The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.
Citation: Baoxing Zhang, Yunkun Zhang, Yuanyuan Xie. Generating irregular fractals based on iterated function systems[J]. AIMS Mathematics, 2024, 9(5): 13346-13357. doi: 10.3934/math.2024651
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The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.
In this paper, we consider the Cauchy problem for the following two-dimensional inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion:
{ρt+u⋅∇ρ=0,(t,x)∈R+×R2,ρ(ut+u⋅∇u)−Δu+∇P+σdiv(∇b⊗∇b−12|∇b|2I)=0,bt+u⋅∇b+1ν(e′(b)−σΔb)=0,divu=0,(ρ,u,b)(t,x)|t=0=(ρ0,u0,b0)(x), | (1.1) |
where the unknowns ρ=ρ(x,t), u=(u1(x,t),u2(x,t)) and b=b(x,t) stand for the density, velocity of the fluid and the spherical part of the elastic strain, respectively. P is a scalar pressure function, which guarantees the divergence-free condition of the velocity field. The coefficients ν and σ are two positive constants. In addition, we suppose that e(⋅) is a smooth convex function about b and e(0)≤0, e′(0)=0, e″(b)≤C0, where C0 is a positive constant depending on the initial data. The class of fluids is the elastic response described by a spherical strain [3]. Compared with [3], we have added the divergence-free condition to investigate the effect of density on viscoelastic rate-type fluids, while the divergence-free condition is for computational convenience.
It is easy to observe that for σ=0, the system (1.1) degenerates two distinct systems involving the inhomogeneous Navier-Stokes equation for the fluid and a transport equation with damped e′(b). Numerous researchers have extensively studied the well-posedness concern regarding the inhomogeneous Navier-Stokes equations; see [1,7,8,9,11,14] and elsewhere. However, the transport equation has a greater effect on the regularity of density than on that of velocity. Additionally, due to the presence of the damped term e′(b), the initial elasticity in system (1.1) exhibits higher regularity compared to the initial velocity.
In the case where σ>0, system (1.1) resembles the inhomogeneous magnetohydrodynamic (MHD) equations, with b as a scalar function in (1.1) that does not satisfy the divergence condition found in MHD equations. It is essential to highlight that the system (1.1) represents a simplified model, deviating from standard viscoelastic rate-type fluid models with stress-diffusion to facilitate mathematical calculations. Related studies on system (1.1) can be found in [3,4,15]. In particular, Bulíček, Málek, and Rodriguez in [5] established the well-posedness of a 2D homogeneous system (1.1) in Sobolev space. Our contribution lies in incorporating the density equation into this established framework.
Inspired by [11,18], we initially establish a priori estimates for the system (1.1). Subsequently, by using a Friedrich's method and the compactness argument, we obtain the existence and uniqueness of the solutions. Our main result is as follows:
Theorem 1.1. Let the initial data (ρ0,u0,b0) satisfy
0<m<ρ0(x)<M<∞,(u0,b0)∈H1(R2)×H2(R2),e(b0)∈L1(R2), | (1.2) |
where m,M are two given positive constants with m<M. Then system (1.1) has a global solution (ρ,u,b) such that, for any given T>0, (t,x)∈[0,T)×R2,
m<ρ(t,x)<M,u∈L∞(0,T;H1(R2))∩L2(0,T;H2(R2)),∂tu∈L∞(0,T;L2(R2))∩L2(0,T;H1(R2)),b∈L∞(0,T;H2(R2))∩L2(0,T;H3(R2)),∂tb∈L∞(0,T;H1(R2))∩L2(0,T;H2(R2)). |
Moreover, if ∇ρ0∈L4(R2), then the solution is unique.
Remark 1.1. Compared to the non-homogeneous MHD equations, handling the damping term e′(b) poses a challenge, so that we cannot obtain the time-weighted energy of the velocity field. To explore the uniqueness of the solution, it is necessary to improve the regularity of the initial density data.
