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Research article Topical Sections

Adjoint curves of special Smarandache curves with respect to Bishop frame

  • In this paper, adjoint curves generated by means of integral curves of special Smarandache curves with respect to the Bishop frame in three-dimensional Euclidean space were introduced. Relations between the main curve and the Bishop apparatus of these adjoint curves were obtained. Some important results were given concerning the slant helix and general helix of these curves. Finally, these findings were illustrated with figures.

    Citation: Esra Damar. Adjoint curves of special Smarandache curves with respect to Bishop frame[J]. AIMS Mathematics, 2024, 9(12): 35355-35376. doi: 10.3934/math.20241680

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  • In this paper, adjoint curves generated by means of integral curves of special Smarandache curves with respect to the Bishop frame in three-dimensional Euclidean space were introduced. Relations between the main curve and the Bishop apparatus of these adjoint curves were obtained. Some important results were given concerning the slant helix and general helix of these curves. Finally, these findings were illustrated with figures.



    In this paper, we consider the Cauchy problem for the following two-dimensional inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion:

    {ρt+uρ=0,(t,x)R+×R2,ρ(ut+uu)Δu+P+σdiv(bb12|b|2I)=0,bt+ub+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,uL(0,T;H1(R2))L2(0,T;H2(R2)),tuL(0,T;L2(R2))L2(0,T;H1(R2)),bL(0,T;H2(R2))L2(0,T;H3(R2)),tbL(0,T;H1(R2))L2(0,T;H2(R2)).

    Moreover, if ρ0L4(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(bb12|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<mM<, the initial data ρ0 satisfies 0<mρ0M<+, 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<mρ(t)M<+, (2.1)
    (ρu,b,u)(t)2L2+t0(u,2b)2L2dτC(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<mρ(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

    12ddtρu2L2+u2L2=σR2Δbbudx, (2.3)

    where we used the fact that

    div(bb12|b|2I)=Δbb.

    Multiplying the third equation of (1.1) by σΔb and integrating by parts, we obtain

    σ2ddtb2L2+σ2νΔb2L2σ2νR2e(b)Δbdx=σR2ubΔbdx. (2.4)

    Thanks to the convexity of e(b), we know

    σ2νR2e(b)Δbdx=σ2νR2(e(b))bdx=σ2νR2e(b)|b|2dx0. (2.5)

    By inserting (2.5) into (2.4), combining the result with (2.3), one yields

    ddt(ρu,b)2L2+u2L2+CΔb2L20.

    Integrating it with respect to time, we have

    (ρu,b)(t)2L2+t0(u,2b)2L2dτC(ρ0u0,b0)2L2. (2.6)

    On the other hand, applying 0<mρM<+, which together with (2.6) implies

    u2L(L2)m1ρu2L(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)2L2dτC, (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

    ddtR2e(b)dx+1νe(b)2L2ubL2e(b)L2+14νe(b)2L2+CΔb2L212νe(b)2L2+Cu2L4b2L4+CΔb2L212νe(b)2L2+CuL2uL2bL22bL2+CΔb2L212νe(b)2L2+C(u2L2+2b2L2).

    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

    12ddtb2L2+b2L2C(b2L2+e(b)2L2),

    after using (2.8) and Grönwall's inequality, we obtain

    bL(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

    12ddtR2(|Δb|2+|u|2+σν|b|2)dx+R2(ρ|ut|2+|bt|2+σν|3b|2)dx=R2ρuuutdxσR2ΔbbutdxR2ubbtdx1νR2e(b)btdxR2Δ(ub)Δbdx1νR2Δe(b)Δbdx6j=1Ij. (2.10)

    Utilizing Gagliardo-Nirenberg's, Hölder's, Young's inequalities (2.2), we estimate the first term as follows:

    I1ρLρutL2uL4uL4116ρut2L2+Cu2L4u2L4116ρut2L2+CuL2u2L22uL2116ρut2L2+1162u2L2+Cu4L2.

