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On instability of Rayleigh–Taylor problem for incompressible liquid crystals under \(L^{1}\)-norm
Boundary Value Problems volume 2021, Article number: 52 (2021)
Abstract
We investigate the nonlinear Rayleigh–Taylor (RT) instability of a nonhomogeneous incompressible nematic liquid crystal in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution. Thus we construct solutions of the linearized problem that grow in time in the Sobolev space \(H^{4}\), then we show that the RT equilibrium state is linearly unstable. With the help of the established unstable solutions of the linearized problem and error estimates between the linear and nonlinear solutions, we establish the nonlinear instability of the density, the horizontal and vertical velocities under \(L^{1}\)-norm.
1 Introduction
The instability arises when steady states of two fluid layers with different densities are accelerated in the direction toward the denser fluid [1]. This phenomenon was first studied by Rayleigh [2] and then Taylor [3], thus is called the Rayleigh–Taylor (RT) instability. In the last decades, this phenomenon has been extensively investigated from both physical and numerical aspects, see [4, 5] for examples. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, such as elasticity [6–8], rotation [9], internal surface tension [10–12], magnetic fields [13–17], and so on. In particular, to the best of our knowledge, the linear Rayleigh–Taylor instability is well understood, see [5, 9], for instance; however, there are only few mathematical analysis results on nonlinear problems in the literature.
In this paper, we further mathematically prove the RT instability in incompressible liquid crystal materials in the presence of a uniform gravitational field in a bounded domain \(\Omega \subset \mathbb{R}^{3}\):
Here the unknown function ρ is the density of the nematic liquid crystals, u the velocity, and p the pressure, d represents the macroscopic average of the nematic liquid crystal orientation field. Also \(\mu >0\) is the coefficient of viscosity, \(g>0\) is the gravitational constant, \(e_{3}=(0,0,1)^{T}\) is the vertical unit vector, and \(-\rho ge_{3}\) is the gravitational force.
In this paper we study the Rayleigh–Taylor (RT) instability of system (1.1). To this purpose, we consider a density profile \(\bar{\rho }:=\bar{\rho }(x_{3})\in {{C^{5}}}(\bar{\Omega })\), which satisfies
and an RT condition
where \(x_{3}^{0}\) denotes the third component of \(x^{0}\). Then we further define a pressure p̄ (unique up to a constant) by the relation
The condition (1.3) means that there is a region in which the RT density profile has a larger density with increasing \(x_{3}\) (height), thus leading to RT instability.
Obviously, \(R_{C}:=(\bar{\rho },0,\bar{p},e_{3})\) is an RT equilibrium-state solution of the system (1.1). Now, we denote the perturbation by
then, \((\varrho ,u,q,\sigma )\) satisfies the following perturbation equations:
with the initial-boundary value conditions:
where Ω is a general bounded domain. In this article, the initial-boundary value problem (1.4)–(1.5) is called the LCRT problem.
If the perturbation is small, we omit the nonlinear terms, and thus get the linearized LCRT equations
The linearized equations (1.6) and the initial-boundary values (1.5) constitute the linearized LCRT problem. It is well-known that the linearized LCRT problem is convenient in mathematical analysis in order to have an insight into the physical and mathematical mechanisms of the instability.
1.1 Main results
Before stating our main result, we shall introduce some mathematical notations of Sobolev spaces:
where \(1< p\leqslant \infty \), k is a nonnegative integer, and the positive constant c may depend on the domain occupied by the fluids and other known physical parameters such as g, ϖ, α, and μ, and vary from line to line.