The key issue to prove the global existence part of Theorem 1.1 is establishing the a priori L∞(0,T;H1(R2)) estimate on (u,∇b) for any positive time T. We cannot directly estimate the L2 estimate of (u,b), which mainly occurs in the velocity term div(∇b⊗∇b−12|∇b|2I). Therefore, we need to estimate the L2 of the ∇b equation. Afterwards, the L2 estimation of equation b was affected by a damping term e′(b), so we made an L2 estimation of equation e′(b). Finally, to show the L∞(0,T;H1(R2)) of u, we also need an estimate of the second derivative of b. In summary, we found that the initial value of the b equation needs to be one derivative higher than the initial value of the u equation.
Concerning the uniqueness of the strong solutions, a common approach is to consider the difference equations between two solutions and subsequently derive some energy estimates for the resulting system differences based on the fundamental natural energy of the system. However, for system (1.1), the presence of a damping term e′(b) of the equation b and density equation prevents the calculation of the time-weighted energy of the velocity field. To research the solution's uniqueness, we need to enhance the regularity of the initial density data.
The paper is structured as follows: Section 2 presents prior estimates for system (1.1). In Section 3, we establish the existence and uniqueness of Theorem 1.1.
Proposition 2.1. Assume that m,M are two given positive constants and 0<m≤M<∞, the initial data ρ0 satisfies 0<m≤ρ0≤M<+∞, and the initial data (√ρ0u0,∇b0)∈L2(R2)×L2(R2). Let (ρ,u,b) be a smooth solution of system (1.1), then there holds for any t>0,
0<m≤ρ(t)≤M<+∞, | (2.1) |
‖(√ρu,∇b,u)(t)‖2L2+∫t0‖(∇u,∇2b)‖2L2dτ≤C‖(√ρ0u0,∇b0)‖2L2, | (2.2) |
where C is a constant depending only on σ, ν.
Proof. First, any Lebesgue norm of ρ0 is preserved through the evolution, and 0<m≤ρ(t)≤M<+∞.
To prove (2.2), taking the L2 inner product of the second equation of (1.1) with u and integrating by parts, then we obtain
12ddt‖√ρu‖2L2+‖∇u‖2L2=−σ∫R2Δb∇b⋅udx, | (2.3) |
where we used the fact that
div(∇b⊗∇b−12|∇b|2I)=Δb∇b. |
Multiplying the third equation of (1.1) by −σΔb and integrating by parts, we obtain
σ2ddt‖∇b‖2L2+σ2ν‖Δb‖2L2−σ2ν∫R2e′(b)Δbdx=σ∫R2u⋅∇bΔbdx. | (2.4) |
Thanks to the convexity of e(b), we know
−σ2ν∫R2e′(b)Δbdx=σ2ν∫R2∇(e′(b))∇bdx=σ2ν∫R2e″(b)|∇b|2dx≥0. | (2.5) |
By inserting (2.5) into (2.4), combining the result with (2.3), one yields
ddt‖(√ρu,∇b)‖2L2+‖∇u‖2L2+C‖Δb‖2L2≤0. |
Integrating it with respect to time, we have
‖(√ρu,∇b)(t)‖2L2+∫t0‖(∇u,∇2b)‖2L2dτ≤C‖(√ρ0u0,∇b0)‖2L2. | (2.6) |
On the other hand, applying 0<m≤ρ≤M<+∞, which together with (2.6) implies
‖u‖2L∞(L2)≤m−1‖√ρu‖2L∞(L2)≤C‖(√ρ0u0,∇b0)‖2L2, |
which, along with inequality (2.6), yields (2.2).
Proposition 2.2. Under the assumptions of Proposition 2.1, the corresponding solution (ρ, u, b) of the system (1.1) admits the following bound for any t>0:
‖(∇u,∇2b,∇b,b)‖2L2+∫t0‖(∇2u,∇3b,√ρuτ,bτ,uτ)‖2L2dτ≤C, | (2.7) |
where C is a positive constant depending on m, M, u0, ρ0, and ∇b0.