    Similarly, by direct calculations, the other terms can be bounded as

    I2116ρut2L2+σ8ν3b2L2+C2b4L2,I314bt2L2+Cu2L2+Cb2L2Δb2L2,I414bt2L2+Ce(b)2L2,I53σ16ν3b2L2+1162u2L2+Cu4L2+C2b4L2,I63σ16ν3b2L2+Cb2L2.

    Next, according to the regularity theory of the Stokes system in Eq (1.1)2, it follows that

    2u2L2ρut2L2+ρuu2L2+σbΔb2L2ρut2L2+Cu2L22uL2+C2b2L23bL2ρut2L2+122u2L2+σ2ν3b2L2+C(u4L2+2b4L2),

    after multiplying by 18, we arrive at

    1162u2L218ρut2L2+σ16ν3b2L2+C(u4L2+2b4L2). (2.11)

    Substituting the estimates I1I6 into (2.10) and combining inequality (2.11), we have

    ddt((u,b,2b)2L2+1)+(ρut,bt,3b,2u)2L2C((u,b,2b)2L2+1)(u,b,2b)2L2+Ce(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)2L2dτC, (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)2L2dτC, (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

    12ddtρut2L2+ut2L2=R2ρtututdxR2ρtuuutdxR2ρutuutdxR2σΔbtbutdxσR2Δbbtutdx. (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

    R2ρtututdx=R2 div(ρu)ututdx=R22ρuututdxCρLuL4utL4utL2Cu12L2u12L2ut32L2ut12L2110ut2L2+Cu2L2ut2L2.

    By using Gagliardo-Nirenberg's, Hölder's, and Young's inequalities and (2.2), we have

    R2ρtuuutdx=R2(ρu)uuutdx=R2ρuuuutdxR2ρuu2uutdxR2uuρuutdxρL(uL6u2L3utL6+u2L62uL2utL6+u2L6uL6utL2)Cu13L2u2L22u23L2ut13L2ut23L2+Cu23L2u43L22uL2ut13L2ut23L2 +Cu23L2u53L22u23L2utL2110ut2L2+C2u2L2+Cu2L2ut2L2.

    Similarly,

    R2ρutuutdxCρLuL2ut2L4110ut2L2+Cu2L2ut2L2

    and

     R2σΔbtbutdxσR2ΔbbtutdxCΔbtL2bL4utL4+CΔbL4btL2utL4CΔbtL2b12L2Δb12L2ut12L2ut12L2+CΔb12L23b12L2btL2ut12L2ut12L2σ16νbt2L2+σ16νΔbt2L2+110ut2L2+C(b2L2+3b2L2)2b2L2ut2L2.

    Inserting these estimates into (2.14), we have

    ddtρut2L2+85ut2L2σ8ν(bt,Δbt)2L2+C2u2L2+C(u,2b,3b)2L2ut2L2. (2.15)

    Now we turn to the b equation of (1.1). Differentiating (1.1)3 with respect to t, we obtain

    btt+utb+ubt+1ν(e(b)btσΔbt)=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+1νR2e(b)(bt)2dx=R2utbbtdx+R2utbΔbtdx+R2ubtΔbtdx+1νR2e(b)btΔbtdxutL4bL4btL2+utL4bL4ΔbtL2+uL4btL4ΔbtL2+CbtL2ΔbtL2σ4νbt2L2+utL2utL2bL2bL2+σ8νΔbt2L2+utL2utL2bL2ΔbL2+σ8νΔbt2L2+CuL2uL2btL22btL2+σ8νΔbt2L2+Cbt2L2σ4νbt2L2+σ2νΔbt2L2+12ut2L2+C(b2L2+Δb2L2)ut2L2+Cbt2L2+Cu2L2bt2L2. (2.16)

    Summing up (2.15) and (2.16) yields that

    ddt(ρut,bt,bt)2L2+(ut,bt,Δbt)2L2C2u2L2+Cbt2L2+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)2L2dτC.

