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Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity

  • We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.

    Citation: Wei-Xi Li, Rui Xu. Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity[J]. Electronic Research Archive, 2021, 29(6): 4243-4255. doi: 10.3934/era.2021082

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  • We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.



    Recently, fractional calculus methods became of great interest, because it is a powerful tool for calculating the derivation of multiples systems. These methods study real world phenomena in many areas of natural sciences including biomedical, radiography, biology, chemistry, and physics [1,2,3,4,5,6,7]. Abundant publications focus on the Caputo fractional derivative (CFD) and the Caputo-Hadamard derivative. Additionally, other generalization of the previous derivatives, such as Ψ-Caputo, study the existence of solutions to some FDEs (see [8,9,10,11,12,13,14]).

    In general, an m-point fractional boundary problem involves a fractional differential equation with fractional boundary conditions that are specified at m different points on the boundary of a domain. The fractional derivative is defined using the Riemann-Liouville fractional derivative or the Caputo fractional derivative. Solving these types of problems can be challenging due to the non-local nature of fractional derivatives. However, there are various numerical and analytical methods available for solving such problems, including the spectral method, the finite difference method, the finite element method, and the homotopy analysis method. The applications of m-point fractional boundary problems can be found in various fields, including physics, engineering, finance, and biology. These problems are useful in modeling and analyzing phenomena that exhibit non-local behavior or involve memory effects (see [15,16,17,18]).

    Pantograph equations are a set of differential equations that describe the motion of a pantograph, which is a mechanism used for copying and scaling drawings or diagrams. The equations are based on the assumption that the pantograph arms are rigid and do not deform during operation, we can simply say that see [19]. One important application of the pantograph equations is in the field of drafting and technical drawing. Before the advent of computer-aided design (CAD) software, pantographs were commonly used to produce scaled copies of drawings and diagrams. By adjusting the lengths of the arms and the position of the stylus, a pantograph can produce copies that are larger or smaller than the original [20], electrodynamics [21] and electrical pantograph of locomotive [22].

    Many authors studied a huge number of positive solutions for nonlinear fractional BVP using fixed point theorems (FPTs) such as SFPT, Leggett-Williams and Guo-Krasnosel'skii (see [23,24]). Some studies addressed the sign-changing of solution of BVPs [25,26,27,28,29].

    In this work, we use Schauder's fixed point theorem (SFPT) to solve the semipostone multipoint Ψ-Caputo fractional pantograph problem

    Dν;ψrϰ(ς)+F(ς,ϰ(ς),ϰ(r+λς))=0, ς in (r,) (1.1)
    ϰ(r)=ϑ1, ϰ()=m2i=1ζiϰ(ηi)+ϑ2, ϑiR, i{1,2}, (1.2)

    where λ(0,r),Dν;ψr is Ψ-Caputo fractional derivative (Ψ-CFD) of order ν, 1<ν2, ζiR+(1im2) such that 0<Σm2i=1ζi<1, ηi(r,), and F:[r,]×R×RR.

    The most important aspect of this research is to prove the existence of a positive solution of the above m-point FBVP. Note that in [30], the author considered a two-point BVP using Liouville-Caputo derivative.

    The article is organized as follows. In the next section, we provide some basic definitions and arguments pertinent to fractional calculus (FC). Section 3 is devoted to proving the the main result and an illustrative example is given in Section 4.

    In the sequel, Ψ denotes an increasing map Ψ:[r1,r2]R via Ψ(ς)0, ς, and [α] indicates the integer part of the real number α.

    Definition 2.1. [4,5] Suppose the continuous function ϰ:(0,)R. We define (RLFD) the Riemann-Liouville fractional derivative of order α>0,n=[α]+1 by

    RLDα0+ϰ(ς)=1Γ(nα)(ddς)nς0(ςτ)nα1ϰ(τ)dτ,

    where n1<α<n.