Proof. First, we obtain by taking L2 inner product of (1.1)3 with e′(b) that
ddt∫R2e(b)dx+1ν‖e′(b)‖2L2≤‖u⋅∇b‖L2‖e′(b)‖L2+14ν‖e′(b)‖2L2+C‖Δb‖2L2≤12ν‖e′(b)‖2L2+C‖u‖2L4‖∇b‖2L4+C‖Δb‖2L2≤12ν‖e′(b)‖2L2+C‖u‖L2‖∇u‖L2‖∇b‖L2‖∇2b‖L2+C‖Δb‖2L2≤12ν‖e′(b)‖2L2+C(‖∇u‖2L2+‖∇2b‖2L2). |
Integrating with respect to time, we obtain
‖e(b)‖L∞(L1)+‖e′(b)‖2L2(L2)≤‖(√ρ0u0,∇b0)‖2L2+‖e(b0)‖L1. | (2.8) |
Similarly, multiplying (1.1)3 by b, we have
12ddt‖b‖2L2+‖∇b‖2L2≤C(‖b‖2L2+‖e′(b)‖2L2), |
after using (2.8) and Grönwall's inequality, we obtain
b∈L∞(0,t;L2(R2))∩L2(0,t;H1(R2)). | (2.9) |
In the following, applying Laplace operator Δ to (1.1)3 and multiplying the resulting equation by Δb; additionally, multiplying (1.1)2 by ut and (1.1)3 by bt, respectively, then integrating them on R2 and adding up all these results together, we obtain
12ddt∫R2(|Δb|2+|∇u|2+σν|∇b|2)dx+∫R2(ρ|ut|2+|bt|2+σν|∇3b|2)dx=−∫R2ρu⋅∇u⋅utdx−σ∫R2Δb∇b⋅utdx−∫R2u⋅∇bbtdx−1ν∫R2e′(b)btdx−∫R2Δ(u⋅∇b)⋅Δbdx−1ν∫R2Δe′(b)⋅Δbdx≜6∑j=1Ij. | (2.10) |
Utilizing Gagliardo-Nirenberg's, Hölder's, Young's inequalities (2.2), we estimate the first term as follows:
I1≤‖√ρ‖L∞‖√ρut‖L2‖u‖L4‖∇u‖L4≤116‖√ρut‖2L2+C‖u‖2L4‖∇u‖2L4≤116‖√ρut‖2L2+C‖u‖L2‖∇u‖2L2‖∇2u‖L2≤116‖√ρut‖2L2+116‖∇2u‖2L2+C‖∇u‖4L2. |
Similarly, by direct calculations, the other terms can be bounded as
I2≤116‖√ρut‖2L2+σ8ν‖∇3b‖2L2+C‖∇2b‖4L2,I3≤14‖bt‖2L2+C‖∇u‖2L2+C‖∇b‖2L2‖Δb‖2L2,I4≤14‖bt‖2L2+C‖e′(b)‖2L2,I5≤3σ16ν‖∇3b‖2L2+116‖∇2u‖2L2+C‖∇u‖4L2+C‖∇2b‖4L2,I6≤3σ16ν‖∇3b‖2L2+C‖∇b‖2L2. |
Next, according to the regularity theory of the Stokes system in Eq (1.1)2, it follows that
‖∇2u‖2L2≤‖ρut‖2L2+‖ρu⋅∇u‖2L2+σ‖∇bΔb‖2L2≤‖√ρut‖2L2+C‖∇u‖2L2‖∇2u‖L2+C‖∇2b‖2L2‖∇3b‖L2≤‖√ρut‖2L2+12‖∇2u‖2L2+σ2ν‖∇3b‖2L2+C(‖∇u‖4L2+‖∇2b‖4L2), |
after multiplying by 18, we arrive at
116‖∇2u‖2L2≤18‖ρut‖2L2+σ16ν‖∇3b‖2L2+C(‖∇u‖4L2+‖∇2b‖4L2). | (2.11) |
Substituting the estimates I1−I6 into (2.10) and combining inequality (2.11), we have
ddt(‖(∇u,∇b,∇2b)‖2L2+1)+‖(√ρut,bt,∇3b,∇2u)‖2L2≤C(‖(∇u,∇b,∇2b)‖2L2+1)‖(∇u,∇b,∇2b)‖2L2+C‖e′(b)‖2L2, |
which, along with Grönwall's inequality (2.2), (2.8), and (2.9), leads to
‖(∇u,∇b,∇2b)‖2L2+∫t0‖(√ρuτ,bτ,∇3b,∇2u)‖2L2dτ≤C, | (2.12) |
which completes the proof of Proposition 2.2.