    What's more, by the same argument of uL(L2) in Proposition 2.1, we have

    utL(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:

    t0uLdτCt23 (2.17)

    and

    sup (2.18)

    Proof. Again, it follows from the regularity of the Stokes system

    \begin{align*} &\Vert \nabla^2 u \Vert _{L^4}+\|\nabla P\|_{L^4} \\& \le \Vert \rho u_t \Vert _{L^4} + \Vert \rho u \cdot \nabla u \Vert _{L^4} + \Vert \Delta b \nabla b \Vert _{L^4} \\ & \le C (\Vert u_t \Vert _{L^4} + \Vert u \Vert _{L^\infty} \Vert \nabla u \Vert _{L^4} + \Vert \Delta b \Vert _{L^4} \Vert \nabla b \Vert _{L^\infty} )\\ & \le C (\Vert u_t \Vert _{L^2}^ \frac{1}{2}\Vert \nabla u_t \Vert _{L^2}^ \frac{1}{2} + \Vert u \Vert _{L^2}^ \frac{1}{2} \Vert \nabla^2 u \Vert _{L^2} \Vert \nabla u \Vert _{L^2}^ \frac{1}{2} + \Vert \nabla^2 b \Vert _{L^2}^ \frac{1}{2} \Vert \nabla^3 b \Vert _{L^2} \Vert \nabla b \Vert _{L^2}^ \frac{1}{2} ). \end{align*}

    By Propositions 2.1–2.3, we obtain

    \begin{align*} \begin{split} \int_{0}^{t} \Vert \nabla^2 u \Vert _{L^4} \mathrm{d} \tau+ &\int_{0}^{t} \Vert \nabla P \Vert _{L^4} \mathrm{d}\tau \le C (\int_{0}^{t} \Vert \nabla^2 u \Vert _{L^4}^2 \mathrm{d} \tau ) ^\frac{1}{2} t^\frac{1}{2} +C (\int_{0}^{t} \Vert \nabla P \Vert _{L^4}^2 \mathrm{d} \tau ) ^\frac{1}{2} t^\frac{1}{2}\\ &\le C\big(\Vert u_t \Vert _{L^2(L^2)}+ \Vert \nabla u_t \Vert _{L^2(L^2)}+ \Vert u \Vert _{L^{\infty}(L^2)}^\frac{1}{2} \Vert \nabla u \Vert _{L^{\infty}(L^2)}^\frac{1}{2} \Vert \nabla^2 u \Vert _{L^2(L^2)} \\ &\qquad + \Vert \nabla b \Vert _{L^{\infty}(L^2)}^\frac{1}{2} \Vert \nabla^2 b \Vert _{L^{\infty}(L^2)}^\frac{1}{2} \Vert \nabla^3 b \Vert _{L^2(L^2)}\big)t^\frac{1}{2}\\ &\le C t^\frac{1}{2}, \end{split}& \end{align*}

    and

    \begin{align*} \begin{split} \int_{0}^{t} \Vert \nabla u \Vert _{L^\infty} \mathrm{d} \tau &\le \int_{0}^{t} \Vert \nabla u \Vert _{L^2}^\frac{1}{3} \Vert \nabla^2 u \Vert_{L^4} ^\frac{2}{3} \mathrm{d} \tau \le C (\int_{0}^{t} \Vert \nabla^2 u \Vert _{L^4}^2 \mathrm{d} \tau ) ^\frac{1}{3} t^\frac{2}{3} \\ &\le \Big(\Vert u_t \Vert _{L^2(L^2)}^{\frac{2}{3}}+ \Vert \nabla u_t \Vert _{L^2(L^2)}^{\frac{2}{3}}+ \Vert u \Vert _{L^{\infty}(L^2)}^\frac{2}{3} \Vert \nabla u \Vert _{L^{\infty}(L^2)}^\frac{2}{3} \Vert \nabla^2 u \Vert _{L^2(L^2)}^{\frac{2}{3}} \\ & \qquad + \Vert \nabla b \Vert _{L^{\infty}(L^2)}^\frac{2}{3} \Vert \nabla^2 b\Vert _{L^{\infty}(L^2)}^\frac{2}{3} \Vert \nabla^3 b \Vert _{L^2(L^2)}^{\frac{2}{3}}\Big)t^\frac{2}{3}\\ &\le C t^\frac{2}{3}, \end{split}& \end{align*}

