+‖∇u‖2L2+‖∇H‖2L2+‖ΔH‖2L2+1). | (3.32) |
Substituting (3.28), (3.29), (3.31) and (3.32) into (3.27) yields
‖ρ1/2˙u‖2L2+ddt(μ2(‖∇u‖2L2+‖∇v‖2L2)+μ+λ2(‖divu‖2L2+‖divv‖2L2)−A(t))≤C2(‖∇θ‖2L2+‖Ht‖2L2+‖ΔH‖2L2)+Cη‖ρ1/2˙u‖2L2+C(‖∇u‖2L2+‖u‖2L∞+1)(‖∇u‖2L2+‖∇H‖2L2+‖ρ1/2θ‖2L2+1), | (3.33) |
where
A(t)≜12∫|H|2divudx+∫Pdivudx−∫H⋅∇u⋅Hdx, | (3.34) |
satisfies
A(t)≤μ4‖∇u‖2L2+C3(‖ρ1/2θ‖2L2+‖curlH‖2L2+1). | (3.35) |
Recalling the inequality (3.26), let
C4=min{2ν−2(C1+1),κ2,1},C5=min{2ν−1(C1+1),cv}, | (3.36) |
adding (3.26) multiplied by C6=max{C−14(C2+1),C−15(C3+1)} into (3.33) and choosing η suitably small, we have
ddt˜A(t)+12‖ρ1/2˙u‖2L2+‖∇θ‖2L2+‖Ht‖2L2+‖ΔH‖2L2≤C(‖ρ1/2θ‖2L2+‖∇u‖2L2+‖∇H‖2L2+1)(‖∇u‖2L2+‖u‖2L∞+‖divu‖L∞+1), | (3.37) |
where
˜A(t)≜C6(cv‖ρ1/2θ‖2L2+2ν−1(C1+1)‖curlH‖2L2)+μ2(‖∇u‖2L2+‖∇v‖2L2)+μ+λ2(‖divu‖2L2+‖divv‖2L2)−A(t), | (3.38) |
satisfies
‖ρ1/2θ‖2L2+μ4‖∇u‖2L2+‖curlH‖2L2−C≤˜A(t)≤C‖ρ1/2θ‖2L2+C‖∇u‖2L2+C‖curlH‖2L2+C. | (3.39) |
Finally, integrating (3.37) over (τ,t), along with (3.39) yields
ψ(t)≤C∫tτ(‖∇u‖2L2+‖u‖2L∞+‖divu‖L∞+1)ψ(s)ds+Cψ(τ), | (3.40) |
where
ψ(t)≜∫t0(‖ρ1/2˙u‖2L2+‖∇θ‖2L2+‖Ht‖2L2+‖ΔH‖2L2)ds+‖ρ1/2θ‖2L2+‖∇u‖2L2+‖curlH‖2L2+1. | (3.41) |
Combined with (3.1), (3.3) and Gronwall inequality implies that for any 0<τ≤t<T∗,
ψ(t)≤Cψ(τ)exp{∫tτ(‖∇u‖2L2+‖u‖2L∞+‖divu‖L∞+1)ds}≤Cψ(τ)exp{∫tτ‖u‖2L∞ds}. | (3.42) |
Utilizing Lemma 2.5, we have
‖u‖2L2(τ,t;L∞)≤C‖u‖2L2(τ,t;H1)ln(e+‖u‖L2(τ,t;W1,b))+C. | (3.43) |
Combining (3.1), (3.2), (3.11), (3.22), (3.23) and Sobolev inequality leads to
‖u‖W1,b≤‖v‖W1,b+C‖w‖W2,2b/(b+2)≤C‖ρ˙u‖L2b/(b+2)+C‖ρθ‖Lb+C‖u‖L2+C‖|H||∇H|‖L2b/(b+2)≤C‖ρ1/2‖Lb‖ρ1/2˙u‖L2+C‖∇θ‖L2+C‖∇u‖L2+C‖H‖Lb‖∇H‖L2+C≤C‖ρ1/2˙u‖L2+C‖∇θ‖L2+C‖∇u‖L2+C‖∇H‖L2+C, | (3.44) |
this implies that
‖u‖L2(τ,t;W1,b)≤Cψ1/2(t). | (3.45) |
Substituting (3.45) into (3.43) indicates
‖u‖2L2(τ,t;L∞)≤C+C‖u‖2L2(τ,t;H1)ln(Cψ(t))≤C+ln(Cψ(t))C7‖u‖2L2(τ,t;H1). | (3.46) |
Using (3.3), one can choose some τ which is close enough to t such that
C7‖u‖2L2(τ,t;H1)≤12, | (3.47) |
which together with (3.42) and (3.46) yields
ψ(t)≤Cψ2(τ)≤C. | (3.48) |
Noticing the definition of ψ in (3.41), we immediately have (3.8). The proof of Lemma 3.3 is completed.
Now, we show some higher order estimates of the solutions which are needed to guarantee the extension of local solution to be a global one under the conditions (1.12)–(1.14) and (3.1).