Proposition 2.3. Under the assumptions of Proposition 2.2, there holds
‖(√ρut,bt,∇bt,ut)‖2L2+∫t0‖(∇uτ,∇bτ,Δbτ)‖2L2dτ≤C, | (2.13) |
where C is a positive constant depending on m, M, u0, ρ0 and b0.
Proof. Taking the derivative of Eq (1.1)2 with respect to time t, then multiplying ut on both sides of the resulting equation and integrating by parts gives
12ddt‖√ρut‖2L2+‖∇ut‖2L2=−∫R2ρtut⋅utdx−∫R2ρtu⋅∇u⋅utdx−∫R2ρut⋅∇u⋅utdx−∫R2σΔbt∇b⋅utdx−σ∫R2Δb∇bt⋅utdx. | (2.14) |
Next, we compute each term on the right-hand side of the equation above one by one using estimates (2.2) and (2.7). The bound of the first term has been estimated as
−∫R2ρtut⋅utdx=∫R2 div(ρu)ut⋅utdx=−∫R22ρuut⋅∇utdx≤C‖ρ‖L∞‖u‖L4‖ut‖L4‖∇ut‖L2≤C‖u‖12L2‖∇u‖12L2‖∇ut‖32L2‖ut‖12L2≤110‖∇ut‖2L2+C‖∇u‖2L2‖ut‖2L2. |
By using Gagliardo-Nirenberg's, Hölder's, and Young's inequalities and (2.2), we have
−∫R2ρtu⋅∇u⋅utdx=∫R2∇⋅(ρu)u⋅∇u⋅utdx=−∫R2ρu⋅∇u⋅∇u⋅utdx−∫R2ρu⋅u⋅∇2u⋅utdx−∫R2u⋅∇u⋅ρu⋅∇utdx≤‖ρ‖L∞(‖u‖L6‖∇u‖2L3‖ut‖L6+‖u‖2L6‖∇2u‖L2‖ut‖L6+‖u‖2L6‖∇u‖L6‖∇ut‖L2)≤C‖u‖13L2‖∇u‖2L2‖∇2u‖23L2‖ut‖13L2‖∇ut‖23L2+C‖u‖23L2‖∇u‖43L2‖∇2u‖L2‖ut‖13L2‖∇ut‖23L2 +C‖u‖23L2‖∇u‖53L2‖∇2u‖23L2‖∇ut‖L2≤110‖∇ut‖2L2+C‖∇2u‖2L2+C‖∇u‖2L2‖ut‖2L2. |
Similarly,
−∫R2ρut⋅∇u⋅utdx≤C‖ρ‖L∞‖∇u‖L2‖ut‖2L4≤110‖∇ut‖2L2+C‖∇u‖2L2‖ut‖2L2 |
and
−∫R2σΔbt∇b⋅utdx−σ∫R2Δb∇bt⋅utdx≤C‖Δbt‖L2‖∇b‖L4‖ut‖L4+C‖Δb‖L4‖∇bt‖L2‖ut‖L4≤C‖Δbt‖L2‖∇b‖12L2‖Δb‖12L2‖ut‖12L2‖∇ut‖12L2+C‖Δb‖12L2‖∇3b‖12L2‖∇bt‖L2‖ut‖12L2‖∇ut‖12L2≤σ16ν‖∇bt‖2L2+σ16ν‖Δbt‖2L2+110‖∇ut‖2L2+C(‖∇b‖2L2+‖∇3b‖2L2)‖∇2b‖2L2‖ut‖2L2. |