    which leads to (2.17). Finally, we recall that the density \rho satisfies

    \begin{align*} \partial_t\rho+u\cdot\nabla\rho = 0. \end{align*}

    Applying the operator \nabla to both sides of the above equation yields

    \begin{align*} \partial_t\nabla\rho+u\cdot\nabla (\nabla\rho) = -\nabla u\cdot\nabla\rho. \end{align*}

    By applying the L^p estimate to the above equation, combined with the divergence free condition implies

    \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t}\|\nabla\rho\|_{L^p}\leq\|\nabla u\|_{L^\infty}\|\nabla \rho\|_{L^p}. \end{align*}

    The Grönwall's inequality implies

    \begin{align*} \|\nabla \rho\|_{L^p}\leq\|\nabla\rho_0\|_{L^p}\exp\int_{0}^{t}\|\nabla u\|_{L^\infty}\mathrm{d}\tau\leq C(t). \end{align*}

    We thus complete the proof of Proposition 2.4.

    The section is to prove Theorem 1.1. For any given \rho_0 and (u_0, b_0)\in H^s(\mathbb{R}^2)\times H^{s+1}(\mathbb{R}^2) , we define the initial data

    \rho_0^{\epsilon} = \rho_0\ast\eta_\epsilon, \quad u_0^{\epsilon} = u_0\ast\eta_\epsilon, \quad b^{\epsilon} = b_0\ast\eta_\epsilon,

    where \eta_\epsilon is the standard Friedrich's mollifier with \epsilon > 0 . With the initial data (\rho_0^{\epsilon}, u_0^{\epsilon}, b_0^{\epsilon}) , the system (1.1) has a unique global smooth solution (\rho^{\epsilon}, u^{\epsilon}, b^{\epsilon}) . From Propositions 2.1 and 2.2, we obtain

    m\leq\rho^{\epsilon}(x, t)\leq M,
    \begin{align*} \Vert (u^{\epsilon}, b^{\epsilon}, \nabla u^{\epsilon}, \nabla b^{\epsilon}, \nabla^2 b^{\epsilon}) \Vert_{L^2}^2 + \int_{0}^{t} \Vert (\sqrt{\rho} u^{\epsilon}_\tau , b^{\epsilon}_\tau, \nabla ^3 b^{\epsilon}, \nabla^2 u^{\epsilon}) \Vert_{L^2}^2\mathrm{d} \tau\le C. \end{align*}

    By standard compactness arguments and Lions-Aubin's Lemma, we can obtain a subsequence denoted again by (u^{\epsilon}, b^{\epsilon}) , that (u^{\epsilon}, b^{\epsilon}) converges strongly to (u, b) in L^2(\mathbb{R}^+; H^{s_1})\times L^2(\mathbb{R}^+; H^{s_2}) , as \epsilon\rightarrow 0 , for s_1 < 2 and s_2 < 3 . By the definition of \rho_0^{\epsilon} and let \epsilon\rightarrow 0 , we find that the limit \rho of \rho^{\epsilon} satisfies m\leq\rho\leq M.

