| In this article, we study the existence of weak solution (or weak time-periodic solu-tion) and Hopf-Bifurcation for the compressible Magnetohydrodynamics, compressible Magnetic equations, compressible quantum Magnetohydrodynamics and incompress-ible Magnetohydrodynamics, respectively.We divide this paper into four parts.The first part is composed by the first chapter, which introduce the derivation, background and recent development of the compressible Magnetohydrodynamics, com-pressible Magnetic equations and compressible quantum Magnetohydrodynamics, re-spectively.The second part is composed by the second. third and fourth chapters. The second part is devoted to studyρt+div(ρu)= 0. divu= 0. (ρu)t+div(ρu(?)u)+▽P=(▽×H)×H+divS(ρ,θ,D(u)).Φt+div(u(Φ’+P))=div((u×H)×H+νH×(▽×H) +uS(ρ,θ,D(u))+q(ρ,θ,▽θ)), Ht-▽×(u×H)=-▽×(ν▽×H), divH= 0, whereρdenotes the density, u∈R3 the velocity, H∈R3 the magnetic field, andθthe temperature;S is the viscous stress tensor depends on the density, the temperature and the symmetric part of the velocity gradient D(u). and the thermal flux q is a function of the density, the temperature and its gradient.Φis the total energy given by with the internal energy e(ρ.θ). (?)ρ|u|2 the kinetic energy, and (?)|H|2 the magnetic energy. D(u) =▽u +▽uT is the symmetric part of the velocity gradient,▽uT is the transpose of the matrix▽×u, and I is the 3×3 identity matrix. v>0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field.We assume that the internal energy e can be decomposed as a sum: e(ρ,θ)=Pe(ρ)+Q(6) S and q are of the form (forρ>0,θ>0, D∈R3x3 symmetric) S(ρ,θ,D)=2μo(ρ,θ,|D|2)D.q(ρ,θ,▽,θ)=κo(ρ,θ)▽θ, and behave as where e∈[0.1], and there exist positive constantsμ,μ.κandκsuch that 0<μ≤μ(ρ.θ)≤μ<+∞, 0<κ≤κ(ρ)≤κ<+∞. Thus, in particular, for allρ>0,θ>0. and D, B∈R3×3 symmetric and for all▽θ∈R3(ρ<ρ*) For simplicity, we impose the boundary conditions u=0. H=0. q ? n=0 on[0.T]×(?)Ω. The initial density is Supposed to be bounded and the initial total energy is integrable, 1.e., and 0<ρ*≤ρ0(x)≤ρ*<+∞a.a.x∈Ω. 0<θ*≤θ0(x)for a.a.x∈Ω.whereρ*,ρ* andθ* are constants.The aim of this chapter is to establish the following result.定理0.11 Under above assumptions and r>2 andα>(?) . There are two positiveconstants cv and ~cZ such that 0<Cυ≤Cυ(θ)≤Cυ<+∞.Then there exists a weak solution to the density-dependent generalized Incompressible Magnetohydrodynarnic Flows in the sense of Definition 2.1 in subsection 2.2.The third chapter is to study the three-dimensional viscous steady compressible magnetorrydrodynamic flows in the Eulerian coordinates div(ρu) = 0, div(ρu(?)u)+▽P(ρ,θ)=(▽×H)×H+divΨ(u) +ρF. div(u(Φ’+P))=div((u×H)×H+νH×(▽×H)+uΨ(u)+κ(θ)▽θ) ,▽×(u×H)=▽×(ν▽×H), divH = 0.where p denotes the density, u∈R3 the velocity. H∈R3 the magnetic field, andθthe temperature: F is the external force:Ψis the viscous stress tensor given byΨ(u)=μD(u)+λdivuI. andΦis the total energy given by with the internal energy e(ρ.6), (?)ρ|u|2 the kinetic energy, and (?)|H|2 the magnetic energy. D(u)=▽u+▽uT is the symmetric part of the velocity gradient,▽uT is the transpose of the matrix▽×u, and I is the 3×3 identity matrix. The viscosity coefficientsλ,μof the flow satisfy 2μ+3λ>0 andμ>0;ν>0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field,κ>0 is the heat conductivity.We assume that e(ρ,θ)=Pe(θ)+Q(θ), where We consider the slip boundary condition u·n=0,Tκ·(T(P,u)n)+fu·Tκ=0.at (?)