Inserting these estimates into (2.14), we have
ddt‖√ρut‖2L2+85‖∇ut‖2L2≤σ8ν‖(∇bt,Δbt)‖2L2+C‖∇2u‖2L2+C‖(∇u,∇2b,∇3b)‖2L2‖ut‖2L2. | (2.15) |
Now we turn to the b equation of (1.1). Differentiating (1.1)3 with respect to t, we obtain
btt+ut⋅∇b+u⋅∇bt+1ν(e″(b)bt−σΔbt)=0. |
Multiplying it by bt and −Δbt, integrating the resulting equation, and summing up these results, due to the divergence-free condition divu=0, we obtain
12ddt‖(bt,∇bt)‖2L2+σν‖(∇bt,Δbt)‖2L2+1ν∫R2e″(b)(bt)2dx=∫R2ut⋅b⋅∇btdx+∫R2ut⋅∇b⋅Δbtdx+∫R2u⋅∇bt⋅Δbtdx+1ν∫R2e″(b)bt⋅Δbtdx≤‖ut‖L4‖b‖L4‖∇bt‖L2+‖ut‖L4‖∇b‖L4‖Δbt‖L2+‖u‖L4‖∇bt‖L4‖Δbt‖L2+C‖bt‖L2‖Δbt‖L2≤σ4ν‖∇bt‖2L2+‖ut‖L2‖∇ut‖L2‖b‖L2‖∇b‖L2+σ8ν‖Δbt‖2L2+‖ut‖L2‖∇ut‖L2‖∇b‖L2‖Δb‖L2+σ8ν‖Δbt‖2L2+C‖u‖L2‖∇u‖L2‖∇bt‖L2‖∇2bt‖L2+σ8ν‖Δbt‖2L2+C‖bt‖2L2≤σ4ν‖∇bt‖2L2+σ2ν‖Δbt‖2L2+12‖∇ut‖2L2+C(‖∇b‖2L2+‖Δb‖2L2)‖ut‖2L2+C‖bt‖2L2+C‖∇u‖2L2‖∇bt‖2L2. | (2.16) |
Summing up (2.15) and (2.16) yields that
ddt‖(√ρut,bt,∇bt)‖2L2+‖(∇ut,∇bt,Δbt)‖2L2≤C‖∇2u‖2L2+C‖bt‖2L2+C‖(∇u,∇2b,∇3b,∇b)‖2L2‖(√ρut,bt,∇bt)‖2L2. |
Applying (2.7) and Grönwall's inequality to the above inequality, we obtain
‖(√ρut,bt,∇bt)‖2L2+∫t0‖(∇uτ,∇bτ,Δbτ)‖2L2dτ≤C. |
What's more, by the same argument of ‖u‖L∞(L2) in Proposition 2.1, we have
‖ut‖L∞(L2)≤C, |
which completes the proof of Proposition 2.3.
