Semi-rigid base materials are used in all pavement grades and are shown to have excellent stability. Furthermore, semi-rigid bases are known for their stiffness and strong frost resistance. However, these semi-rigid base materials are starting to require significant levels of maintenance and repair due to wear and tear. Maintenance is performed primarily to add pavement and resurfacing material, with the hope that the resurfacing layer can withstand large tensile and shear forces at the cracks of the base, thus increasing the overall durability of the pavement. Furthermore, crushing technology used for cement panel maintenance can be used to eliminate cracks in the overlay layer thus improving overall service life expectancy. The studies below study the application of crushing technology to a semi-rigid base and the subsequent alternation to the structural design of the pavement. Furthermore, the mechanical index sensitivities of pavement structures post-rubblization were analyzed using finite elements. Mechanical indexes with high sensitivity were subsequently adopted as structural damage control indexes. These indexes were used to optimize overlay structure and provide a reference for structural design post-rubblization.
Citation: Liting Yu, Lenan Wang, Jianzhong Pei, Rui Li, Jiupeng Zhang, Shihui Cheng. Structural optimization study based on crushing of semi-rigid base[J]. Electronic Research Archive, 2023, 31(4): 1769-1788. doi: 10.3934/era.2023091
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Semi-rigid base materials are used in all pavement grades and are shown to have excellent stability. Furthermore, semi-rigid bases are known for their stiffness and strong frost resistance. However, these semi-rigid base materials are starting to require significant levels of maintenance and repair due to wear and tear. Maintenance is performed primarily to add pavement and resurfacing material, with the hope that the resurfacing layer can withstand large tensile and shear forces at the cracks of the base, thus increasing the overall durability of the pavement. Furthermore, crushing technology used for cement panel maintenance can be used to eliminate cracks in the overlay layer thus improving overall service life expectancy. The studies below study the application of crushing technology to a semi-rigid base and the subsequent alternation to the structural design of the pavement. Furthermore, the mechanical index sensitivities of pavement structures post-rubblization were analyzed using finite elements. Mechanical indexes with high sensitivity were subsequently adopted as structural damage control indexes. These indexes were used to optimize overlay structure and provide a reference for structural design post-rubblization.
The Navier-Stokes equations have been derived, more than one century ago, by the engineer C. L. Navier to describe the motion of an incompressible Newtonian fluid. Later, they have been reformulated by the mathematician-physicist G. H. Stokes. Since that time, these equations continue to attract a great deal of attention due to their mathematical and physical importance. In a seminal paper [16], Leray proved the global existence of a weak solution with finite energy. It is well known that weak solutions are unique and regular in two spatial dimensions. In three dimensions, however, the question of regularity and uniqueness of weak solutions is an outstanding open problem in mathematical fluid mechanics, we refer to excellent monographs [15], [17] and [20].
More recently, stochastic versions of the Navier-Stokes equations have been considered in the literature; first by introducing a stochastic forcing term which comes from a Brownian motion (see, e.g., the first results in [2,22]). The addition of the white noise driven term to the basic governing equations is natural for both practical and theoretical applications to take into account for numerical and empirical uncertainties, and have been proposed as a model for turbulence. Later on other kinds of noises have been studied.
In this paper we consider the the Navier-Stokes equations with a stochastic forcing term modelled by a fractional Brownian motion
{∂tu(t,x)=(νΔu(t,x)−[u(t,x)⋅∇]u(t,x)−∇π(t,x))dt+Φ∂tWH(t,x)div u(t,x)=0u(0,x)=u0(x) | (1.1) |
We fix a smooth bounded domain
When
Our aim is to deal with
We shall prove a local existence and uniqueness result. Some remarks on global solutions will also be given. Let us recall also that results on the local existence of mild
In more details, in Section 2 we shall introduce the mathematical setting, in Section 3 we shall deal with the linear problem and in Section 4 we shall prove our main result.
In this section we introduce the functional setting to rewrite system (1.1) in abstract form.
Let
Lpσ= the closure in [Lp(D)]d of {u∈[C∞0(D)]d, div u=0} |
and
Gp={∇q,q∈W1,p(D)}. |
We then have the following Helmholtz decomposition
[Lp(D)]d=Lpσ⊕Gp, |
where the notation
Let us recall some results on the Stokes operator (see, e.g., [20]).