    Definition 2.2. [4,5] The Ψ-Riemann-Liouville fractional integral (Ψ-RLFI) of order α>0 of a continuous function ϰ:[r,]R is defined by

    Iα;Ψrϰ(ς)=ςr(Ψ(ς)Ψ(τ))α1Γ(α)Ψ(τ)ϰ(τ)dτ.

    Definition 2.3. [4,5] The CFD of order α>0 of a function ϰ:[0,+)R is defined by

    Dαϰ(ς)=1Γ(nα)ς0(ςτ)nα1ϰ(n)(τ)dτ, α(n1,n),nN.

    Definition 2.4. [4,5] We define the Ψ-CFD of order α>0 of a continuous function ϰ:[r,]R by

    Dα;Ψrϰ(ς)=ςr(Ψ(ς)Ψ(τ))nα1Γ(nα)Ψ(τ)nΨϰ(τ)dτ, ς>r, α(n1,n),

    where nΨ=(1Ψ(ς)ddς)n,nN.

    Lemma 2.1. [4,5] Suppose q,>0, and ϰinC([r,],R). Then ς[r,] and by assuming Fr(ς)=Ψ(ς)Ψ(r), we have

    1) Iq;ΨrI;Ψrϰ(ς)=Iq+;Ψrϰ(ς),

    2) Dq;ΨrIq;Ψrϰ(ς)=ϰ(ς),

    3) Iq;Ψr(Fr(ς))1=Γ()Γ(+q)(Fr(ς))+q1,

    4) Dq;Ψr(Fr(ς))1=Γ()Γ(q)(Fr(ς))q1,

    5) Dq;Ψr(Fr(ς))k=0, k=0,,n1, nN, qin(n1,n].

    Lemma 2.2. [4,5] Let n1<α1n,α2>0, r>0, ϰL(r,), Dα1;ΨrϰL(r,). Then the differential equation

    Dα1;Ψrϰ=0

    has the unique solution

    ϰ(ς)=W0+W1(Ψ(ς)Ψ(r))+W2(Ψ(ς)Ψ(r))2++Wn1(Ψ(ς)Ψ(r))n1,

    and

    Iα1;ΨrDα1;Ψrϰ(ς)=ϰ(ς)+W0+W1(Ψ(ς)Ψ(r))+W2(Ψ(ς)Ψ(r))2++Wn1(Ψ(ς)Ψ(r))n1,

    with WR, {0,1,,n1}.

    Furthermore,

    Dα1;ΨrIα1;Ψrϰ(ς)=ϰ(ς),

    and

    Iα1;ΨrIα2;Ψrϰ(ς)=Iα2;ΨrIα1;Ψrϰ(ς)=Iα1+α2;Ψrϰ(ς).

    Here we will deal with the FDE solution of (1.1) and (1.2), by considering the solution of

    Dν;ψrϰ(ς)=h(ς), (2.1)

    bounded by the condition (1.2). We set

    Δ:=Ψ()Ψ(r)Σm2i=1ζi(Ψ(ηi)Ψ(r)).

    Lemma 2.3. Let ν(1,2] and ς[r,]. Then, the FBVP (2.1) and (1.2) have a solution ϰ of the form

    ϰ(ς)=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+rϖ(ς,τ)h(τ)Ψ(τ)dτ,

    where

    ϖ(ς,τ)=1Γ(ν){[(Ψ()Ψ(r))ν1Σm2j=iζj(Ψ(ηj)Ψ(τ))ν1]Ψ(ς)Ψ(r)Δ(Ψ(ς)Ψ(τ))ν1,τς,ηi1<τηi,[(Ψ()Ψ(τ))ν1Σm2j=iζj(Ψ(ηj)Ψ(τ))ν1]Ψ()Ψ(r)Δ,ςτ,ηi1<τηi, (2.2)

    i=1,2,...,m2.

