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Research article

The qualitative analysis of solution of the Stokes and Navier-Stokes system in non-smooth domains with weighted Sobolev spaces

  • This study typically emphasizes analyzing the geometrical singularities of weak solutions of the mixed boundary value problem for the stationary Stokes and Navier-Stokes system in two-dimensional non-smooth domains with corner points and points at which the type of boundary conditions change. The existence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. The solvability of the underlying boundary value problem is analyzed in weighted Sobolev spaces and the regularity theorems are formulated in the context of these spaces. To compute the singular terms for various boundary conditions, the generalized form of the boundary eigenvalue problem for the stationary Stokes system is derived. The emerging eigenvalues and eigenfunctions produce singular terms, which permits us to evaluate the optimal regularity of the corresponding weak solution of the Stokes system. Additionally, the obtained results for the Stokes system are further extended for the non-linear Navier-Stokes system.

    Citation: Yasir Nadeem Anjam. The qualitative analysis of solution of the Stokes and Navier-Stokes system in non-smooth domains with weighted Sobolev spaces[J]. AIMS Mathematics, 2021, 6(6): 5647-5674. doi: 10.3934/math.2021334

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  • This study typically emphasizes analyzing the geometrical singularities of weak solutions of the mixed boundary value problem for the stationary Stokes and Navier-Stokes system in two-dimensional non-smooth domains with corner points and points at which the type of boundary conditions change. The existence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. The solvability of the underlying boundary value problem is analyzed in weighted Sobolev spaces and the regularity theorems are formulated in the context of these spaces. To compute the singular terms for various boundary conditions, the generalized form of the boundary eigenvalue problem for the stationary Stokes system is derived. The emerging eigenvalues and eigenfunctions produce singular terms, which permits us to evaluate the optimal regularity of the corresponding weak solution of the Stokes system. Additionally, the obtained results for the Stokes system are further extended for the non-linear Navier-Stokes system.



    The limit theorem plays a pivotal role in the study of probability theory. Furthermore, the almost sure convergence is integral to the development of the limit theorem, a subject many scholars have studied. So far, a lot of excellent results have been obtained under the condition that the model holds with certainty. However, many uncertain phenomena of quantum mechanics and risk management cannot be explained by additive probability or expectation. To deal with this issue, many scholars have made great attempts and efforts. In particular, Peng [1,2] proposed the theory frame of the sub-linear expectations under a generic function space to solve this distributional uncertainty. In recent years, based on Peng, more and more scholars in the industry have done extensive research and obtained many related results; the study of the almost sure convergence has remained a hot-button issue. For example, Chen [3], Cheng [4], and Feng and Lan [5] obtained the SLLN (strong law of large numbers) of i.i.d.r.v. (independent identically distributed random variables), and Cheng [6] studied the SLLN of independent r.v. with supi1ˆE[|Xi|l(|Xi|)]<. Through further research, Wu and Jiang [7] obtained the SLLN of the extended independent and identically distributed r.v.; Chen and Liu [8], Gao et al. [9], and Liang and Wu [10] proved the SLLN of ND (negatively dependent) r.v.; Zhang [11] built the exponential inequality and the law of logarithm of independent and ND r.v.; Wang and Wu [12] and Feng [13] offered the almost sure covergence for weighted sums of ND r.v.; Zhang [14] derived the SLLN of the extended independent and END (extended negatively dependent) r.v.; Wang and Wu [15] obtained the almost sure convergence of END r.v.; Lin [16] achieved the SLLN of WND (widely negative dependence) r.v.; and Hwang [17] investigated the almost sure convergence of WND r.v..

    Anna [18] proposed the definition of WNOD r.v. for the first time and obtained the limiting conclusions for WNOD r.v. in Peng's theory frame. Based on Yan's results [19], this paper promotes them to the sub-linear expectation space. Compared to the previously mentioned ND and END r.v., dominating coefficients g(n) have been added to the definition of WNOD r.v., leading to a broader range. Besides, the sub-additivity property of the sub-linear expectation and capacity is added, making the research more meaningful and complex. Finally, the conclusions of almost sure convergence for WNOD r.v. are achieved. This paper contributes to the relevant research results of limiting behavior of WNOD r.v. in Peng's theory frame.

    Our essay is arranged as follows: Section 2 recommends interrelated definitions and properties as well as some important lemmas in the frame. Section 3 gives the conclusions including two theorems and two corollaries. Section 4 shows that the process of proving the conclusions is given in detail.

    Running through this essay, we point out that c will be a positive constant, its value is not important, and it may take different values according to the situation. axbx means limxaxbx=1. anbn means there must be a positive number c, satifying ancbn when n is large enough. Denote log(y)=ln(max{e,y}).