Lemma 3.4. Under the assumptions of Theorem 1.2 and (3.1), it holds that for any t∈[0,T∗),
sup0≤s≤t(‖ρ‖W1,q+‖θ‖H2+‖u‖H2+‖H‖H2)≤C. | (3.49) |
Proof. First, it follows from (3.8), Gagliardo-Nirenberg and Poincaré inequalities that for 2≤q<∞,
‖u‖Lq+‖H‖Lq≤C. | (3.50) |
Combining (1.1)4, (3.3), (3.8) and (3.18) yields
‖H‖H2+‖∇H‖2L4≤C‖∇u‖L4+C‖Ht‖L2+C. | (3.51) |
Furthermore, it can be deduced from (3.8), (3.24), (3.50) and (3.51) that
‖∇u‖L4≤C‖ρ1/2˙u‖L2+C‖∇θ‖L2+C‖Ht‖L2+C. | (3.52) |
Then, according to (3.11) and Sobolev inequality, we get
‖θ‖2L∞≤ε‖∇2θ‖2L2+C‖∇θ‖2L2+C, | (3.53) |
which combined with (1.1)3, (3.8), and choosing ε suitably small yield
‖θ‖2H2≤C‖ρ1/2˙θ‖2L2+C‖∇θ‖2L2+C‖∇u‖4L4+C‖∇H‖4L4+C. | (3.54) |
Therefore, the combination of (3.51) and (3.52) yields
sup0≤s≤t(‖θ‖Lr+‖∇θ‖L2+‖∇u‖L4+‖H‖H2+‖∇H‖L4)≤C,∀r≥1. | (3.55) |
Together with (3.53) and (3.54) gives
sup0≤s≤t(‖θ‖H2+‖θ‖L∞)≤C. | (3.56) |
Now, we bound ‖∇ρ‖W1,q and ‖u‖H2. For r∈[2,q], it holds that
ddt‖∇ρ‖Lr≤C‖∇ρ‖Lr(‖∇u‖L∞+1)+C‖∇2u‖Lr≤C‖∇ρ‖Lr(‖∇v‖L∞+‖∇w‖L∞+1)+C‖∇2v‖Lr+C‖∇2w‖Lr≤C‖∇ρ‖Lr(‖∇v‖L∞+‖∇w‖L∞+1)+C‖∇2w‖Lr+C, | (3.57) |
where in the last inequality one has applied the following fact
‖∇2v‖Lr≤C‖∇ρ‖Lr+C. | (3.58) |
Taking (3.2), (3.56), (3.58) and Lemmas 2.2–2.4, we get
‖∇v‖L∞≤Cln(e+‖∇ρ‖Lr)+C. | (3.59) |
Putting (3.59) into (3.57), it can be deduced from Gronwall inequality that
‖∇ρ‖Lr≤C. | (3.60) |
Finally, let r=2 in (3.60), according to Lemma 2.2, (3.50), (3.55) and (3.58) yields
‖u‖H2≤C. | (3.61) |
Therefore, together with (3.55), (3.56), (3.60) and (3.61), we get (3.49). The proof of Lemma 3.4 is completed.
With the priori estimates in Lemmas 3.1–3.4, we can prove Theorem 1.2.
Proof of Theorem 1.2. Assume that (1.16) is false, namely, (3.1) holds. Notice that the general constant C in Lemmas 3.1–3.4 is independent of t, that is, all the priori estimates attained in Lemmas 3.1–3.4 are uniformly bounded for any t≤T∗. Therefore, the function
(ρ,θ,u,H)(x,T∗)≜limt→T∗(ρ,θ,u,H)(x,t) |
satisfies the initial conditions (1.12) at t=T∗.
Due to
(ρ˙u,ρ˙θ)(x,T∗)=limt→T∗(ρ˙u,ρ˙θ)∈L2, |
therefore
−μ△u−(μ+λ)∇divu+R∇(ρθ)−H⋅∇H+12∇|H|2|t=T∗=ρ1/2(x,T∗)g1(x),−κ△θ−2μ|D(u)|2−λ(divu)2−ν(curlH)2|t=T∗=ρ1/2(x,T∗)g2(x), |
with
g1(x)≜{ρ−1/2(x,T∗)(ρ˙u)(x,T∗),forx∈{x|ρ(x,T∗)>0},0,forx∈{x|ρ(x,T∗)=0}, |
and
g2(x)≜{ρ−1/2(x,T∗)(cvρ˙θ+Rθρdivu)(x,T∗),forx∈{x|ρ(x,T∗)>0},0,forx∈{x|ρ(x,T∗)=0}, |
satisfying g1,g2∈L2. Thus, (ρ,θ,u,H)(x,T∗) also satisfies (1.13) and (1.14).
Hence, Theorem 1.1 shows that we could extend the local strong solutions beyond T∗, while taking (ρ,θ,u,H)(x,T∗) as the initial data. This contradicts the hypothesis of Theorem 1.2 that T∗ is the maximum existence time of the strong solution. This completes the proof of theorem 1.2.
This paper concerns the blow-up criterion for the initial boundary value problem of the two-dimensional full compressible magnetohydrodynamic equations in the Eulerian coordinates. When the initial density allowed to vanish, and the magnetic field H satisfies the perfect conducting boundary condition H⋅n=curlH=0, we prove the blow-up criterion limt→T∗(‖H‖L∞(0,t;Lb)+‖divu‖L1(0,t;L∞))=∞ for any b>2, which depending on both H and divu.
The author sincerely thanks the editors and anonymous reviewers for their insightful comments and constructive suggestions, which greatly improved the quality of the paper. The research was partially supported by the National Natural Science Foundation of China (No.11971217).
The author declares no conflict of interest in this paper.
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