    Proof. According to the Lemma 2.2 the solution of Dν;ψrϰ(ς)=h(ς) is given by

    ϰ(ς)=1Γ(ν)ςr(Ψ(ς)Ψ(τ))ν1h(τ)Ψ(τ)dτ+c0+c1(Ψ(ς)Ψ(r)), (2.3)

    where c0,c1R. Since ϰ(r)=ϑ1 and ϰ()=m2i=1ζiϰ(ηi)+ϑ2, we get c0=ϑ1 and

    c1=1Δ(1Γ(ν)m2i=1ζiηjr(Ψ(ηi)Ψ(τ))ν1h(τ)Ψ(τ)dτ+1Γ(ν)r(Ψ()Ψ(τ))ν1h(τ)Ψ(τ)dτ+ϑ1[m2i=1ζi1]+ϑ2).

    By substituting c0,c1 into Eq (2.3) we find,

    ϰ(ς)=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+(Ψ(ς)Ψ(r))Δϑ21Γ(ν)(ςr(Ψ(ς)Ψ(τ))ν1h(τ)Ψ(τ)dτ+(Ψ(ς)Ψ(r))Δm2i=1ζiηjr(Ψ(ηi)Ψ(τ))ν1h(τ)Ψ(τ)dτΨ(ς)Ψ(r)Δr(Ψ()Ψ(τ))ν1h(τ)Ψ(τ)dτ)=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+(Ψ(ς)Ψ(r))Δϑ2+rϖ(ς,τ)h(τ)Ψ(τ)dτ,

    where ϖ(ς,τ) is given by (2.2). Hence the required result.

    Lemma 2.4. If 0<m2i=1ζi<1, then

    i) Δ>0,

    ii) (Ψ()Ψ(τ))ν1m2j=iζj(Ψ(ηj)Ψ(τ))ν1>0.

    Proof. i) Since ηi<, we have

    ζi(Ψ(ηi)Ψ(r))<ζi(Ψ()Ψ(r)),
    m2i=1ζi(Ψ(ηi)Ψ(r))>m2i=1ζi(Ψ()Ψ(r)),
    Ψ()Ψ(r)m2i=1ζi(Ψ(ηi)Ψ(r))>Ψ()Ψ(r)m2i=1ζi(Ψ()Ψ(r))=(Ψ()Ψ(r))[1m2i=1ζi].

    If 1Σm2i=1ζi>0, then (Ψ()Ψ(r))Σm2i=1ζi(Ψ(ηi)Ψ(r))>0. So we have Δ>0.

    ii) Since 0<ν11, we have (Ψ(ηi)Ψ(τ))ν1<(Ψ()Ψ(τ))ν1. Then we obtain

    m2j=iζj(Ψ(ηj)Ψ(τ))ν1<m2j=iζj(Ψ()Ψ(τ))ν1(Ψ()Ψ(τ))ν1m2i=1ζi<(Ψ()Ψ(τ))ν1,

    and so

    (Ψ()Ψ(τ))ν1m2j=iζj(Ψ(ηj)Ψ(τ))ν1>0.

    Remark 2.1. Note that rϖ(ς,τ)Ψ(τ)dτ is bounded ς[r,]. Indeed

    r|ϖ(ς,τ)|Ψ(τ)dτ1Γ(ν)ςr(Ψ(ς)Ψ(τ))ν1Ψ(τ)dτ+Ψ(ς)Ψ(r)Γ(ν)Δm2i=1ζiηir(Ψ(ηj)Ψ(τ))ν1Ψ(τ)dτ+Ψ(ς)Ψ(r)ΔΓ(ν)r(Ψ()Ψ(τ))ν1Ψ(τ)dτ=(Ψ(ς)Ψ(r))νΓ(ν+1)+Ψ(ς)Ψ(r)ΔΓ(ν+1)m2i=1ζi(Ψ(ηi)Ψ(r))ν+Ψ(ς)Ψ(r)ΔΓ(ν+1)(Ψ()Ψ(r))ν(Ψ()Ψ(r))νΓ(ν+1)+Ψ()Ψ(r)ΔΓ(ν+1)m2i=1ζi(Ψ(ηi)Ψ(r))ν+(Ψ()Ψ(r))ν+1ΔΓ(ν+1)=M. (2.4)

    Remark 2.2. Suppose Υ(ς)L1[r,], and w(ς) verify

    {Dν;ψrw(ς)+Υ(ς)=0,w(r)=0, w()=Σm2i=1ζiw(ηi), (2.5)

    then w(ς)=rϖ(ς,τ)Υ(τ)Ψ(τ)dτ.