Next we state the instability result in the LCRT problem
Theorem 1
Let Ω be a \(C^{5}\)-bounded domain, and let the density profile \(\bar{\rho }\in C^{5}(\bar{\Omega })\) satisfy (1.2)–(1.3). Then, the equilibrium \((\bar{\rho },0,\bar{q},e_{3})\) to LCRT problem (1.4)–(1.5) is unstable in Hadamard sense, that is, there are positive constants Λ, \(m_{0}\), ε, \(\delta _{0}\), and
such that, for any given \(\delta \in (0,\delta _{0})\), there is a unique classical solution \((\varrho ,u,\sigma )\in C^{0}(\bar{I}_{T}, H^{4}(\Omega )\times H_{ \sigma }^{4}(\Omega )\times H^{4}(\Omega ))\), with a unique associated (perturbation) pressure \(q\in C^{0}(\bar{I}_{T},\underline{H}^{3})\), to the LCRT problem with the initial data
but the solution satisfies
for some escape time \(T^{\delta }:=\frac{1}{\Lambda }\ln \frac{2\varepsilon }{m_{0}\delta }\in I_{T}\). In addition, the initial data \(\varrho ^{0}\), \(u^{0}\), \(q^{0}\), and \(\sigma ^{0}\) satisfy the compatibility conditions:
The proof of Theorem 1 is based on a bootstrap instability method, which has its origin in [18, 19]. We mention that many authors have established various versions of the bootstrap methods for mathematical proofs of various flow instabilities, see [20–22], for example. We complete the proof of Theorem 1 in four steps. Firstly, we introduce unstable solutions to the linearized LCRT problem, in view of the linearized LCRT problem, we can obtain a growing mode ansatz of solutions, i.e., for some \(\Lambda >0\), \({{(\varrho ,u,q,\sigma ):=e^{\Lambda t}(-\bar{\rho }' \tilde{u}_{3}/\Lambda ,\tilde{u},\tilde{q},0)}}\), see Proposition 1. Secondly, by using the standard energy method, we establish a Gronwall-type energy inequality of the local-in-time solution of the LCRT problem, see Proposition 2. Thirdly, we use initial data of solutions of the linearized LCRT problem to construct initial data for solutions of the LCRT problem, so that the modified initial data \((\varrho _{0}^{\delta },u_{0}^{\delta },q_{0}^{\delta }):=\delta ( \tilde{\varrho }^{0},\tilde{u}^{0},\tilde{q}^{0})+\delta ^{2}(0,u^{r},q^{r})\) belongs to \({{H^{4}}}\times H_{\sigma }^{4}\times \underline{H}^{3}\) and satisfies necessary compatibility condition, see Proposition 4. Finally, we introduce the error estimates between the solutions of the linearized and nonlinear LCRT problems, and then prove the nonlinear solution is unstable under \(L^{1}\)-norm.
Now, we will introduce some well-known mathematical results, which will be used in the proof of Theorem 1.
Lemma 1
(1) Embedding inequalities (see [23, Theorem 4.12]):
(2) Estimates of the product of functions in Sobolev spaces (denoted as product estimates):
which can be easily verified by Hölder’s inequality and the embedding inequality (1.11)–(1.12).
(3) Interpolation inequality in \(H^{j}\) (see [23, Theorem 5.2]):
2 Linear instability
Proposition 1
Under the assumptions of Theorem 1, the LCRT equilibrium state \(R_{C}\) is linearly unstable, that is, there is an unstable solution in the form
to (1.5)–(1.6), where \(({{\tilde{u}}},\tilde{q})\in H_{\sigma }^{4}\times \underline{H}^{3}\) solves the following boundary problem:
with the constant growth rate Λ defined by
Moreover, ũ satisfies
Proof
Please refer to the proof of [24, Theorem 1.1]. □
3 Gronwall-type energy inequality of nonlinear solutions
We derive that any small solution of the LCRT problem enjoys a Gronwall-type energy inequality. We will derive such an inequality by the a priori estimate method for simplicity. Let \((\varrho ,u,\sigma )\) be a solution of LCRT problem such that
Moreover, the solution enjoys fine regularity, which makes valid the procedure of formal deduction. In addition, we rewrite (1.4) with the boundary-value condition in (1.5) as a nonhomogeneous form:
Lemma 2
Under the assumption (3.1) with sufficiently small δ, it holds that
Proof
Multiplying (3.2)2 by u in \(L^{2}\) and using integration by parts, we get
By (3.1) and product estimate, it holds that
Thus we immediately derive (3.3) from (3.4) and (3.5) by using the Young’s and Friedrichs’s inequalities. □
Lemma 3
Under the assumption (3.1) with sufficiently small δ, it holds that
Proof
Let \(\alpha {{:}}=(\alpha _{1},\alpha _{2},\alpha _{3})\) be a multiindex of order \(|\alpha |:=\alpha _{1}+\alpha _{2}+\alpha _{3}\leqslant 4\), and \(\partial ^{\alpha }:=\partial _{1}^{\alpha _{1}}\partial _{2}^{ \alpha _{2}}\partial _{3}^{\alpha _{3}}\).