Now we fix
Now, we define the Stokes operator
In particular, for
D(Aβ)={v∈L2σ: ‖v‖2D(Aβ)=∞∑j=1λ2βj|(v,ej)|2<∞}, |
Aβv=∞∑j=1λβj(v,ej)ej. |
For negative exponents, we get the dual space:
‖A−β‖2γ(L2σ,L2σ):=∞∑j=1‖A−βej‖2L2σ=∞∑j=1λ−2βj∼∞∑j=1j−2β2d |
and the latter series in convergent for
We also recall (see, e.g., [23]) that for any
‖S(t)u‖Lpσ≤Mtd2(1r−1p)‖u‖Lrσ for 1<r≤p<∞ | (2.1) |
‖AαS(t)u‖Lrσ≤Mtα‖u‖Lrσ for 1<r<∞,α>0 | (2.2) |
for any
Lemma 2.1. We have
‖S(t)‖γ(Hd2;L2σ)≤M(2−lnt)∀t∈(0,1) |
and for
‖S(t)‖γ(Hq;L2σ)≤Mtd4−q2∀t>0 | (2.3) |
Proof The Hilbert-Schmidt norm of the semigroup can be computed. Recall that
‖S(t)‖2γ(Hq,L2σ)=∞∑j=1‖S(t)ejλq/2j‖2L2σ=∞∑j=11λqj‖e−λjtej‖2L2σ=∞∑j=1e−2λjtλqj. |
Since
‖S(t)‖2γ(Hq,L2σ)≤C∞∑j=1e−2j2dtj2qd. |
Therefore we analyse the series
● When
sd2(t)=∞∑j=1j−1e−2j2dt=e−2t+∞∑j=2j−1e−2j2dt≤e−2t+∫∞11xe−2x2dtdx. |
The integral is computed by means of the change of variable
∫∞11xe−2x2dtdx=∫∞√tdye−2y2dy. |
Hence, for
sd2(t)≤e−2t+d∫1√t1ydy+∫∞1e−2y2dy≤1−d2lnt+C. |
● When
∞∑j=1e−2j2dtj2qd≤∫∞0e−2x2dtx2qddx. |
Again, by the change of variable
∞∑j=1e−2j2dtj2qd≤tq−d2d∫∞0yd−2q−1e−2y2dy. |
The latter integral is convergent since
● When
sq(tn)≡∞∑j=1atn(j)≤∫n1atn(x)dx+atn(n)+∫∞natn(x)dx=∫∞1x−2qde−2x2dtndx+n−2qde−2n2dtn≤d(∫∞0yd−1−2qe−2y2dy) tq−d2n+Cqtqn |
where we have computed the integral by means of the change of variable
sq(tn)≤˜Ctq−d2n for any n |
and therefore for
sq(t)≤Ctd2−q. |
This proves (2.3) when
Let us define the nonlinear term by
⟨B(u,v),z⟩=−⟨B(u,z),v⟩,⟨B(u,v),v⟩=0 | (2.4) |
Then one specifies that
‖B(u,v)‖H−1≤‖u‖L4σ‖v‖L4σ |
and thus
Since
For short we shall write
First, we recall that a real fractional Brownian motion (fBm)
E[BH(t)BH(s)]:=RH(t,s)=12(t2H+s2H−|t−s|2H),s,t≥0 | (2.5) |
For more details see [18].
We are interested in the infinite dimensional fractional Brownian motion. We consider the separable Hilbert space
WH(t)=∞∑j=1ejβHj(t) | (2.6) |
where
We need to define the integral of the form
Applying the projection operator
{du(t)+Au(t) dt=B(u(t)) dt+ΦdWH(t),t>0u(0)=u0 | (2.7) |
We consider its mild solution on the time interval
Definition 2.2. A measurable function
●
● for all
u(t)=S(t)u0+ ∫t0S(t−s)B(u(s)) ds+∫t0S(t−s)ΦdWH(s) | (2.8) |
Now we consider the linear problem associated to the Navier-Stokes equation (2.7), that is
dz(t)+Az(t) dt=ΦdWH(t) | (3.1) |
When the initial condition is
z(t)=∫t0S(t−s)Φ dWH(s). | (3.2) |
To analyze its regularity we appeal to the following result.