    Next we recall the Schauder fixed point theorem.

    Theorem 2.1. [23] [SFPT] Consider the Banach space Ω. Assume bounded, convex, closed subset in Ω. If ϝ: is compact, then it has a fixed point in .

    We start this section by listing two conditions which will be used in the sequel.

    (Σ1) There exists a nonnegative function ΥL1[r,] such that rΥ(ς)dς>0 and F(ς,ϰ,v)Υ(ς) for all (ς,ϰ,v)[r,]×R×R.

    (Σ2) G(ς,ϰ,v)0, for (ς,ϰ,v)[r,]×R×R.

    Let =C([r,],R) the Banach space of CFs (continuous functions) with the following norm

    ϰ=sup{|ϰ(ς)|:ς[r,]}.

    First of all, it seems that the FDE below is valid

    Dν;ψrϰ(ς)+G(ς,ϰ(ς),ϰ(r+λς))=0, ς[r,]. (3.1)

    Here the existence of solution satisfying the condition (1.2), such that G:[r,]×R×RR

    G(ς,z1,z2)={F(ς,z1,z2)+Υ(ς), z1,z20,F(ς,0,0)+Υ(ς), z10 or z20, (3.2)

    and ϰ(ς)=max{(ϰw)(ς),0}, hence the problem (2.5) has w as unique solution. The mapping Q: accompanied with the (3.1) and (1.2) defined as

    (Qϰ)(ς)=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+rϖ(ς,τ)G(ς,ϰ(τ),ϰ(r+λτ))Ψ(τ)dτ, (3.3)

    where the relation (2.2) define ϖ(ς,τ). The existence of solution of the problems (3.1) and (1.2) give the existence of a fixed point for Q.

    Theorem 3.1. Suppose the conditions (Σ1) and (Σ2) hold. If there exists ρ>0 such that

    [1+Σm2i=1ζi1Δ(Ψ()Ψ(r))]ϑ1+Ψ()Ψ(r)Δϑ2+LMρ,

    where Lmax{|G(ς,ϰ,v)|:ς[r,], |ϰ|,|v|ρ} and M is defined in (2.4), then, the problems (3.1) and (3.2) have a solution ϰ(ς).

    Proof. Since P:={ϰ:ϰρ} is a convex, closed and bounded subset of B described in the Eq (3.3), the SFPT is applicable to P. Define Q:P by (3.3). Clearly Q is continuous mapping. We claim that range of Q is subset of P. Suppose ϰP and let ϰ(ς)ϰ(ς)ρ, ς[r,]. So

    |Qϰ(ς)|=|[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+rϖ(ς,τ)G(τ,ϰ(τ),ϰ(r+λτ))Ψ(τ)dτ|[1+Σm2i=1ζi1Δ(Ψ()Ψ(r))]ϑ1+Ψ()Ψ(r)Δϑ2+LMρ,

    for all ς[r,]. This indicates that Qϰρ, which proves our claim. Thus, by using the Arzela-Ascoli theorem, Q: is compact. As a result of SFPT, Q has a fixed point ϰ in P. Hence, the problems (3.1) and (1.2) has ϰ as solution.

    Lemma 3.1. ϰ(ς) is a solution of the FBVP (1.1), (1.2) and ϰ(ς)>w(ς) for every ς[r,] iff the positive solution of FBVP (3.1) and (1.2) is ϰ=ϰ+w.