    Next, we shall prove the uniqueness of the solutions. Assume that (\rho_i, u_i, b_i)\, (i = 1, 2) be two solutions of system (1.1), which satisfy the regularity propositions listed in Theorem 1.1. We denote

    \begin{align*} (\tilde{\rho}, \tilde{u}, \tilde{b}, \tilde{P})\overset{def}{ = }(\rho_2-\rho_1, u_2-u_1, b_2-b_1, P_2-P_1). \end{align*}

    Then the system for (\tilde{\rho}, \tilde{u}, \tilde{b}, \tilde{P}) reads

    \begin{equation} \begin{cases} \tilde{\rho}_t+u_2\cdot\nabla\tilde{\rho} = -\tilde{u}\cdot\nabla\rho_1, \\ \rho_2\tilde{u}_t+\rho_2 u_2\cdot\nabla\tilde{u}-\Delta\tilde{u}+\nabla\tilde{P} = \tilde{F}, \\ \tilde{b}_t+u_2\cdot\nabla\tilde{b}+\frac{1}{\nu}(e'(b_2)-e'(b_1)-\sigma\Delta \tilde{b}) = -\tilde{u}\cdot\nabla b_1, \\ \mathrm{div}\, \tilde{u} = 0, \\ (\tilde{\rho}, \tilde{u}, \tilde{b})(t, x)|_{t = 0} = (0, 0, 0), \end{cases} \end{equation} (3.1)

    where

    \tilde{F} = -\sigma\Delta\tilde{b}\nabla b_2-\sigma\Delta b_2 \nabla\tilde{b}-\tilde{\rho}\partial_tu_1-\tilde{\rho}u_1\cdot\nabla u_1-\rho_2\tilde{u}\cdot\nabla u_1.

    Setting \nu = \sigma = 1 in what follows.

    Step 1: Taking L^2 inner product to the second equation of (3.1) with \tilde{u} , we have

    \begin{align} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\sqrt{\rho_2}\tilde{u}\|_{L^2}^2+\|\nabla\tilde{u}\|_{L^2}^2 = &-\int_{\mathbb{R}^2}\Delta\tilde{b}\nabla b_2\cdot \tilde{u}\mathrm{d}x -\int_{\mathbb{R}^2} \Delta b_2 \nabla\tilde{b}\cdot\tilde{u}\mathrm{d}x-\int_{\mathbb{R}^2}\tilde{\rho}\partial_tu_1\cdot\tilde{u}\mathrm{d}x\\ &-\int_{\mathbb{R}^2}\tilde{\rho}u_1\cdot\nabla u_1\cdot\tilde{u}\mathrm{d}x-\int_{\mathbb{R}^2}\rho_2\tilde{u}\cdot\nabla u_1\cdot\tilde{u}\mathrm{d}x. \end{align} (3.2)

    By Hölder's and interpolation inequalities, we have

    \begin{equation} \begin{split} &-\int_{\mathbb{R}^2}\Delta\tilde{b}\nabla b_2\cdot \tilde{u}\mathrm{d}x -\int_{\mathbb{R}^2} \Delta b_1 \nabla\tilde{b}\cdot\tilde{u}\mathrm{d}x\\ &\leq C\|\Delta\tilde{b}\|_{L^2}\|\nabla b_2\|_{L^4}\|\tilde{u}\|_{L^4}+C\|\Delta b_1 \|_{L^4}\|\nabla\tilde{b}\|_{L^2}\|\tilde{u}\|_{L^4}\\ &\leq C\|\Delta\tilde{b}\|_{L^2}\|\nabla b_2\|_{L^2}^{\frac{1}{2}}\|\nabla^2 b_2\|_{L^2}^{\frac{1}{2}}\|\tilde{u}\|_{L^2}^{\frac{1}{2}}\|\nabla\tilde{u}\|_{L^2}^{\frac{1}{2}}\\ &\quad+C\|\Delta b_1 \|_{L^2}^{\frac{1}{2}}\|\nabla^3 b_1\|_{L^2}^{\frac{1}{2}}\|\nabla\tilde{b}\|_{L^2}\|\tilde{u}\|_{L^2}^{\frac{1}{2}}\|\nabla \tilde{u}\|_{L^2}^{\frac{1}{2}}\\ &\leq \frac{1}{8}\|(\Delta\tilde{b}, \nabla\tilde{b})\|_{L^2}^2+\frac{1}{8}\|\nabla\tilde{u}\|_{L^2}^2+C(\|\nabla b_2\|_{L^2}^{2}\|\nabla^2 b_2\|_{L^2}^{2}+\|\nabla^3 b_1\|_{L^2}^2\|\nabla^2 b_1\|_{L^2}^{2})\|\tilde{u}\|_{L^2}^2. \end{split} \end{equation} (3.3)