Using integration by parts, we can get
thus, applying \(\partial ^{\alpha }\) to (3.2)1, and then multiplying the resulting identity by \(\partial ^{\alpha }\varrho \) in \(L^{2}\), we get
By (3.1) and product estimate, it holds that
Thus putting (3.8) into (3.7), we immediately derive (3.6). □
Lemma 4
Under the assumption (3.1) with sufficiently small δ, it holds that
Proof
Let \(1\leqslant i\leqslant 2\). Applying \(\partial _{t}^{i}\) to (1.4)2, (1.4)4, and (3.2)6, we get
Multiplying (3.11)1 with \(i=2\) by \(u_{tt}\) in \(L^{2}\), and using the integration by parts and (3.2)1, we can get that
By (3.1) and product estimate, it holds that
Moreover, by using (3.1), integration by parts, and product estimate, we can obtain
Putting the above two estimates into (3.12), and then using Friedrichs’s and Young’s inequalities, we get (3.10). Similarly, we can easily derive (3.9) from (3.11) with \(i=1\). □
Lemma 5
Under the assumption (3.1) with sufficiently small δ, it holds that
Proof
Let \(1\leqslant i\leqslant 2\), applying \(\partial _{t}^{i}\) to (3.2)3, we get
multiplying (3.17) by σ in \(L^{2}\) with \(i=0\), then using integration by parts, we get
By (3.1), integration by parts and product estimate, it holds that
Putting the above estimate into (3.18), and then using Young’s inequality, we get (3.14). Similarly, we can easily derive (3.15) and (3.16) from (3.17) with \(i=1\) and \(i=2\), respectively. □
Lemma 6
Under the assumption (3.1) with sufficiently δ, it holds that
Proof
Applying \(\partial _{t}^{i}\) to (3.2)2, we have
By (3.11)2, (3.11)3, and (3.23), for \(i=0\) and \(i=1\), we get the following Stokes problem:
where we have defined
Applying the classical Stokes estimate to (3.24) yields
and
By (3.1), we can estimate that
In addition, using (3.1) and (3.2)1, we have
By using Young’s inequality and the four estimates above, we get (3.19)–(3.22) from (3.26) and (3.27). □
Lemma 7
Under the assumption (3.1) with sufficiently δ, it holds that
Proof
Applying \(\partial _{t}^{i}\) to (3.2)3 and (3.2)6, we get the following elliptic equations:
Applying the classical regularity theory for an elliptic problem (3.36) with \({{i=0,1,2,3}}\), we have
(1) Taking \((i,k)=(2,0)\) in (3.37), we immediately get
Now we estimate \(\|\mathcal{N}_{3}\|_{2}\) and, by the definition of \(\mathcal{N}_{3}\), get
thus, by using (3.1), we get (3.32) from (3.38) and (3.39).
(2) Taking \((i,k)=(3,0)\) in (3.37), we immediately get
and similarly to (3.39), we get
thus (3.33) follows by putting (3.41) into (3.40).
(3) Taking \((i,k)=(2,1)\) in (3.37), we can obtain that
By a simple calculation, we get
thus by putting (3.43) into (3.42), we obtain (3.34).
(4) Taking \((i,k)=(3,1)\) in (3.37), we obtain that
similarly, we get that
thus, (3.35) follows form the above two estimates. □
Lemma 8
Under the assumption (3.1) with sufficiently δ, we have that
Proof
By (3.19), (3.20), (3.32), and (3.34), to get (3.46), it suffices to derive, for sufficiently small δ,
Next, we verify (3.47).