    This article uses the theory frame and concepts proposed by Peng [1,2]. Suppose (Ω,F) is a given measurable space and H is a linear space of real functions defined on (Ω,F) so that if X1,X2,,XnH, then φ(X1,X2,,Xn)H for every φCl, Lip (Rn), where φCl, Lip (Rn) shows the linear space of (local Lipschitz) functions φ satisfying

    |φ(x)φ(y)|c(1+|x|m+|y|m)|xy|,x,yRn,

    for some c>0,mN depending on φ. H is considered as a space of random variable. In this circumstance, we denote XH.

    Definition 2.1. (Peng [1]). A sub-linear expectation ˆE on H is a function ˆE:H[,] satisfying the following properties: for all X,YH, we have

    (a) Monotonicity: if XY, then ˆE(X)ˆE(Y);

    (b) Constant preserving: ˆE(c)=c;

    (c) Sub-additivity: ˆE(X+Y)ˆE(X)+ˆE(Y);

    (d) Positive homogeneity: ˆE(λX)=λˆE(X),λ0.

    The triple (Ω,H,ˆE) is known as a sub-linear expectation space.

    Next, give the definition of the conjugate expectation ˆE of ˆE by

    ˆE(X):=ˆE(X),XH.

    By the above definitions of ˆE and ˆE, the following inequality is feasible for all X,YH,

    ˆE(X)ˆE(X),ˆE(XY)ˆE(X)ˆE(Y),ˆE(X+c)=ˆE(X)+c,|ˆE(XY)|ˆE|XY|.

    When we are talking about ˆE and ˆE in the course of the proof, we often use the above formula.

    Definition 2.2. (Peng [1]). Make GF, a function V:G[0,1] is described to be a capacity, when

    V()=0,V(Ω)=1 and V(A)V(B) for AB,A,BG.

    Similar to sub-linear expectations, it is known as sub-additive when V(AB)V(A)+V(B) for every A,BG. Now, represent V and V, respectively corresponing to ˆE and ˆE, using

    V(A):=inf{ˆE[ξ],IAξ,ξH},V(A):=1V(Ac),AF,

    where Ac denotes the complement set of A.

    From the definion and sub-additivity property of (V, V), the following formulas are true

    ˆEζV(C)ˆEη,ˆEζV(C)ˆEη,ifζI(C)η,ζ,ηH.

    And now we have Markov inequality:

    V(|Y|y)ˆE|Y|P/yp,y>0,p>0.

    Definition 2.3. (Peng [1]). The Choquet integrals (CV) is defined as follows

    CV(X)=0V(Xx)dx+0[V(Xx)1]dx,

    where V and V can replace V when required.

    Definition 2.4. (Zhang [11]). (ⅰ) ˆE is referred to be countably sub-additive, when

    ˆE(X)n=1ˆE(Xn), whenever Xn=1Xn,X,XnH,X0,Xn0,n1.

    (ⅱ)V is referred to be countably sub-additive when

    V(n=1An)n=1V(An),AnF.

    Definition 2.5. {Xn,n1} is a sequence of r.v. and it is known to be stochastically dominated by a random variable X if for a positive number c, there has

    ˆE[f(|Xn|)]cˆE[f(|X|)],forn1,0fCl,Lip(R).

    Definition 2.6. (Anna [18]. Widely negative orthant dependent (WNOD)) {Xn,n1} is called to be widely negative orthant dependent if there is a finite positive array {g(n),n1} satisfying for every n1,

    ˆE(ni=1φi(Xi))g(n)ni=1ˆE (φi(Xi)), 

    where φiCb,Lip(R),φi0,1in and all functions φi are uniformly monotonous. Where the coefficients g(n) (n1) are known as dominating coefficients.

    It is visible that, when {Xn,n1} is widely negative orthant dependent and all functions fk(x)Cl,Lip(R) (where k=1,2,,n) are uniformly monotonous, then {fn(Xn),n1} is also widely negative orthant dependent.

    Definition 2.7. (Seneta [20]). (ⅰ) A positive function l(x) defined on [a,),a>0 is known to be a slowly varying function, satisfying

    limxl(tx)l(x)=1,foreacht0.

    (ⅱ) Each slowly varying function l(x) can be expressed as

    l(x)=C(x)exp{x1f(y)ydy},

    whenever limxC(x)=c>0, as well as limyf(y)=0.

    In this article, we want to research the almost sure convergence of WNOD sequence under sub-linear expectations. Since V is only sub-addictive, the definition of almost sure convergence is a little different and is described in detail in Wu and Jiang [7].