Ω, where Tκ,κ=1.2 are two perpendicular tangent vectors to (?)Ω, n is the outer normal vector and T(P.u)=-PI+(?)(u) is the stress tensor. The friction coefficient f is non-negative (if (?)=0 we assume additionally that U is not axially symmetric).For temperature. We assume thatκ(θ)(?)+L(θ)(θ-θo)=0,at (?)Ω, whereθo:(?)Ω'R+ is a strictly positive sufficiently smooth given function and there existθ∈θR+ such that 0<θ≤θo≤θ<+∞for almost all (a.a)x∈Ω, and L(θ)=c(1+θl).l∈R+ We also add the prescribed mass of the gas and the heat conductivity depending on the temperatureκ(θ)=κ0(1+θm),κ0,m>0.Under above assumption, we have定理0.12 LetΩ(?)R3 be a bounded domain of class Cf which is not axially symmetric if f=0. Assume that the pressure P(ρ,θ) is given by (3.1.6), where pe,pθ∈C1[0,∞) and Pe(0)=0,Pθ(0)=0, p’e(ρ)≥a1ργ-1,P’θ(ρ)≥0. Pe(ρ)≤a2ργ,Pθ(ρ)≤a3ρ(?) with some constantγ>(?),a1>0,a2>0 and a3>0.Let F∈L∞(Ω) and (?) m=1+l<m+. There exist two positive constants (?) and ~cZ such that 0<Cν≤Cν(θ)≤Cν<+∞Then the steady full compressible MHD has a weak solution (ρ,u,θ,H) such that for 1≤p<∞ρ∈L∞(Ω).u∈W1.p(Ω),θ∈W1.p(Ω).H∈W1.2(Ω) Moreover, the temperatureθ>0 a.e. inΩ.The fourth chapter is devoted to study (?)tρ+div(ρu)=0. (?)t(ρu)+div(ρu(?)u)+▽P(ρ) =(▽×H)×H+μΔu+(μ+λ)▽(divu)+ρf(t.x). (?)tH-▽×(u×H)+▽×(ν▽×H)=0, divH=0,A natural question of the existence of time-periodic solution arises when the external forces are time-periodic, i.e. f(t+w.x)=f(t.x). for a.e. x∈T3. t∈R1. with some constant uj > 0 (periodic). The corresponding periodic solution of equations should satisfyρ(t +w,x)=ρ(t,x), u(t +w,x)=u(t,x), H(t+w,x)=H(t,x), for all t∈R1,x∈T3We imposed so-called no-stick boundary condition of the velocity for all i = 1,2,3, u(t,x)·ν(x)=0, (?)t∈R1.x∈(?)D. In fact, the non-stick boundaty condition can read as Ui=0 on the opposite faces {xi=0,xj∈[0,π].j≠i}∪{xi=π,xj∈[0,π],j≠i}, The density satisfies the physical requirements: for a given positive mass m.The main result of this chapter state as follows:定理0.13 Assume that /’ e LOC(Q) and satisfying fi(t,Yi(x))=-fi(t.x,fi(t,Yj(x))=f(t,x),j≠i.i,j=1.2.3 and the pressure function p(ρ)=aρ1 with a>0 andγ>(?). Then, for a given m≥0, MHD under periodic force has a time-periodic weak solution (ρ(t,x).u(t.x).H(t.x)) in the sense of 1-5 in subsection 4-2. The third part is to study the existence and large time behavior of weak solution for compressible quantum Magnetohydrodynamics flow, the existence of weak time-periodic solution for compressible Magnetic flows.In the fifth chapter, we consider the dissipative three dimensional compressible quantum magnetohydrodynamic (QMHD) system: (?)tρ+div(ρu)= 0, =(▽×H)×H+μdiv(ρD(u)), (?)lH—▽×(u×H)+▽×(ν▽×H)=0, divH=0, where p denotes the particle density, u∈R3 the particle velocity, H∈R3 the magnetic field. P(ρ)=αpr the pressure with constantα>0 and the adiabatic exponentγ>1; h denotes the Planck constant; D(u)=(?)(▽u+▽uT) is the symmetric part of the velocity gradient:the viscosity coefficients of the flow satisfyμ>0;ν>0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, and all these kinetic coefficients and the magnetic diffusivity are independent of the magnitude and direction of the magnetic field. The symbol(?)denotes the Kronecker tensor product. The expression (?) can be interpreted as a quantum potential, the called Bohm potential. More precisely. The term(?) can be written in different ways:Here, we solve above system by introducing so called effective velocity v=u+μ▽logρ. This leads to the quantum Euler-Maxwell system: (?)tρ+div(ρv)=μ△ρ. with condition divH=0, where ho2=h2-μ2.The first result shows that the global existence for the viscous quantum Magne-tohydrodynamic model:定理0.14 Assume that constantsμ.