Multiplying (3.23) with \(i=1\) by \(u_{tt}\) in \(L^{2}\), we infer that
Using (1.2), (3.1), (3.28), (3.31), and Young’s inequality, we can derive from the above identity that
Next we shall estimate for \(\|u_{t}\|_{2}\) and \(\|\sigma _{t}\|_{2}\).
By using (3.29), we can obtain from (3.2)2 that
and
where n⃗ denotes the unit outer normal vector on ∂Ω. Applying the classical elliptic regularity theory to (3.50) yields that
Inserting the above estimate into (3.49), and then using interpolation inequality, we arrive at
Next we shall estimate \(u_{t}\).
We multiply (3.2)2 by \(u_{t}\) in \(L^{2}\), and then use the integration by parts to obtain
Using (3.1) and Young’s inequality, we can derive from the above identity that
Thus, we derive from (3.48), (3.51), and (3.52) that, for sufficiently small δ,
Next we estimate \(\|\sigma _{t}\|_{2}\).
By (3.39), we can derive from (3.2)3 that
Now, we estimate \(\sigma _{tt}\). Multiplying (3.36)1 with \(i=1\) by \(\sigma _{tt}\) in \(L^{2}\), we infer that
Using (3.1), (3.43), and Young’s inequality, we derive from the above identity
Thus, by (3.52), (3.54), and (3.56), we can get
Finally, by (3.2)1, we have
Thus, we get (3.46) from (3.52) and (3.53), (3.57), and above two estimates. □
Proposition 2
There exist a constant \(\delta _{1}\in (0,1)\) and \(C>0\) such that, for any \(\delta \leqslant \delta _{1}\), if the solution \((\varrho ,u,\sigma )\) of LCRT problem satisfies (3.1), then for any \(t\in (0,T)\), the solution \((\varrho ,v,\sigma )\) satisfies the Gronwall-type energy inequality
Proof
We derive from Lemmas 2–5 that, for sufficiently large constant \(c_{1}\),
where we have defined
Integrating (3.61) over \((0,t)\), and then using interpolation and Young’s inequalities, we get, for any given \(\varepsilon >0\),
where the positive constant c depends on ε.
Noting that, by (3.31), (3.58), (3.58), (3.59), and Lemmas 6–8, we easily derive that there exists a constant c such that, for sufficiently small δ,
Consequently, we immediately derive (3.60) from (3.62)–(3.64). □
Proposition 3
(1) Let \({{\bar{\rho }}}\in C^{5}(\bar{\Omega })\). Then there are a sufficiently small \(\delta _{2}\in (0,1)\) and \(K_{1}>0\) such that if \(({{\varrho ^{0},u^{0},\sigma ^{0}}})\) satisfies
and the compatibility conditions
then there exist a local existence time \(T^{\mathrm{max}}>0\) (depending on \(\delta _{2}\), the domain and the known parameters) and a unique local-in-time classical solution \((\varrho ,u,\sigma ,q)\in C^{0}({{H^{4}}}\times H_{\sigma }^{4} \times H_{0}^{4}\times \underline{H}^{3})\) to the LCRT problem.
(2) In addition, if the solution \((\varrho ,u,\sigma )\) further satisfies
then \((\varrho ,u,\sigma )\) enjoys the equivalent estimate (3.46) and the Gronwall-type energy inequality (3.60).
Remark 1
For any given initial data \(({{\varrho ^{0},u^{0},\sigma ^{0}}})\in {{H^{4}}} \times H_{\sigma }^{4}\times H_{0}^{4}\) satisfying (3.65)–(3.67) with sufficiently small \(\delta _{2}\), there exists a unique local-in-time strong solution \((\varrho ,u,\sigma ,q)\in C^{0}([0,T),H^{2}\times H_{\sigma }^{2} \times H_{0}^{2}\times H^{1})\). Moreover, the initial date of q is a weak solution to
If the condition (3.68) is further satisfied, i.e., \((\varrho ^{0},u^{0},\sigma ^{0})\) satisfies
then we can improve the regularity of \((\varrho ,u,\sigma )\) so that it is a classical solution for sufficiently small \(\delta _{2}\).