Proposition 2.4. Under the assumption of Proposition 2.3, it holds that for any t>0:
∫t0‖∇u‖L∞dτ≤Ct23 | (2.17) |
and
supt>0‖∇ρ(t)‖Lp≤C(t). | (2.18) |
Proof. Again, it follows from the regularity of the Stokes system
‖∇2u‖L4+‖∇P‖L4≤‖ρut‖L4+‖ρu⋅∇u‖L4+‖Δb∇b‖L4≤C(‖ut‖L4+‖u‖L∞‖∇u‖L4+‖Δb‖L4‖∇b‖L∞)≤C(‖ut‖12L2‖∇ut‖12L2+‖u‖12L2‖∇2u‖L2‖∇u‖12L2+‖∇2b‖12L2‖∇3b‖L2‖∇b‖12L2). |
By Propositions 2.1–2.3, we obtain
∫t0‖∇2u‖L4dτ+∫t0‖∇P‖L4dτ≤C(∫t0‖∇2u‖2L4dτ)12t12+C(∫t0‖∇P‖2L4dτ)12t12≤C(‖ut‖L2(L2)+‖∇ut‖L2(L2)+‖u‖12L∞(L2)‖∇u‖12L∞(L2)‖∇2u‖L2(L2)+‖∇b‖12L∞(L2)‖∇2b‖12L∞(L2)‖∇3b‖L2(L2))t12≤Ct12, |
and
∫t0‖∇u‖L∞dτ≤∫t0‖∇u‖13L2‖∇2u‖23L4dτ≤C(∫t0‖∇2u‖2L4dτ)13t23≤(‖ut‖23L2(L2)+‖∇ut‖23L2(L2)+‖u‖23L∞(L2)‖∇u‖23L∞(L2)‖∇2u‖23L2(L2)+‖∇b‖23L∞(L2)‖∇2b‖23L∞(L2)‖∇3b‖23L2(L2))t23≤Ct23, |
which leads to (2.17). Finally, we recall that the density ρ satisfies
∂tρ+u⋅∇ρ=0. |
Applying the operator ∇ to both sides of the above equation yields
∂t∇ρ+u⋅∇(∇ρ)=−∇u⋅∇ρ. |
By applying the Lp estimate to the above equation, combined with the divergence free condition implies
ddt‖∇ρ‖Lp≤‖∇u‖L∞‖∇ρ‖Lp. |
The Grönwall's inequality implies
‖∇ρ‖Lp≤‖∇ρ0‖Lpexp∫t0‖∇u‖L∞dτ≤C(t). |
We thus complete the proof of Proposition 2.4.
The section is to prove Theorem 1.1. For any given ρ0 and (u0,b0)∈Hs(R2)×Hs+1(R2), we define the initial data
ρϵ0=ρ0∗ηϵ,uϵ0=u0∗ηϵ,bϵ=b0∗ηϵ, |
where ηϵ is the standard Friedrich's mollifier with ϵ>0. With the initial data (ρϵ0,uϵ0,bϵ0), the system (1.1) has a unique global smooth solution (ρϵ,uϵ,bϵ). From Propositions 2.1 and 2.2, we obtain
m≤ρϵ(x,t)≤M, |
‖(uϵ,bϵ,∇uϵ,∇bϵ,∇2bϵ)‖2L2+∫t0‖(√ρuϵτ,bϵτ,∇3bϵ,∇2uϵ)‖2L2dτ≤C. |
By standard compactness arguments and Lions-Aubin's Lemma, we can obtain a subsequence denoted again by (uϵ,bϵ), that (uϵ,bϵ) converges strongly to (u,b) in L2(R+;Hs1)×L2(R+;Hs2), as ϵ→0, for s1<2 and s2<3. By the definition of ρϵ0 and let ϵ→0, we find that the limit ρ of ρϵ satisfies m≤ρ≤M.
Next, we shall prove the uniqueness of the solutions. Assume that (ρi,ui,bi)(i=1,2) be two solutions of system (1.1), which satisfy the regularity propositions listed in Theorem 1.1. We denote
(˜ρ,˜u,˜b,˜P)def=(ρ2−ρ1,u2−u1,b2−b1,P2−P1). |
Then the system for (˜ρ,˜u,˜b,˜P) reads
{˜ρt+u2⋅∇˜ρ=−˜u⋅∇ρ1,ρ2˜ut+ρ2u2⋅∇˜u−Δ˜u+∇˜P=˜F,˜bt+u2⋅∇˜b+1ν(e′(b2)−e′(b1)−σΔ˜b)=−˜u⋅∇b1,div˜u=0,(˜ρ,˜u,˜b)(t,x)|t=0=(0,0,0), | (3.1) |
where
˜F=−σΔ˜b∇b2−σΔb2∇˜b−˜ρ∂tu1−˜ρu1⋅∇u1−ρ2˜u⋅∇u1. |
Setting ν=σ=1 in what follows.