    Next, we give some lemmas before reaching our conclusions.

    Lemma 2.1. (Seneta [20]). For α>0, there is a non-decreasing function φ(x) and a non-increasing function ξ(x) such that

    xαl(x)φ(x),xαl(x)ξ(x),x,

    where l(x) is a slowly varying function.

    In the following section, we assume l(x),x>0 is a non-decreasing slowly varying function that can be expressed as l(x)=cexp{x1f(y)ydy}, where c>0, limxf(x)=0.

    Let

    τn=nl(n)1,n1. (2.1)
    Xn=τ1/pnI(Xn<τ1/pn)+XnI(|Xn|τ1/pn)+τ1/pnI(Xn>τ1/pn). (2.2)
    Xn=XnXn=(Xn+τ1/pn)I(Xn<τ1/pn)+(Xnτ1/pn)I(Xn>τ1/pn). (2.3)

    Lemma 2.2. Assume XH,0<p<2,τn defined by Eq (2.1).

    (ⅰ) For every c>0,

    CV(|X|p)<n=1l1(n)V(|X|p>cτn)<. (2.4)

    (ⅱ) When CV(|X|p)<, and now for every c>0,

    k=12kl(2k)V(|X|p>cτ2k)<. (2.5)

    Proof. (ⅰ) Obviously,

    CV(|X|p)<CV(|X|p/c)<.
    CV(|X|p/c)1V(|X|p>cx)dx1l(y)yl(y)f(y)yl2(y)V(|X|p>cyl(y))dy(makex=yl(y))11l(y)V(|X|p>cτy)dy.

    So

    CV(|X|p)<n=1l1(n)V(|X|p>cτn)<.

    (ⅱ) For every positive c, using the conclusion of (ⅰ), because of the monotonically increasing property of l(x),

    >n=1l1(n)V(|X|p>cτn)=k=12k1n<2kl1(n)V(|X|p>cτn)k=12k1n<2kl1(2k)V(|X|p>cτ2k)=21k=12kl1(2k)V(|X|p>cτ2k).

    As such, we have completed the proof of (ⅱ).

    Lemma 2.3. {Xn,n1} is a sequence of random variables, as well as stochastically dominated by a r.v. X and CV(|X|p)<,1p<2, ˆE has countable sub-additivity, then

    n=1τ2/pnl1(n)ˆE(Xn)2<, (2.6)

    moreover, when 1<p<2,

    n=1τ1/pnl1(n)ˆE|Xn|<. (2.7)

    Where Xn,Xn are respectively defined by Eqs (2.2) and (2.3).

    Proof. For 0<μ<1, assume an even function h(x)Cl,Lip(R) and h(x) when x>0, so that the value of h(x) is [0,1], for xR and h(x)1 when |x|μ, h(x)0 when |x|>1. We have

    I(|x|μ)h(|x|)I(|x|1),I(|x|>1)1h(x)I(|x|>μ). (2.8)

    For α=1,2,

    |Xk|α=|Xk|αI(|Xk|τ1/pk)+τα/pkI(|Xk|>τ1/pk)|Xk|αh(μ|Xk|τ1/pk)+τα/pk(1h(|Xk|τ1/pk)). (2.9)
    |Xk|α=|Xk+τ1/pk|αI(Xk<τ1/pk)+|Xkτ1/pk|αI(Xk>τ1/pk)=||Xk|+τ1/pk|αI(Xk<τ1/pk)+||Xk|τ1/pk|αI(Xk>τ1/pk)=||Xk|τ1/pk|αI(|Xk|>τ1/pk)|Xk|αI(|Xk|>τ1/pk)|Xk|α(1h(|Xk|τ1/pk)). (2.10)

    So, by (2.8) and Definition 2.7,

    ˆE|Xk|αˆE|Xk|αh(μ|Xk|τ1/pk)+τα/pkˆE(1h(|Xk|τ1/pk))ˆE|X|αh(μ|X|τ1/pk)+τα/pkˆE(1h(|X|τ1/pk))ˆE|X|αh(μ|X|τ1/pk)+τα/pkV(|X|>μτ1/pk). (2.11)
    ˆE|Xk|αˆE|X|α(1h(|X|τ1/pk)). (2.12)

    Assume that hj(x)Cl,Lip(R),j1, consider that the value of hj(x) is [0,1] for xR. hj(x)1 when τ1/p2j1<|x|τ1/p2j; hj(x)0 when |x|μτ1/p2j1 or |x|>(1+μ)τ1/p2j. The following formulas can be derived,