Proposition 1. Let
If there exist
‖S(t)Φ‖γ(L2σ,L2σ)≤Ctλ∀t>0 | (3.3) |
and
λ+α2<H | (3.4) |
then
Proof. This is a well known result for
Now we use this result with
We have our regularity result for the stochastic convolution by assuming that
Proposition 2. Let
d2(1−1p)−q2<H | (3.5) |
then the process
Proof. According to Proposition 1 we have to estimate the Hilbert-Schmidt norm of the operator
Bearing in mind Lemma 2.1, when
‖S(t)Φ‖γ(L2σ,L2σ)≤‖S(t)‖γ(Hq,L2σ)‖Φ‖L(L2σ,Hq)≤Ctd4−q2 | (3.6) |
and when
‖S(t)Φ‖γ(L2σ,L2σ)≤‖S(t)‖γ(Hd2,L2σ)‖Φ‖L(L2σ,Hd2)≤Cta | (3.7) |
for any
Otherwise, when
‖S(t)Φ‖γ(L2σ,L2σ)≤‖S(t)‖L(L2σ,L2σ)‖Φ‖γ(L2σ,L2σ)≤C | (3.8) |
for all
‖S(t)A12(q−d2)Φ‖γ(L2σ,L2σ)≤‖AεS(t)‖L(L2σ,L2σ)‖A−d4−ε‖γ(L2σ,L2σ)‖Aq2‖L(Hq,L2σ)‖Φ‖L(L2σ;Hq)≤Mtε |
According to Proposition 1, choosing
∫t0S(t−s)A12(q−d2)Φ dWH(s),t∈[0,T] |
has a
ε+12[d(12−1p)−(q−d2)]<H<1 |
i.e. choosing
d2(1−1p)−q2<H<1. |
Since
Remark 1. Instead of appealing to the Sobolev embedding
There are results providing the regularity in Banach spaces; see e.g. Corollary 4.4. in the paper [4] by Čoupek, Maslowski, and Ondreját. They involve the
According to [4], assuming
‖S(t)Φ‖γ(L2σ,Lpσ)≤Ctλ∀t>0 |
Given
‖S(t)Φ‖γ(L2σ,Lpσ)≤‖Φ‖L(L2σ,Hq)‖S(t)‖γ(Hq,Lpσ). |
The
[∫D(∞∑j=1|S(t)ej(x)λq/2j|2)p2dx]1/p |
since
Therefore, we estimate the integral. Let us do it for
∫D(∞∑j=1|S(t)ej(x)λq/2j|2)p2dx=∫D(∞∑j=1λ−qje−2λjt|ej(x)|2)p2dx=∫DΠp/2n=1(∞∑jn=1λ−qjne−2λjnt|ejn(x)|2)dx |
Using the Hölder inequality, we get
∫D|ej1(x)|2|ej2(x)|2⋯|ejp/2(x)|2dx≤‖ej1‖2Lp‖ej2‖2Lp⋯‖ejp/2‖2Lp |
Hence
∫D(∞∑j=1|S(t)ej(x)λq/2j|2)p2dx≤(∞∑j=1λ−qje−2λjt‖ej‖2Lp)p/2 |
How to estimate
Finally, let us point out that for
In this section we study the Navier-Stokes initial problem (2.7) in the space
Following [7], we set
{dvdt(t)+Av(t)=B(v(t)+z(t)),t>0v(0)=u0 | (4.1) |
and we get an existence result for
Theorem 4.1. Let
Given
d2(1−1p)−q2<H | (4.2) |
then there exists a local mild
Proof. From Proposition 2 we know that
Now we observe that to find a mild solution (2.8) to equation (2.7) is equivalent to find a mild solution
v(t)=S(t)u0+ ∫t0S(t−s)B(v(s)+z(s))ds |
to equation (4.1).
We work pathwise and define a sequence by iterations: first
vj+1(t)=S(t)u0+ ∫t0S(t−s)B(z(s)+vj(s)) ds,t∈[0,T] |
for
Let us denote by
K0=max(‖u0‖Lpσ,supt∈[0,T]‖z(t)‖Lpσ). |
We shall show that there exists a random time
‖vj+1(t)‖Lpσ≤‖S(t)u0‖Lpσ+∫t0‖S(t−s)B(vj(s)+z(s))‖Lpσ ds |
We observe that from (2.1) and (2.2) we get
‖S(t)u0‖Lpσ≤‖u0‖Lpσ | (4.3) |
and
∫t0‖S(t−s)B((vj(s)+z(s))‖Lpσds≤∫t0‖A12S(t−s)A−12P div ((vj(s)+z(s))⊗(vj(s)+z(s)))‖Lpσ ds,≤∫t01(t−s)12 ‖S(t−s)A−12P div ((vj(s)+z(s))⊗(vj(s)+z(s)))‖Lpσ ds≤∫t0M(t−s)12+d2p ‖A−12P div ((vj(s)+z(s))⊗(vj(s)+z(s)))‖Lp/2σ ds≤∫t0M(t−s)12+d2p ‖(vj(s)+z(s))⊗(vj(s)+z(s))‖Lp/2σ ds≤∫t0M(t−s)12+d2p ‖vj(s)+z(s)‖2Lpσ ds | (4.4) |