Step 1: Taking L2 inner product to the second equation of (3.1) with ˜u, we have
12ddt‖√ρ2˜u‖2L2+‖∇˜u‖2L2=−∫R2Δ˜b∇b2⋅˜udx−∫R2Δb2∇˜b⋅˜udx−∫R2˜ρ∂tu1⋅˜udx−∫R2˜ρu1⋅∇u1⋅˜udx−∫R2ρ2˜u⋅∇u1⋅˜udx. | (3.2) |
By Hölder's and interpolation inequalities, we have
−∫R2Δ˜b∇b2⋅˜udx−∫R2Δb1∇˜b⋅˜udx≤C‖Δ˜b‖L2‖∇b2‖L4‖˜u‖L4+C‖Δb1‖L4‖∇˜b‖L2‖˜u‖L4≤C‖Δ˜b‖L2‖∇b2‖12L2‖∇2b2‖12L2‖˜u‖12L2‖∇˜u‖12L2+C‖Δb1‖12L2‖∇3b1‖12L2‖∇˜b‖L2‖˜u‖12L2‖∇˜u‖12L2≤18‖(Δ˜b,∇˜b)‖2L2+18‖∇˜u‖2L2+C(‖∇b2‖2L2‖∇2b2‖2L2+‖∇3b1‖2L2‖∇2b1‖2L2)‖˜u‖2L2. | (3.3) |
Similarly,
−∫R2˜ρ∂tu1⋅˜udx−∫R2˜ρu1⋅∇u1⋅˜udx≤‖˜ρ‖L2(‖∂tu1‖L4+‖u1⋅∇u1‖L4)‖˜u‖L4≤‖˜ρ‖L2(‖∂tu1‖L2+‖∇∂tu1‖L2+‖u1‖L∞‖Δu1‖L2+‖u1‖L∞‖∇u1‖L2)×(‖˜u‖L2+‖∇˜u‖L2)≤18‖∇˜u‖2L2+F1(t)‖˜ρ‖2L2+C‖˜u‖2L2, | (3.4) |
where
F1(t)=‖∂tu1‖2L2+‖∇∂tu1‖2L2+‖u1‖2L∞‖Δu1‖2L2+‖u1‖2L∞‖∇u1‖2L2. |
Hölder's inequality implies
−∫R2ρ2˜u⋅∇u1⋅˜udx≤‖∇u1‖L∞‖√ρ2˜u‖2L2. | (3.5) |
By substituting above estimates (3.3)–(3.5) into (3.2), we have
ddt‖√ρ2˜u‖2L2+‖∇˜u‖2L2≤14‖Δ˜b‖2L2+14‖∇˜b‖2L2+CF2(t)‖˜u‖2L2+F1(t)‖˜ρ‖2L2, | (3.6) |
where
F2(t)=‖∇b2‖2L2‖∇2b2‖2L2+‖∇3b1‖2L2‖∇2b1‖2L2+‖∇u1‖L∞+1. |
Step 2: Taking L2 inner product to the third equation of (3.1) with ˜b−Δ˜b, we obtain
12ddt‖(˜b,∇˜b)‖2L2+‖(∇˜b,Δ˜b)‖2L2+∫R2[e′(b2)−e′(b1)]˜bdx=∫R2u2⋅∇˜b⋅Δ˜bdx−∫R2˜u⋅∇b1⋅(˜b−Δ˜b)dx+∫R2[e′(b2)−e′(b1)]Δ˜bdx. | (3.7) |
We shall estimate each term on the right-hand side of (3.7). For the first term of (3.7), using Hölder's inequality, we have
∫R2u2⋅∇˜b⋅Δ˜bdx≤‖u2‖L∞‖∇˜b‖L2‖Δ˜b‖L2≤18‖Δ˜b‖2L2+C‖u2‖2L∞‖∇˜b‖2L2. | (3.8) |
Meanwhile, we have
∫R2[e′(b2)−e′(b1)]˜bdx=∫R2e″(ξ)˜b2dx>0, | (3.9) |
where ξ is a function between b2 and b1.