Remark 2
For any classical solution \((\varrho ,u,\sigma )\) constructed by Proposition 3, and for any given \(t_{0}\in (0,T^{\mathrm{max}})\), we take \((\varrho ,u,\sigma )|_{t=t_{0}}\) as a new initial datum. Then the new initial data can define a unique local-in-time classical solution \((\tilde{\varrho },\tilde{u},\tilde{\sigma },\tilde{q})\) constructed by Proposition 3, moreover, the initial data of q̃ is equal to \(q|_{t=t_{0}}\) by unique solvability of (3.69).
4 Construction of initial data for the nonlinear problem
For any given \(\delta >0\), let
where \(({{\tilde{\varrho }^{0},\tilde{u}^{0},\tilde{q}^{0}):=(- \bar{\rho }'\tilde{u}_{3}/\Lambda ,\tilde{u},\tilde{q})}}\), and \({{(\tilde{u},\tilde{q})\in H_{\sigma }^{4}\times \underline{H}^{3}}}\) comes from Proposition 1. Then \((\varrho ^{\mathrm{a}},u^{\mathrm{a}},q^{\mathrm{a}})\) is a solution to the linearized RTLC equations, and enjoys the estimate, for any \(i\geqslant 0\),
Moreover, by (2.4),
Next we shall modify the initial data of the linear solutions.
Proposition 4
Let \((\tilde{\varrho }^{0},\tilde{u}^{0},\tilde{q}^{0})\) be the same as in (4.1), then is a constant \(\delta _{3}\), such that for any \(\delta \in (0,\delta _{3})\), there exists \((u^{r},q^{r})\in H_{\sigma }^{4}\times \underline{H}^{3}\) enjoying the following properties:
(1) The modified initial data
belongs to \(\underline{H}_{0}^{4}\times H_{\sigma }^{4}\times \underline{H}^{3}\), and satisfies the compatibility conditions (3.69)1 and (3.70) with \((u_{0}^{\delta },q_{0}^{\delta })\) in place of \((u^{0},q^{0})\).
(2) The uniform estimate holds:
where the constant \(C_{2}\geqslant 1\) depends on the domain, the density profile, and the known parameters, but is independent of δ.
Proof
Recalling the construction of \((\tilde{u}^{0},\tilde{q}^{0})\), we can see that \((\tilde{u}^{0},\tilde{q}^{0})\) satisfies
If \((u^{r},\sigma ^{r},q^{r})\in H_{\sigma }^{4}\times H_{0}^{4}\times \underline{H}^{3}\) satisfies, for any given δ,
where \((\varrho _{0}^{\delta },u_{0}^{\delta },\sigma _{0}^{\delta },q_{0}^{ \delta })\) is given in the mode (4.3), then, by (4.5), it is easy to check that \((\varrho _{0}^{\delta },u_{0}^{\delta },\sigma _{0}^{\delta },q_{0}^{ \delta })\) \(H_{4}\times H_{\sigma }^{4}\times H_{0}^{4}\times \underline{H}^{3}\), and it satisfies the compatibility conditions (3.69)1 and (3.70) with \((u_{0}^{\delta },q_{0}^{\delta })\) in place of \((u^{0},q^{0})\). Next we construct such \((u^{r},\sigma ^{r},q^{r})\) which satisfy (4.5) for sufficiently small δ.
Since \(H^{2}\hookrightarrow L^{\infty }\), there exists a constant \(\delta _{4}>0\) such that
Moreover,
(1) There exists \((\Upsilon ,q)\in H_{0}^{2}\times \underline{H}^{1}\) such that
and
(2) There exists \((u^{r},q^{r})\in H_{0}^{4}\times \underline{H}^{3}\) such that
and \(\|(u^{r},q^{r}{{)}}\|_{S,2}\leqslant c\), where ϒ is constructed in (4.8). We mention that the constant c above is independent of δ. Thus we can get Proposition 4 from the above arguments. □
5 Error estimates and existence of escape times
Let
and \((u_{0}^{\delta },q_{0}^{\delta })\) be constructed by Proposition 4.