From (4.3) and (4.4) we deduce that
‖vj+1(t)‖Lpσ≤K0+∫t0M(t−s)12+d2p ‖vj(s)+z(s)‖2Lpσ ds≤K0+∫t02M(t−s)12+d2p ‖z(s)‖2Lpσ ds+∫t02M(t−s)12+d2p ‖vj(s)‖2Lpσ ds |
Thus, when
supt∈[0,T]‖vj+1(t)‖Lpσ≤K0+2M T12−d2p12−d2p supt∈[0,T]‖z(t)‖2Lpσ+2M T12−d2p12−d2p (supt∈[0,T]‖vj(t)‖Lpσ)2≤K0+4pMp−dT12−d2pK20+4pMp−d T12−d2p (supt∈[0,T]‖vj(t)‖Lpσ)2 |
Now we show that if
supt∈[0,T]‖vj+1(t)‖Lpσ≤K0+4pMp−dT12−d2pK20+4pMp−dT12−d2p4K20=2K0(12+1220pMp−dT12−d2pK0). |
Hence, when
20pMp−dT12−d2pK0≤1 |
we obtain the required bound. Therefore we define the stopping time
τ=min{T,(p−d20pMK0)2pp−d} | (4.5) |
so that
20pMp−dτ12−d2pK0≤1 | (4.6) |
and obtain that
supt∈[0,τ]‖vj(t)‖Lpσ≤2K0∀j. | (4.7) |
Now, we shall show the convergence of the sequence
B(vj+1+z)−B(vj+z)=−Pdiv ((vj+1−vj)⊗vj+1+vj⊗(vj+1−vj)+(vj+1−vj)⊗z+z⊗(vj+1−vj)). |
We proceed as in (4.4) and get
‖vj+2(t)−vj+1(t)‖Lpσ≤∫t0‖S(t−s)(B(vj+1(s)+z(s))−B(vj(s)+z(s)))‖Lpσds≤∫t0M(t−s)12+d2p(‖vj+1(s)‖Lpσ+‖vj(s)‖Lpσ+2‖z(s)‖Lpσ) ‖vj+1(s)−vj(s)‖Lpσds |
Hence, using (4.7) we get
supt∈[0,τ]‖vj+2(t)−vj+1(t)‖Lpσ≤∫τ0M6K0(t−s)12+d2pds (sups∈[0,τ]‖vj+1(s)−vj(s)‖Lpσ)=12pMK0p−d τ12−d2p (supt∈[0,τ]‖vj+1(t)−vj(t)‖Lpσ) |
Setting
supt∈[0,τ]‖vj+2(t)−vj+1(t)‖Lpσ≤C0supt∈[0,τ]‖vj+1(t)−vj(t)‖Lpσ≤Cj+10supt∈[0,τ]‖v1(t)−v0(t)‖Lpσ |
Therefore
Since
Remark 2. We briefly discuss the case of cylindrical noise, i.e.
d2(1−1p)<H<1. | (4.8) |
When
1−1p<H<1 | (4.9) |
This means that
Now we show pathwise uniqueness of the solution given in Theorem 4.1.
Theorem 4.2. Let
Given
d2(1−1p)−q2<H |
then the local mild
Proof. Let
u(t)−˜u(t)= ∫t0S(t−s)(B(u(s))−B(˜u(s))) ds. |
Writing
‖u(t)−˜u(t)‖Lpσ≤ ∫t0‖S(t−s)(B(u(s))−B(˜u(s)))‖Lpσ ds≤∫t0M(t−s)12+d2p(‖u(s)‖Lpσ+‖˜u(s)‖Lpσ)‖u(s)−˜u(s)‖Lpσ ds |
Thus
sup[0,τ]‖u(t)−˜u(t)‖Lpσ≤4K0Mτ12−d2p12−d2p supt∈[0,τ]‖u(t)−˜u(t)‖Lpσ. |
Keeping in mind the definition (4.5) of
sup[0,τ]‖u(t)−˜u(t)‖Lpσ≤25sup[0,τ]‖u(t)−˜u(t)‖Lpσ |
which implies
Let us recall that [6] proved global existence an uniqueness of an
Let us begin with the case
Let us multiply equation (4.1) by
12ddt‖v(t)‖2L2σ+‖∇v(t)‖2L2=⟨B(v(t)+z(t),z(t)),v(t)⟩≤‖v(t)+z(t)‖L4σ‖z(t)‖L4σ‖∇v(t)‖L2≤12‖∇v(t)‖2L2+C2‖z(t)‖4L4σ‖v(t)‖2L2σ+C2‖z(t)‖4L4σ |
Hence
ddt‖v(t)‖2L2σ≤C‖z(t)‖4L4σ‖v(t)‖2L2σ+C‖z(t)‖4L4σ. |
As soon as
Notice that for
Similarly one proceeds when
C. Olivera is partially supported by FAPESP by the grants 2017/17670-0 and 2015/07278-0. B. Ferrario is partially supported by INdAM-GNAMPA, by PRIN 2015 "Deterministic and stochastic evolution equations" and by MIUR -Dipartimenti di Eccellenza Program (2018-2022) - Dept. of Mathematics "F. Casorati", University of Pavia.
The authors declare no conflicts of interest in this paper.
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