    I(τ1/p2j1<|x|τ1/p2j)hj(|x|)I(μτ1/p2j1<|x|(1+μ)τ1/p2j). (2.13)
    |X|rh(|X|τ1/p2k)1+kj=1|X|rhj(|X|),r>0. (2.14)
    |X|r(1h(|X|τ1/p2k))j=k|X|rhj(|X|μ),r>0. (2.15)

    First, prove (2.6). For 1p<2, by (2.11) and (2.4),

    H1:=n=1τ2/pnl1(n)ˆE(Xn)2n=1τ2/pnl1(n)[ˆE(X2h(μ|X|τ1/pn))+τ2/pnV(|X|>μτ1/pn)]=n=1τ2/pnl1(n)ˆE[X2h(μ|X|τ1/pn)]+n=1l1(n)V(|X|>μτ1/pn)n=1τ2/pnl1(n)ˆE[X2h(μ|X|τ1/pn)].

    Then, because h(x) is decreasing in (0,), according to Lemma 2.1, τ2/pnl1(n) is decreasing in (0,). So,

    H1k=12k1n<2kτ2/pnl1(n)ˆE[X2h(μ|X|τ1/pn)]k=12k1n<2kτ2/p2k1l1(2k1)ˆE[X2h(μ|X|τ1/p2k)]k=12k1n<2kτ2/p2kl1(2k)ˆE[X2h(μ|X|τ1/p2k)]k=12kτ2/p2kl1(2k)ˆE[X2h(μ|X|τ1/p2k)].

    Last by (2.14), (2.13), and (2.5),

    H1k=12kτ2/p2kl1(2k)+k=12kτ2/p2kl1(2k)kj=1ˆE(X2hj(μ|X|))j=1k=j2kτ2/p2kl1(2k)ˆE(X2hj(μ|X|))j=12jτ2/p2jl1(2j)τ2/p2jV(|X|>τ1/p2j1)j=12jl(2j)V(|X|>τ1/p2j)<.

    Therefore, (2.6) holds.

    Next, our proof of (2.7) is similar to (2.6). For 1<p<2, by (2.12) and the monotonically decreasing propety of h(x) in (0,), according to Lemma 2.1, τ1/pnl1(n) is decreasing in (0,), we have,

    H2:=n=1τ1/pnl1(n)ˆE|Xn|n=1τ1/pnl1(n)ˆE[|X|(1h(|X|τ1/pn))]=k=12k1n<2kτ1/pnl1(n)ˆE[|X|(1h(|X|τ1/pn))]k=12kτ1/p2k1l1(2k1)ˆE[|X|(1h(|X|τ1/p2k1))]k=12kτ1/p2kl1(2k)ˆE[|X|(1h(|X|τ1/p2k))].

    Then, from (2.15), (2.13), and (2.5), countable sub-additivity of ˆE,

    H2k=12kτ1/p2kl1(2k)j=kˆE(|X|hj(|X|μ))=j=1jk=12kτ1/p2kl1(2k)ˆE(|X|hj(|X|μ))j=12jτ1/p2jl1(2j)τ1/p2jV(|X|>μ2τ1/p2j1)j=12jl(2j)V(|X|>μ2τ1/p2j)<.

    Therefore, (2.7) holds.

    Lemma 2.4. (Zhang [11] Borel-Cantelli Lemma) Suppose {Bn;n1} is an array of matters in F. Suppose V has countable sub-additivity. We can obtain V(Bn;i.o.)=0 provided that n=1V(Bn)<, where (Bn;i.o.)=n=1m=nBm.

    Theorem 3.1. Suppose {Xn,n1} is a sequence of WNOD r.v., and its dominating coefficients are g(n). The sequence is stochastically dominated by a r.v. X. ˆE and V both have countable sub-additivity, and satisfying

    CV(|X|p)<,1<p<2. (3.1)

    Make {ank,1kn,n1} be a positive sequence according to

    max1knank=O(τ1/pnl1(n)),n, (3.2)

    where τn is defined by (2.1).

    If for some 0<δ<1,

    n=1e(δ2)l(n)g(n)<, (3.3)

    then,

    lim supnnk=1ank(XkˆEXk)0a.s.V, (3.4)
    lim infnnk=1ank(XkˆEXk)0a.s.V, (3.5)

    in particular, when ˆEXk=ˆEXk, then

    limnnk=1ank(XkˆEXk)=0a.s.V. (3.6)

    Remark 3.1. Theorem 3.1 under sub-linear expectations space is an extension of Theorem 2.1 of Yan [19] of the classical probability space.