Noting that
then
Thus, by the first assertion in Proposition 3, one sees that there is a (nonlinear) solution \((u,q)\) of the problem defined in some time interval \(I_{T^{\mathrm{max}}}\) with the initial value \((\varrho _{0}^{\delta },u_{0}^{\delta })\). Moreover, we have \(\int {{q}}\,dx=\int {{q_{0}^{\delta }}}\,dx=0\).
Let \(\varepsilon _{0}\in (0,1)\) be a constant, which will be defined in (5.9). We define
where \(T^{\mathrm{max}}\) denotes the maximal time of existence of the solution \((\varrho ,u)\in C([0,T^{\max }),H^{4}\times H_{\sigma }^{4} )\). Obviously, \(T^{*}T^{**}>0\) and
We denote \(T^{\min }:=\min \{T^{\delta },T^{*},T^{**}\}\). By the definition of \(T^{**}\), we can deduce from the estimate (3.60) that, for all \(t< T^{\min }\),
Applying Gronwall’s inequality to the above estimate, we arrive at, for some constant \(C_{4}\),
In addition, we have the following error estimate between the nonlinear solution \((\varrho ,u)\) and the linear solution \((\varrho ^{a},u^{a})\).
Lemma 9
Let \(\varepsilon _{0}\leqslant {{\delta _{4}}}/C_{4}\). There exists a constant \(C_{5}\) such that, for any \(\delta \in (0,1)\) and any \(t\in I_{T^{min}}\),
where \((\varrho ^{d},u^{d}):=(\varrho ,u)-(\varrho ^{a},u^{a})\), \(\aleph =L^{1} \textit{ or } L^{2}\), and \(C_{5}\) is independent of \(T^{\min }\).
Proof
Please refer to [25, Lemma 3.1]. We mention that the condition \(\varepsilon _{0}\leqslant \delta _{4}/C_{4}\) makes sure that \(\inf_{x\in \bar{\Omega }}\{\varrho +\bar{\rho }\}\geqslant \inf_{x \in \bar{\Omega }}\{\bar{\rho }\}/2>0\). □
Now we define that
It is easy to see that \(m_{0}>0\) by (5.10). Now, we assert that
which can be proved by contradiction as follows:
(1) If \(T^{\min }=T^{*}< T^{\delta }\), then \(T^{*}<\infty \). Moreover, \(T^{*}\leqslant T^{\max }\) by Proposition 3. Note that we can deduce from (5.3), (5.7), and (5.9) that
which contracts (5.4). Hence \(T^{\min }\neq T^{*}\).
(2) If \(T^{\min }=T^{**}< T^{\delta }\), then \(T^{**}< T^{*}\leqslant T^{\mathrm{max}}\). Moreover, making use of (4.1), (5.1), (5.3), (5.8), and (5.9), we see that
which also contradicts (5.5). Therefore, \(T^{\min }\neq T^{**}\). We immediately see that (5.11) holds. This completes the proof of claim (5.11).
Since \(T^{\delta }< T^{*}\leqslant T^{\max }\), we can use (4.1), (5.8), and (5.9) to deduce that
Similarly, we also have
and
where \(u_{i}\) denote the ith component of \(u(T^{\delta })\) for \(1\leqslant i\leqslant 3\). This completes the proof of Theorem 1 by defining \(\varepsilon :=m_{0}\varepsilon _{0}/2\).
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The authors would like to thank the anonymous referee for invaluable suggestions, which improved the presentation of this paper.
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This work was carried out in collaboration between both authors. XL designed the study and guided the research. ML performed the analysis and wrote the first draft of the manuscript. XL and ML managed the analysis of the study. Both authors read and approved the final manuscript.
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Liu, M., Lin, X. On instability of Rayleigh–Taylor problem for incompressible liquid crystals under \(L^{1}\)-norm. Bound Value Probl 2021, 52 (2021). https://doi.org/10.1186/s13661-021-01528-3
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DOI: https://doi.org/10.1186/s13661-021-01528-3