Moreover,
−∫R2˜u⋅∇b1⋅(˜b−Δ˜b)dx≤C‖˜u‖L4‖∇b1‖L4(‖˜b‖L2+‖Δ˜b‖L2)≤18‖Δ˜b‖2L2+C‖˜b‖2L2+C‖˜u‖L2‖∇˜u‖L2‖∇b1‖L2‖Δb1‖L2≤18‖Δ˜b‖2L2+C‖˜b‖2L2+18‖∇˜u‖2L2+C‖˜u‖2L2‖∇b1‖2L2‖Δb1‖2L2, | (3.10) |
and
∫R2[e′(b2)−e′(b1)]Δ˜bdx=∫R2e″(ξ)˜bΔ˜bdx≤C0‖˜b‖L2‖Δ˜b‖L2≤14‖Δ˜b‖2L2+C‖˜b‖2L2. | (3.11) |
By inserting (3.8)–(3.11) into (3.7), one yields
12ddt‖(˜b,∇˜b)‖2L2+12‖(∇˜b,Δ˜b)‖2L2≤C‖u2‖2L∞‖∇˜b‖2L2+C‖˜b‖2L2+18‖∇˜u‖2L2+C‖˜u‖2L2‖∇b1‖2L2‖Δb1‖2L2. | (3.12) |
Step 3: We will derive the estimate of ‖˜ρ‖L2 as follows:
12ddt‖˜ρ‖2L2≤‖˜u⋅∇ρ1‖L2‖˜ρ‖L2≤‖˜u‖L4‖∇ρ1‖L4‖˜ρ‖L2≤‖∇˜u‖12L2‖˜u‖12L2‖∇ρ1‖L4‖˜ρ‖L2≤14‖∇˜u‖2L2+C‖∇ρ1‖43L4(‖˜ρ‖2L2+‖˜u‖2L2). | (3.13) |
Step 4: Summing up the above estimates, that is, (3.6), (3.12), and (3.13), we obtain
12ddt‖(˜ρ,√ρ2˜u,˜b,∇˜b)‖2L2+‖(∇˜u,∇˜b,Δ˜b)‖2L2≤CF5(t)‖∇˜b‖2L2+C‖˜b‖2L2+CF4(t)‖˜u‖2L2+F3(t)‖˜ρ‖2L2≤C(1+F3(t)+F4(t)+F5(t))‖(˜ρ,√ρ2˜u,˜b,∇˜b)‖2L2, | (3.14) |
where
F3(t)=‖∂tu1‖2L2+‖∇∂tu1‖2L2+‖u1‖2L∞‖Δu1‖2L2+‖u1‖2L∞‖∇u1‖2L2+‖∇ρ1‖43L4,F4(t)=‖∇b2‖2L2‖∇2b2‖2L2+‖∇3b1‖2L2‖∇2b1‖2L2+‖∇u1‖L∞+‖∇b1‖2L2‖Δb1‖2L2+‖∇ρ1‖43L4+1,F5(t)=‖u2‖2L∞. |
Noticing the fact that ∫t0(1+F3(τ)+F4(τ)+F5(τ))dτ≤Ct+C and that ‖f‖2L∞≤‖f‖L2‖∇2f‖L2, we can obtain that there exists a small ϵ0 such that
‖(˜ρ,√ρ2˜u,˜b,∇˜b)‖L∞(L2)=0, |
for t∈[0,ϵ0]. Therefore, we obtain ˜ρ(t)=˜u(t)=˜b(t)≡0 for any t∈[0,ϵ0]. The uniqueness of such strong solutions on the whole time interval [0,+∞) then follows by a bootstrap argument.
Moreover, the continuity with respect to the initial data may also be obtained by the same argument in the proof of the uniqueness, which ends the proof of Theorem 1.1.
This paper focuses on two-dimensional inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion. We have established its global solution, and the uniqueness of the solution in specific situations is also proved in this paper.
Xi Wang and Xueli Ke: Conceptualization, methodology, validation, writing-original draft, writing-review & editing. All authors have read and approved the final version of the manuscript for publication.
The authors would like to thank the anonymous referees for their suggestions which make the paper more readable.
The authors have no relevant financial or non-financial interests to disclose. The authors have no competing interests to declare that are relevant to the content of this article.
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