    Remark 3.2. If g(n)=M, for each n1, then the sequence is simplified to END. When let l(n)=logn,n1, for 0<δ<1,

    n=1e(δ2)l(n)g(n)=Mn=1n(2δ)<,

    condition (3.3) is satisfied. By Theorem 3.1, Eqs (3.4)–(3.6) hold.

    Remark 3.3. We can obtain different conclusions by taking different forms of slowly varying function l(x). By taking l(n)=logn and l(n)=exp{(logn)ν}(0<ν<1), we will get the following two corollaries.

    Corollary 3.1. Suppose {Xn,n1} is a sequence of WNOD r.v., and its dominating coefficients are g(n). The sequence is stochastically dominated by a r.v. X. Besides, the sequence is satisfied (3.1). ˆE and V both have countable sub-additivity. Make sure {ank,1kn,n1} is a positive sequence according to

    max1knank=O(1n1/plog11/pn),n. (3.7)

    For some 0<b<1δ,

    g(n)nbc, (3.8)

    then (3.4)–(3.6) hold.

    Corollary 3.2. Suppose {Xn,n1} is a sequence of WNOD random variables, and its dominating coefficients are g(n). The sequence is stochastically dominated by a r.v. X. Besides, the sequence is satisfied (3.1). ˆE and V both have countable sub-additivity. Make sure {ank,1kn,n1} is a positive sequence according to

    max1knank=O(n1/pe(1+1/p)(logn)ν),n, (3.9)

    where 0<ν<1.

    For some m>0,

    g(n)nmc, (3.10)

    then (3.4)–(3.6) hold.

    Then, we will think about the situation of p=1.

    Theorem 3.2. Suppose {Xn,n1} is a sequence of WNOD r.v., and its dominating coefficients are g(n) and are satisfied (3.3). The sequence is stochastically dominated by a r.v. X. ˆE and V both have countable sub-additivity, and satisfying

    CV(|X|log|X|)<. (3.11)

    Suppose {ank,1kn,n1} is a positive sequence according to

    max1knank=O(n1),n, (3.12)

    then (3.4)–(3.6) hold.

    Because the sequence {Xk,k1} fulfills the criterion of Theorem 3.1, making {Xk,k1} as a substitute for {Xk,k1} in formula (3.4), by ˆEX=ˆE(X), there is

    0lim supnnk=1ank((Xk)ˆE(Xk))=lim supnnk=1ank((Xk)+ˆEXk)=lim supnnk=1ank((XkˆEXk)).
    lim infnnk=1ank(XkˆEXk)0.

    Therefore, (3.5) holds. Then, by ˆEXk=ˆEXk, (3.4) and (3.5), we can get (3.6). So we just need to prove (3.4).

    We denote Xn, Xn respectively by equations (2.2) and (2.3). By Definition 2.6, {XkˆEXk,k1} is also WNOD. Denote ˜Xk:=XkˆEXk.

    Therefore,

    nk=1ank(XkˆEXk)=nk=1ank˜Xk+nk=1ankXk+nk=1ank(ˆEXkˆEXk):=I1+I2+I3.

    So, if we want to prove (3.4), just prove

    lim supnIi0a.s.V,i=1,2,andlimnI3=0. (4.1)

    By (3.2) and the formula ex1+x+x22e|x|,x[,], for all t>0,1kn as well as large enough n,

    exp{tank˜Xk}1+tank˜Xk+t2a2nk(˜Xk)22exp{tank|˜Xk|}1+tank˜Xk+cτ2/pnl2(n)t2(˜Xk)2exp{ctl1(n)}. (4.2)

    By Definition 2.6, let φi(x)=etXi,i1, we can get for WNOD r.v.,

    ˆEexp{tni=1Xi}g(n)ni=1ˆEexp{tXi}. (4.3)

    By (4.2), (4.3), and the inequality 1+xex,xR, for all t>0 as well as large enough n,

    ˆEexp{tnk=1ank˜Xk}g(n)nk=1ˆEexp{tank˜Xk}g(n)nk=1ˆE[1+tank˜Xk+cτ2/pnl2(n)t2(˜Xk)2exp{ctl1(n)}]g(n)nk=1[1+cτ2/pnl2(n)t2exp{ctl1(n)}ˆE(˜Xk)2]g(n)exp{cτ2/pnl2(n)t2exp{ctl1(n)}nk=1ˆE(˜Xk)2].

    For ε>0, let t=2ε1l(n). According to Markov inequality, we can get

    V{nk=1ank˜Xk>ε}eεtˆEexp{tnk=1ank˜Xk}eεtg(n)exp{cτ2/pnl2(n)t2exp{ctl1(n)}nk=1ˆE(˜Xk)2]e2l(n)g(n)exp{cε2exp{cε1}l(n)τ2/pnl1(n)nk=1ˆE(˜Xk)2}.