    Similarly,

    \begin{equation} \begin{split} &-\int_{\mathbb{R}^2}\tilde{\rho}\partial_tu_1\cdot\tilde{u}\mathrm{d}x-\int_{\mathbb{R}^2}\tilde{\rho}u_1\cdot\nabla u_1\cdot\tilde{u}\mathrm{d}x\\ &\leq \|\tilde{\rho}\|_{L^2}(\|\partial_tu_1\|_{L^4}+\|u_1\cdot\nabla u_1\|_{L^4})\|\tilde{u}\|_{L^4}\\ &\leq \|\tilde{\rho}\|_{L^2}\big(\|\partial_tu_1\|_{L^2}+\|\nabla\partial_tu_1\|_{L^2}+\|u_1\|_{L^\infty}\|\Delta u_1\|_{L^2}+\| u_1\|_{L^\infty}\|\nabla u_1\|_{L^2}\big)\\ &\quad\times(\|\tilde{u}\|_{L^2}+\|\nabla\tilde{u}\|_{L^2}) \\ &\leq \frac{1}{8}\|\nabla\tilde{u}\|_{L^2}^2+\mathcal{F}_1(t)\|\tilde{\rho}\|_{L^2}^2+C\|\tilde{u}\|_{L^2}^2, \end{split} \end{equation} (3.4)

    where

    \mathcal{F}_1(t) = \|\partial_tu_1\|_{L^2}^2+\|\nabla\partial_tu_1\|_{L^2}^2+\|u_1\|_{L^\infty}^2\|\Delta u_1\|_{L^2}^2+\|u_1\|_{L^\infty}^2\|\nabla u_1\|_{L^2}^2.

    Hölder's inequality implies

    \begin{align} -\int_{\mathbb{R}^2}\rho_2\tilde{u}\cdot\nabla u_1\cdot\tilde{u}\mathrm{d}x\leq\|\nabla u_1\|_{L^\infty}\|\sqrt{\rho_2}\tilde{u}\|_{L^2}^2. \end{align} (3.5)

    By substituting above estimates (3.3)–(3.5) into (3.2), we have

    \begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\|\sqrt{\rho_2}\tilde{u}\|_{L^2}^2+\|\nabla\tilde{u}\|_{L^2}^2\leq\frac{1}{4}\|\Delta\tilde{b}\|_{L^2}^2+\frac{1}{4}\|\nabla\tilde{b}\|_{L^2}^2+C\mathcal{F}_2(t)\|\tilde{u}\|_{L^2}^2+\mathcal{F}_1(t)\|\tilde{\rho}\|_{L^2}^2, \end{align} (3.6)

    where

    \mathcal{F}_2(t) = \|\nabla b_2\|_{L^2}^{2}\|\nabla^2 b_2\|_{L^2}^{2}+\|\nabla^3 b_1\|_{L^2}^2\|\nabla^2 b_1\|_{L^2}^{2}+\|\nabla u_1\|_{L^\infty}+1.