λ,T>0.ho>μ,the pressure function p(ρ)= aργwith a>0.γ>3,the initial dataρo>0 and E(ρo.uo+μ∨▽Vlogρo.Ho) is finite. Then, for a given m≥0. the QMHD has a weak solution (ρ(t,x). u(t,x),H(t.x)) with the regularityρ∈L∞(0,T;Lγ(T3)∩L2(0.T;W1.3(T3))∩H1(0.T;L2(T3)): (?)∈L∞(0,T;H1(T3)∩L2(0.T;H2(T3)),(?)υ∈L∞(0,T;L2(T3)),ρυ∈L2(0,T;W1.(?)(T3),(?)|▽υ|∈L2(0,T;L2(T3), H∈L2(0,T;L2(T3)∩L2(0.T;H1.(T3)) Furthermore, for all smooth test functions w∈D(R3×R) satisfying w(.T) = 0. there holds with the total energy andAnother result shows that the global existence for the viscous quantum Euler-Maxwell model:定理0.15 Assume thatμ,λ,T>0. ho>μ, the pressure function p(ρ)=aργwith a>0.γ>3,the initial dataρo≥0 and E(ρo.υo.Ho) is finite. Then, for a given m≥0, QEM has a weak solution (ρ(t,x).u(t,x).H(t,x)) with the regularityρ∈L∞(0,T;Lγ(T3)∩L2(0.T;W1.3(T3))∩H1(0.T;L2(T3)): (?)∈L∞(0,T;H1(T3)∩L2(0.T;H2(T3)),(?)υ∈L∞(0,T;L2(T3)),ρυ∈L2(0,T;W1.(?)(T3),(?)|▽υ|∈L2(0,T;L2(T3), H∈L2(0,T;L2(T3)∩L2(0.T;H1.(T3)) Furthermore, for all smooth test functions w€P(M3 x R) satisfying w(-. T) = 0. there holds with the total energy and Here, the product A : B means summation over both indices of the matrices A and B.The following results show the large-time behavior of solutions to QMHD and the quantum Euler-Maxwell system.定理0.16Assume that (ρ.u.H) is the. finite energy weak solution to QMHD obtained in Theorem 0.14. Then there exist a stationary state of density ps which is a positive constant, a stationary state of velocity us = 0 and a stationary state of magnetic field Hs=0 such that as t'∞;ρ(x,t)'ρs strongly in Lγ(T3), u(x,t)'0 strongly in L2(T3), H(x,t)'0 strongly in L2(T3).定理0.17 Assume that (p.v.H) is the finite energy weak solution to QEM obtained in Theorem 0.15. Then there exist a stationary state of density ps which is a positive constant, a stationary state of velocityυs=μ▽logρs and a stationary state of magnetic field Hs=0 such that ast'∞,ρ(x,t)'ρs strongly in Lγ(T3),υ(x,t)'μ▽logρs strongly in L2(T3), H(x,t)'0 strongly in L2(T3).In the sixth chapter, we consider the compressible magnetic fluid under periodic force: dtp + div(pu) = 0. (?)(ρu)+div(ρu(?)u)-μΔu-(λ+μ)▽(divu)+▽P(ρ,M))=R+ρf(t,x), (?)(ρΩ)+div(ρu(?)Ω)-μ’ΔΩ-(λ’+μ’)▽(divΩ)=S+ρg(t,x), (?)M+div(u(?)M)-σΔM+1/(?)(M-χ0H)=Ω×M+ρl(t,x), H=▽ψ,div(H+M)=F, where (t,x)∈R1×T3,ρ∈R+1 denotes the density, u∈R3 denotes the fluid velocity. M∈R3 denotes the magnetization, H∈R3 denotes the magnetic field,Ω∈R3 denotes the angular velocity, F is a given function defined on R1×T3 with∫T3 Fdx=0, for all t∈R1, R=μ0M·▽H-ζ▽×(▽×u-2Ω). S=μ0M×H+2ζ(▽×u-2Ω). the pressure P(ρ,M) obeys the state law [84] P(ρ,M)=Pe(ρ)+Pm(M) with the isentropic pressure Pe(ρ)=aργ,a>0 and the adiabatic exponentγ>3/2 are constants, the magnetic pressure Pm(M)=(μ0)/2|M|2. The parametersλ,μ,λ’,μ’,χ0,μ0,ζ(?) andτare positive and their physical meaning can be found in [84]. f(t,x), g(t,x) and l(t,x) denote the time periodic external forces with a periodω> 0. Moreover, the functions f(t,x), g(t,x) and l(t,x) have bounded and measurable components with no restriction on their amplitudes. The symbol (?) denotes the Kronecker tensor product.Assume that f(t+ω,x)=f(t,x).g(t+ω,x)=g(t,x).l(t+w,x)=l(t,x), for a.e.x∈T3,t∈R1, with some constantω>0 (periodic). The corresponding periodic solution of equations should satisfyρ(t+ω,x)=ρ(t,x).u(t+ω,x)=u(t,x).