    Combining ˆE(˜Xk)24ˆE(Xk)2, (2.6), and Kronecker's Lemma,

    τ2/pnl1(n)nk=1ˆE(˜Xk)20,n.

    So, for 0<δ<1, and large enough n, l(n) is non-decreasing in (0,), we can get

    cε2exp{cε1}τ2/pnl1(n)nk=1ˆE(˜Xk)2l(n)δl(1)δl(n).

    Therefore, by (3.3),

    n=1V{nk=1ank˜Xk>ε}cn=1e2l(n)g(n)eδl(n)=cn=1e(δ2)l(n)g(n)<.

    Because V has countable sub-additivity, and for every ε>0, we obtain from Lemma 2.4,

    lim supnI10,a.s.V. (4.4)

    For each n, there must be a m such that 2m1n<2m, by (2.12) and (3.2), h(x) is decreasing in (0,), according to Lemma 2.1, τ1/pnl1(n) is decreasing in (0,),

    H3:=nk=1ank|ˆEXkˆEXk|nk=1ankˆE|Xk|nk=1ankˆE[|X|(1h(|X|τ1/pk))]τ1/pnl1(n)nˆE[|X|(1h(|X|τ1/pn))]2mτ1/p2m1l(2m1)ˆE[|X|(1h(|X|τ1/p2m1))]2mτ1/p2ml(2m)ˆE[|X|(1h(|X|τ1/p2m))].

    Then, by (2.15) and (2.13), ˆE is countably sub-additive,

    H32mτ1/p2ml(2m)j=mˆE[|X|hj(|X|μ)]2mτ1/p2ml(2m)j=mτ1/p2jV(|X|>μ2τ1/p2j1)j=m2jτ1/p2jl(2j)τ1/p2jV(|X|>μ2τ1/p2j)=j=m2jl(2j)V(|X|>μ2τ1/p2j).

    Combining (2.5), we get

    limnI3=0. (4.5)

    If we want to prove (3.4), just prove

    lim supnI20,a.s.V. (4.6)

    Using (3.2) as well as the Lemma 2.1,

    max2mn<2m+1|nk=1ankXk|cmax2mn<2m+1τ1/pnl1(n)nk=1|Xk|cτ1/p2ml1(2m)2m+1k=1|Xk|,

    for ε>0, by (2.7) and Markov inequality,

    m=1V(max2mn<2m+1|nk=1ankXk|>ε)m=1V(cτ1/p2ml1(2m)2m+1k=1|Xk|>ε)cm=1τ1/p2ml1(2m)2m+1k=1ˆE|Xk|=ck=1ˆE|Xk|m:2m+1kτ1/p2ml1(2m)k=1τ1/pkl1(k)ˆE|Xk|<.

    By Lemma 2.4, for ε>0,

    lim supmmax2mn<2m+1|nk=1ankXk|ε,a.s.V.

    Combining |nk=1ankX|max2mn<2m+1|nk=1ankXk| and the arbitrariness of ε, (4.6) holds. So far, Theorem 3.1 has been proved.

    Let l(n)=log(n), for 0<b<1δ, by (3.8), we have

    n=1e(δ2)l(n)g(n)=n=1nδ2g(n)=n=1nδ2+bg(n)nbcn=1nδ2+b<.

    Then, (3.4) holds. From Theorem 3.1, Eqs (3.4)–(3.6) hold.

    Let l(n)=exp{(logn)ν},0<ν<1. For q>0, we have

    (logn)νqloglogn,

    so,

    exp{(logn)ν}eqloglogn=logqnqlogn.

    By (3.10), 0<δ<1, when q>m+12δ, we have

    n=1e(δ2)l(n)g(n)=n=1exp{(δ2)exp{logνn}}g(n)n=1exp{(δ2)qlogn}g(n)=n=1n(δ2)q+mg(n)nmcn=1n(δ2)q+m<.

    Then, (3.4) holds. From Theorem 3.1, Eqs (3.4)–(3.6) hold.

    When p=1, CV(|X|)CV(|X|log|X|)<, thus (4.4) and (4.5) are still valid, we just need to prove (4.6). Imitating the proof of Lemma 2.2, from CV(|X|log|X|)<, we can obtain

    k=12kkl(2k)V(|X|>cτ2k)<. (4.7)

    Combining (2.12) and the monotonically decreasing property of h(x) in (0,),

    H4:=n=11nˆE|Xn|n=11nˆE|X|(1h(|X|τn))=k=12k1n<2k1nˆE|X|(1h(|X|τn))k=12k112k1ˆE|X|(1h(|X|τ2k1))k=1ˆE|X|(1h(|X|τ2k)).