    Step 2: Taking L^2 inner product to the third equation of (3.1) with \tilde{b}-\Delta\tilde{b} , we obtain

    \begin{align} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|(\tilde{b}, \nabla\tilde{b})\|_{L^2}^2+\|(\nabla\tilde{b}, \Delta\tilde{b})\|_{L^2}^2+\int_{\mathbb{R}^2}[e'(b_2)-e'(b_1)]\tilde{b}\mathrm{d}x\\ & = \int_{\mathbb{R}^2}u_2\cdot\nabla \tilde{b}\cdot\Delta\tilde{b}\mathrm{d}x-\int_{\mathbb{R}^2}\tilde{u}\cdot\nabla b_1\cdot(\tilde{b}-\Delta\tilde{b})\mathrm{d}x+\int_{\mathbb{R}^2}[e'(b_2)-e'(b_1)]\Delta\tilde{b}\mathrm{d}x. \end{align} (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

    \begin{align} \int_{\mathbb{R}^2}u_2\cdot\nabla \tilde{b}\cdot\Delta\tilde{b}\mathrm{d}x\leq\|u_2\|_{L^\infty}\|\nabla\tilde{b}\|_{L^2}\|\Delta\tilde{b}\|_{L^2}\leq\frac{1}{8}\|\Delta\tilde{b}\|_{L^2}^2+C\|u_2\|_{L^\infty}^2\|\nabla\tilde{b}\|_{L^2}^2. \end{align} (3.8)

    Meanwhile, we have

    \begin{align} \int_{\mathbb{R}^2}[e'(b_2)-e'(b_1)]\tilde{b}\mathrm{d}x = \int_{\mathbb{R}^2}e''(\xi)\tilde{b}^2\mathrm{d}x > 0, \end{align} (3.9)

    where \xi is a function between b_2 and b_1 .

    Moreover,

    \begin{align} &-\int_{\mathbb{R}^2}\tilde{u}\cdot\nabla b_1\cdot(\tilde{b}-\Delta\tilde{b})\mathrm{d}x\leq C\|\tilde{u}\|_{L^4}\|\nabla b_1\|_{L^4}(\|\tilde{b}\|_{L^2}+\|\Delta\tilde{b}\|_{L^2})\\ &\leq \frac{1}{8}\|\Delta\tilde{b}\|_{L^2}^2+C\|\tilde{b}\|_{L^2}^2+C\|\tilde{u}\|_{L^2}\|\nabla\tilde{u}\|_{L^2}\|\nabla b_1\|_{L^2}\|\Delta b_1\|_{L^2}\\ &\leq \frac{1}{8}\|\Delta\tilde{b}\|_{L^2}^2+C\|\tilde{b}\|_{L^2}^2+\frac{1}{8}\|\nabla\tilde{u}\|_{L^2}^2+C\|\tilde{u}\|_{L^2}^2\|\nabla b_1\|_{L^2}^2\|\Delta b_1\|_{L^2}^2, \end{align} (3.10)

    and

    \begin{align} \int_{\mathbb{R}^2}[e'(b_2)-e'(b_1)]\Delta\tilde{b}\mathrm{d}x& = \int_{\mathbb{R}^2}e''(\xi)\tilde{b}\Delta\tilde{b}\mathrm{d}x\\ &\le C_0 \|\tilde{b}\|_{L^2}\|\Delta\tilde{b}\|_{L^2}\le\frac{1}{4}\|\Delta\tilde{b}\|_{L^2}^2+C\|\tilde{b}\|_{L^2}^2. \end{align} (3.11)

    By inserting (3.8)–(3.11) into (3.7), one yields

    \begin{equation} \begin{split} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|(\tilde{b}, \nabla\tilde{b})\|_{L^2}^2+\frac{1}{2}\|(\nabla\tilde{b}, \Delta\tilde{b})\|_{L^2}^2\\ &\leq C\|u_2\|_{L^\infty}^2\|\nabla\tilde{b}\|_{L^2}^2+C\|\tilde{b}\|_{L^2}^2+\frac{1}{8}\|\nabla\tilde{u}\|_{L^2}^2+C\|\tilde{u}\|_{L^2}^2\|\nabla b_1\|_{L^2}^2\|\Delta b_1\|_{L^2}^2. \end{split} \end{equation} (3.12)