Ω(t+w,x)=Ω(t,x). M(t+ω,x)=M(t,x),H(t+ω,x)=H(t,x), for all t∈R1,x∈T3.We impose so-called no-stick boundary conditions to the fluid velocity, the angular velocity, the magnetization field and the magnetic field: u(t,x)·ν(x)=0,Ω(t,x)·ν(x)=0,(?)t∈R1,x∈(?)D, M(t,x)·ν(x)=0,H(t,x)·ν(x)=0 (?)t∈R1,x∈(?)D, (▽×M)×ν(x)=0,(?)t∈R1,x∈(?)D, where v(x) denotes the outer normal vector.this no-stick boundary conditions on the fluid velocity. the angular velocity, the magnetization field and the magnetic field equal to ui=0 on the opposite faces {xi=0,xj∈[0,π],j≠i}∪{xi=π,xj∈[0,π],j≠i},Ωi=0 on the opposite faces {xi=0,xj∈[0,π],j≠i}∪{xi=π,xj∈[0,π],j≠i}, Mi=0 on the opposite faces {xi=0,xj∈[0,π],j≠i}∪{xi=π,xj∈[0,π],j≠i}, (?)=0 for i≠j on {xi=0,xj∈[0,π],j≠i}∪{xi=π,xj∈[0,π],j≠i}, Hi=0 on the opposite faces {xi=0,xj∈[0,π],j≠i}∪{xi=π,xj∈[0,π],j≠i}, The density satisfies the physical requirements: for a given positive mass m.定理0.18 Assume that fi,gi,li∈L∞(Q) and satisfy fi(t,Yi(x))=-fi(t,x),fi(t,Yj(x))=f(t,x),j≠i,i,j=1.2.3 gi(t,Yi(x))=-gi(t,x),gi(t,Yj(x))=g(t,x),j≠i,i,j=1.2.3 Pm(M) with Pe(ρ)=aρ2 and Pm(M) =(?)|M|2. where a,μo> 0 andγ>(?). Then, for a given m≥0, the compressible Magnetic fluid has a time-periodic weak solution (ρ(t,x).u(t,x),Ω(t,x). M(t.x).H(t,x)) in the sense of 1-6 in subsection 6.2.In the last chapter, we consider the 3D incompressible magneto-hydrodynamics (MHD) equations under external time-independent force U1-νΔU+U·▽U=-▽P-(?)▽H2+H·▽H+fαHt-ηΔH+U·▽H=H·▽U.▽·H=▽·U=0 where U is the flow velocity vector. H is the magnetic field vector, the kinematic viscosity v and the magnetic diffusivity k are positive constants. P(x. f) is a scalar pressure, fa and ha are external time independent forces.By [96]. we know that external forces fa and ha can be chosen suitably so that (Uα(x)+Uc1.Hα(x)+HC1.Pα(x)) is the solution of the steady Magneto-Hydrodynamics equation-νΔU+U·▽U=-▽P-(?)H2+H·▽+fα-ηΔH+U·▽H=H·▽U▽·H=▽·U=0 with Uc=(c1.0.0)T, Hc=(c2.0.0)T and (?)Uα(x)=0,(?)Hα(x)=0 where 0=(0.0.0)T.To overcome the essential spectrum of operator—(N + G) up to the imaginary axis, we need the following assumption: (H1 For anyα∈[αc-αo.αc+αo],(0.0) is not an eigenvalue of N+G.(H2)Forα=αc, the operator -N+G) has two pair eigenvalues (λo+,μo+) and (λo-,μo-)satisfyingλo±(αc)=μo±(αc)=±iξ≠0,forξo>0, (?)Re(λo±(α))|α=αc·(?)Re(μo±(α))|α=αc>0.(H3) The rest eigenvalue of -(N+G) is strictly bounded away from the imagi-nary axis in the left half plane for allα∈[αc-αo,αc+αo].Under the generic assumption the cubic coefficient terms a1.a2≠0 in subsection 7.3, Hopf-bifurcation result about MHD is stated:定理0.19 Assume that (H1-(H3 hold. Then incompressible MHD has a one di-mensional family of small time-periodic solutions, i.e. U(x.t)=U(x,t+2π/ξ1),H(x,t)=H(x.t+2π/ξ2) withα=αc+ε,ε∈(0.αo). Moreover,ξ1=ξ0+O(ε),ξ2=ξ0+O(ε), and‖U(x,t)‖Cbo(R3×[0.2π/ξ1])=O(ε)‖(x,t)‖Cbo(R3×[0.2π/ξ1])=O(ε). Above result also holds in a three dimensional torus T3 and a bounded domain.定理0.20 Let q>3. Assume that (H2) holds. If incompressible MHD has a smooth time periodic solution (Uα.α)(x,t),x∈T3 then (Uα,Hα)(x.t) is (Lq.Lq) nonlinearly stable in the sense of Lyapunov. |
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