    Then, by (2.15) and (4.7),

    H4k=1j=kˆE|X|hj(|X|μ)j=1jτ2jV(|X|>μ2τ2j1)j=12jjl(2j)V(|X|>μ2τ2j)<.

    For ε>0, by (3.12) and Markov inequality,

    m=1V(max2mn<2m+1|nk=1ankXk>ε|)cm=1max2mn<2m+11nnk=1ˆE|Xk|cm=112m2m+1k=1ˆE|Xk|=ck=1ˆE|Xk|m:2m+1>k12mck=11kˆE|Xk|<.

    By Lemma 2.4, for ε>0,

    lim supmmax2mn<2m+1|nk=1ankXk|ε,a.s.V.

    Combining |nk=1ankX|max2mn<2m+1|nk=1ankXk| and the arbitrariness of ε, (4.6) holds. So far, Theorem 3.2 has been proved.

    Almost sure convergence of WNOD r.v. in Peng's theory frame is built through this essay. It is based on the corresponding definition of stochastic domination in the sub-linear expectation space, as well as the properties of WNOD r.v. and the related proving methods. Compared with the previous research of ND, END, and so on, the research in this paper is suitable for a wider range of r.v.. So, broader conclusions are reached. In future research work, we will further consider investigating more meaningful conclusions.

    Baozhen Wang: Conceptualization, Formal analysis, Investigation, Methodology, Writing-original draft, Writing-review & editing; Qunying Wu: Funding acquisition, Formal analysis, Writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This paper was supported by the National Natural Science Foundation of China (12061028) and Guangxi Colleges and Universities Key Laboratory of Applied Statistics.

    In this article, all authors disclaim any conflict of interest.