    Step 3: We will derive the estimate of \|\tilde{\rho}\|_{L^2} as follows:

    \begin{align} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\tilde{\rho}\|_{L^2}^2&\leq \|\tilde{u}\cdot\nabla\rho_1\|_{L^2}\|\tilde{\rho}\|_{L^2}\\ &\leq\|\tilde{u}\|_{L^4}\|\nabla\rho_1\|_{L^4}\|\tilde{\rho}\|_{L^2}\\ &\leq \|\nabla\tilde{u}\|_{L^2}^\frac{1}{2}\|\tilde{u}\|_{L^2}^{\frac{1}{2}}\|\nabla\rho_1\|_{L^4}\|\tilde{\rho}\|_{L^2}\\ &\leq \frac{1}{4}\|\nabla\tilde{u}\|_{L^2}^2+C\|\nabla\rho_1\|_{L^4}^{\frac{4}{3}}(\|\tilde{\rho}\|_{L^2}^2+\|\tilde{u}\|_{L^2}^2). \end{align} (3.13)

    Step 4: Summing up the above estimates, that is, (3.6), (3.12), and (3.13), we obtain

    \begin{align} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|(\tilde{\rho}, \sqrt{\rho_2}\tilde{u}, \tilde{b}, \nabla\tilde{b})\|_{L^2}^2+\|(\nabla\tilde{u}, \nabla\tilde{b}, \Delta\tilde{b})\|_{L^2}^2\\ &\leq C\mathcal{F}_5(t)\|\nabla\tilde{b}\|_{L^2}^2+C\|\tilde{b}\|_{L^2}^2+C\mathcal{F}_4(t)\|\tilde{u}\|_{L^2}^2+\mathcal{F}_3(t)\|\tilde{\rho}\|_{L^2}^2\\ &\leq C(1+\mathcal{F}_3(t)+\mathcal{F}_4(t)+\mathcal{F}_5(t))\|(\tilde{\rho}, \sqrt{\rho_2}\tilde{u}, \tilde{b}, \nabla\tilde{b})\|_{L^2}^2, \end{align} (3.14)

    where

    \begin{align*} &\mathcal{F}_3(t) = \|\partial_tu_1\|_{L^2}^2+\|\nabla\partial_tu_1\|_{L^2}^2+\|u_1\|_{L^\infty}^2\|\Delta u_1\|_{L^2}^2+\|u_1\|_{L^\infty}^2\|\nabla u_1\|_{L^2}^2+\|\nabla\rho_1\|_{L^4}^{\frac{4}{3}}, \\ &\mathcal{F}_4(t) = \|\nabla b_2\|_{L^2}^{2}\|\nabla^2 b_2\|_{L^2}^{2}+\|\nabla^3 b_1\|_{L^2}^2\|\nabla^2 b_1\|_{L^2}^{2}+\|\nabla u_1\|_{L^\infty}+\|\nabla b_1\|_{L^2}^2\|\Delta b_1\|_{L^2}^2\\ &\qquad\quad+\|\nabla\rho_1\|_{L^4}^{\frac{4}{3}}+1, \\ &\mathcal{F}_5(t) = \|u_2\|_{L^\infty}^2. \end{align*}

    Noticing the fact that \int_0^t \big(1+\mathcal{F}_3(\tau)+\mathcal{F}_4(\tau)+\mathcal{F}_5(\tau)\big)\mathrm{d}\tau\le Ct+C and that \|f\|_{L^\infty}^2\leq\|f\|_{L^2}\|\nabla^2f\|_{L^2} , we can obtain that there exists a small \epsilon_0 such that

    \begin{equation*} \begin{split} \|(\tilde{\rho}, \sqrt{\rho_2}\tilde{u}, \tilde{b}, \nabla\tilde{b})\|_{L^\infty(L^2)} = 0, \end{split} \end{equation*}

    for t\in [0, \epsilon_0] . Therefore, we obtain \tilde{\rho}(t) = \tilde{u}(t) = \tilde{b}(t)\equiv 0 for any t\in [0, \epsilon_0] . The uniqueness of such strong solutions on the whole time interval [0, +\infty) 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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