    Proof. Let ϰ(ς) be a solution of FBVP (3.1) and (1.2). Then

    ϰ(ς)=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+(Ψ(ς)Ψ(r))Δϑ2+1Γ(ν)rϖ(ς,τ)G(τ,ϰ(τ),ϰ(r+λτ))Ψ(τ)dτ=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+1Γ(ν)rϖ(ς,τ)(F(τ,ϰ(τ),ϰ(r+λτ))+p(τ))Ψ(τ)dτ=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+1Γ(ν)rϖ(ς,τ)F(τ,(ϰw)(τ),(ϰw)(r+λτ))Ψ(τ)dτ+1Γ(ν)rϖ(ς,τ)p(τ)Ψ(τ)dτ=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+1Γ(ν)rϖ(ς,τ)G(τ,(ϰw)(τ),(ϰw)(r+λτ))Ψ(τ)dτ+w(ς).

    So,

    ϰ(ς)w(ς)=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+1Γ(ν)rϖ(ς,τ)F(τ,(ϰw)(τ),(ϰw)(r+λτ))Ψ(τ)dτ.

    Then we get the existence of the solution with the condition

    ϰ(ς)=[1+Σm2i=1ζi1Δ(Ψ(ς)Ψ(r))]ϑ1+Ψ(ς)Ψ(r)Δϑ2+1Γ(ν)rϖ(ς,τ)F(τ,ϰ(τ),ϰ(r+λτ))Ψ(τ)dτ.

    For the converse, if ϰ is a solution of the FBVP (1.1) and (1.2), we get

    Dν;ψr(ϰ(ς)+w(ς))=Dν;ψrϰ(ς)+Dν;ψrw(ς)=F(ς,ϰ(ς),ϰ(r+λς))p(ς)=[F(ς,ϰ(ς),ϰ(r+λς))+p(ς)]=G(ς,ϰ(ς),ϰ(r+λς)),

    which leads to

    Dν;ψrϰ(ς)=G(ς,ϰ(ς),ϰ(r+λς)).

    We easily see that

    ϰ(r)=ϰ(r)w(r)=ϰ(r)0=ϑ1,

    i.e., ϰ(r)=ϑ1 and

    ϰ()=m2i=1ζiϰ(ηi)+ϑ2,
    ϰ()w()=m2i=1ζiϰ(ηi)m2i=1ζjw(ηi)+ϑ2=m2i=1ζi(ϰ(ηi)w(ηi))+ϑ2.

    So,

    ϰ()=m2i=1ζiϰ(ηi)+ϑ2.

    Thus ϰ(ς) is solution of the problem FBVP (3.1) and (3.2).

    We propose the given FBVP as follows

    D75ϰ(ς)+F(ς,ϰ(ς),ϰ(1+0.5ς))=0, ς(1,e), (4.1)
    ϰ(1)=1, ϰ(e)=17ϰ(52)+15ϰ(74)+19ϰ(115)1. (4.2)

    Let Ψ(ς)=logς, where F(ς,ϰ(ς),ϰ(1+12ς))=ς1+ςarctan(ϰ(ς)+ϰ(1+12ς)).

    Taking Υ(ς)=ς we get e1ςdς=e212>0, then the hypotheses (Σ1) and (Σ2) hold. Evaluate Δ0.366, M3.25 we also get |G(ς,ϰ,v)|<π+e=L such that |ϰ|ρ, ρ=17, we could just confirm that

    [1+Σm2i=1ζi1Δ(Ψ()Ψ(r))]ϑ1+Ψ()Ψ(r)Δϑ2+LM16.3517. (4.3)

    By applying the Theorem 3.1 there exit a solution ϰ(ς) of the problem (4.1) and (4.2).

    In this paper, we have provided the proof of BVP solutions to a nonlinear Ψ-Caputo fractional pantograph problem or for a semi-positone multi-point of (1.1) and(1.2). What's new here is that even using the generalized Ψ-Caputo fractional derivative, we were able to explicitly prove that there is one solution to this problem, and that in our findings, we utilize the SFPT. The results obtained in our work are significantly generalized and the exclusive result concern the semi-positone multi-point Ψ-Caputo fractional differential pantograph problem (1.1) and (1.2).

    The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small Groups (RGP.1/350/43).

    The authors declare no conflict of interest.



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