    [1] R. A. Adams, J. J. Fournier. Sobolev spaces, Vol. 140, Academic Press, 2003.
    [2] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Commun. Pure Appl. Math., 12 (1959), 623–727. doi: 10.1002/cpa.3160120405
    [3] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Commun. Pure Appl. Math., 17 (1964), 35–92. doi: 10.1002/cpa.3160170104
    [4] Y. N. Anjam, Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments, AIMS Mathematics, 5 (2020), 440–466. doi: 10.3934/math.2020030
    [5] I. Babuˇska, B. Q. Guo, J. E. Osborn, Regularity and numerical solution of eigenvalue problems with piecewise analytic data, SIAM J. Num. Comp., 26 (1989), 1534–1560. doi: 10.1137/0726090
    [6] M. Beneˇs, P. Kuˇcera, Solutions of the Navier-Stokes equations with various types of boundary conditions, Arch. Math., 98 (2012), 487–497. doi: 10.1007/s00013-012-0387-x
    [7] H. J. Choi, J. R. Kweon, The stationary Navier-Stokes system with no-slip boundary condition on polygons: corner singularity and regularity, Commun. Part. Diff. Eq., 38 (2013), 1235–1255. doi: 10.1080/03605302.2012.752386
    [8] N. Chorfi, Geometric singularities of the Stokes problem, Abstr. Appl. Anal., 2014 (2014), 1–8.
    [9] P. G. Ciarlet, Mathematical elasticity (studies in mathematics and its applications), Elsevier, 1988.
    [10] M. Dauge, Stationary Stokes and Navier-Stokes systems on two or three-dimensional domains with corners. part I. linearized equations, SIAM J. Math. Anal., 20 (1989), 74–97. doi: 10.1137/0520006
    [11] M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions, Vol. 1341, 2006.
    [12] M. Durand, Singularities in elliptic problems. In: Singularities and constructive methods for their treatment, Springer, Berlin, Heidelberg, 1985.
    [13] J. Fabricius, Stokes flow with kinematic and dynamic boundary conditions, Quart. Appl. Math., 77 (2019), 525–544. doi: 10.1090/qam/1534
    [14] S. Fucik, O. John, A. Kufner, Function spaces, Springer, Netherlands, 1977.
    [15] V. Girault, P. A. Raviart, Finite element methods for Navier-Stokes equations: theory and algorithms, Vol. 5, Springer Science and Business Media, 2012.
    [16] P. Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III, (1976), 207–274.
    [17] P. Grisvard, Elliptic problems in nonsmooth domains, Vol. 2, 2–2, Pitman Advanced Pub. Program, Boston, 1985.
    [18] B. Q. Guo, I. Babuˇska, On the regularity of elasticity problems with piecewise analytic data, Adv. Appl. Math., 14 (1993), 307–347. doi: 10.1006/aama.1993.1016
    [19] B. Q. Guo, C. Schwab, Analytic regularity of Stokes flow on polygonal domains in countably weighted Sobolev spaces, J. Comput. Appl. Math., 190 (2006), 487–519. doi: 10.1016/j.cam.2005.02.018
    [20] Y. Hou, S. Pei, On the weak solutions to steady Navier-Stokes equations with mixed boundary conditions, Math. Zeit., 291 (2019), 47–54. doi: 10.1007/s00209-018-2072-7
    [21] S. Itoh, N. Tanaka, A. Tani, On some boundary value problem for the stokes equations in an infinite sector, Anal. Appl., 4 (2006), 357–375. doi: 10.1142/S0219530506000826
    [22] V. V. Katrakhov, S. V. Kiselevskaya, A singular elliptic boundary value problem in domains with corner points. I. Function spaces, Diff. Eq., 42 (2006), 395–403.
    [23] R. B. Kellogg, J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Func. Anal., 21 (1976), 397–431. doi: 10.1016/0022-1236(76)90035-5
    [24] V. A. Kondratiˊev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Mos. Matem. Obsh., 16 (1967), 209–292.
    [25] V. A. Kondrat\acute{i}ev, O. A. Oleinik, Boundary-value problems for partial differential equations in non-smooth domains, Russ. Math. Surveys, 38 (1983), 1–86.
    [26] V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Elliptic boundary value problems in domains with point singularities, American Mathematical Society, 1997.
    [27] V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Vol. 85, American Mathematical Society, 2001.
    [28] S. Kr\breve{a}cmar, J. Neustupa, A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions, Nonlinear Anal-Theor, 47 (2001), 4169–4180. doi: 10.1016/S0362-546X(01)00534-X
    [29] S. G. Krejn, V. P. Trofimov, Holomorphic operator-valued functions of several complex variables, Funct. Anal. i Prilo\breve{z}en., 3 (1969), 85–86.
    [30] P. Ku\check{c}era, Basic properties of solution of the non-steady Navier-Stokes equations with mixed boundary conditions in a bounded domain, Annali dell' Univ. di Ferrara. Sezione VII. Scie. Matem., 55 (2009), 289–308. doi: 10.1007/s11565-009-0082-4
    [31] J. R. Kweon, Regularity of solutions for the Navier-Stokes system of incompressible flows on a polygon, J. Diff. Eq., 235 (2007), 166–198. doi: 10.1016/j.jde.2006.12.008
    [32] J. R. Kweon, Edge singular behavior for the heat equation on polyhedral cylinders in R^{3}, Potential Anal., 38 (2013), 589–610. doi: 10.1007/s11118-012-9288-7
    [33] J. R. Kweon, The compressible Stokes flows with no-slip boundary condition on non-convex polygons, J. Math. Fluid Mech., 19 (2017), 47–57. doi: 10.1007/s00021-016-0264-7
    [34] O. S. Kwon, J. R. Kweon, Compressible Navier-Stokes equations in a polyhedral cylinder with inflow boundary condition, J. Math. Fluid Mech., 20 (2018), 581–601. doi: 10.1007/s00021-017-0336-3
    [35] I. Lasiecka, K. Szulc, A. \dot{Z}ochowski, Boundary control of small solutions to fluid-structure interactions arising in coupling of elasticity with Navier-Stokes equation under mixed boundary conditions, Nonlinear Anal-Real, 44 (2018), 54–85. doi: 10.1016/j.nonrwa.2018.04.004
    [36] V. Maza, J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Rati. Mech. Anal., 194 (2009), 669–712. doi: 10.1007/s00205-008-0171-z
    [37] S. A. Nazarov, A. Novotny, K. Pileckas, On steady compressible Navier-Stokes equations in plane domains with corners, Math. Ann., 304 (1996), 121–150. doi: 10.1007/BF01446288
    [38] M. Orlt, A. M. S\ddot{a}ndig, Regularity of viscous Navier-Stokes flows in nonsmooth domains. Boundary value problems and integral equations in nonsmooth domains, Lecture Notes in Pure and Applied Mathematics, 167 (1995), 185–201.
    [39] A. M. S\ddot{a}ndig, Some applications of weighted Sobolev spaces, Vieweg+Teubner Verlag, 1987.
    [40] A. E. Taylor, Introduction to functional analysis, John Wiley and Sons, London, 1958.
    [41] R. Temam, Navier-Stokes equations: Theory and numerical analysis, Elsevier North-Holland, 1979.
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