Abstract
One of the most delicate issues in the analysis of singular limits for the Navier-Stokes-Fourier system in the low Mach number regime is the influence of acoustic waves. If the physical domain is bounded and the complete slip boundary conditions imposed, the acoustic waves, being reflected by the boundary, inevitably develop high frequency oscillations resulting in the weak convergence of the velocity field, in particular, its gradient part converges to zero only in the sense of integral means. This rather unpleasant phenomenon creates additional problems when handling the convective term in the momentum equation. Here, we focus on the mechanisms by which the acoustic energy may be dissipated, and the ways how the dissipation may be used in order to show strong (pointwise) convergence of the velocities in the incompressible limit.
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As we have seen in the previous chapters, one of the most delicate issues in the analysis of singular limits for the Navier-Stokes-Fourier system in the low Mach number regime is the influence of acoustic waves. If the physical domain is bounded and the complete slip boundary conditions (cf. (5.15)) imposed, the acoustic waves, being reflected by the boundary, inevitably develop high frequency oscillations resulting in the weak convergence of the velocity field, in particular, its gradient part converges to zero only in the sense of integral means. This rather unpleasant phenomenon creates additional problems when handling the convective term in the momentum equation (cf. Sects. 5.4.7, 6.6.3 above). In this chapter, we focus on the mechanisms sofar neglected by which the acoustic energy may be dissipated, and the ways how the dissipation may be used in order to show strong (pointwise) convergence of the velocities.
The principal mechanism of dissipation in the Navier-Stokes-Fourier system is of course viscosity, here imposed through Newton’s rheological law. At a first glance, the presence of the viscous stress \(\mathbb{S}\) in the momentum equation does not seem to play any significant role in the analysis of acoustic waves. In the situation described in Sect. 4.4.1, the acoustic equation can be written in the form
Replacing for simplicity \(\mathrm{div}_{x}\mathbb{S}_{\varepsilon }\) by \(\Delta \mathbf{V}_{\varepsilon }\), we examine the associated eigenvalue problem:
Applying the divergence operator to the second equation and using the first one to express all quantities in terms of r, we arrive at the eigenvalue problem
Under the periodic boundary conditions, meaning \(\Omega = \mathcal{T}^{3}\), the corresponding eigenvalues are given as
where \(\Lambda _{n}\) are the (real non-negative) eigenvalues of the Laplace operator supplemented with the periodic boundary conditions. It is easy to check that
Moreover, the corresponding eigenfunctions read
where r n are the eigenfunctions of the Laplacian supplemented with the periodic boundary conditions.
The same result is obtained provided the velocity field satisfies the complete slip boundary conditions (1.19), (1.27) leading to the Neumann boundary conditions for r, namely
In particular, the eigenfunctions differ from those of the limit problem with ɛ = 0 only by a multiplicative constant approaching 1 for ɛ → 0.
Physically speaking, the complete slip boundary conditions correspond to the ideal mechanically smooth boundary of the physical space. As suggested by the previous arguments, the effect of viscosity in this rather hypothetical situation does not change significantly the asymptotic analysis in the low Mach number limit.
The dissipation of the acoustic energy caused by viscosity in domains with mechanically smooth boundaries is irrelevant in the low Mach number regime. The decay of the acoustic waves is exponential with a rate independent of ɛ.
On the other hand, the decay rate of the acoustic waves may change substantially if the fluid interacts with the boundary, meaning, if some kind of “dissipative” (in terms of the acoustic energy) boundary conditions is imposed on the velocity field. Thus, for instance, the no-slip boundary conditions (1.28) give rise to
Accordingly, system (7.2), supplemented with (7.3), becomes a singularly perturbed eigenvalue problem. In particular, if the (overdetermined) limit problem
admits only the trivial solution for λ ≠ 0, we can expect that a boundary layer is created in the limit process ɛ → 0 resulting in a faster decay of the acoustic waves. This can be seen by means of the following heuristic argument. Suppose that problem (7.2), (7.3) admits a family of eigenfunctions {r ɛ , w ɛ } ɛ > 0 with the associated set of eigenvalues {λ ɛ } ɛ > 0. Multiplying (7.2) on \(\overline{r}_{\varepsilon }\), \(\overline{\mathbf{w}}_{\varepsilon }\), where the bar stands for the complex conjugate, integrating the resulting expression over \(\Omega\), and using (7.3), we obtain
where Re denotes the real part of a complex number. Normalizing {r ɛ , w ɛ } ɛ > 0 in \(L^{2}(\Omega ) \times L^{2}(\Omega; \mathbb{R}^{3})\) we easily observe that
since otherwise {w ɛ } ɛ > 0 would be bounded in \(W^{1,2}(\Omega; \mathbb{R}^{3})\) and any weak accumulation point (r, w) of {r ɛ , w ɛ } ɛ > 0 would represent a nontrivial solution of the overdetermined limit system (7.4).
Sticky boundaries in combination with the viscous effects may produce a decay rate of the acoustic waves that is considerably faster than their frequency in the low Mach number regime. In particular, the mechanical energy is converted into heat and the acoustic waves are anihilated at a time approaching zero in the low Mach number limit.
Finally, we claim that a similar effect may be produced even if the complete slip boundary conditions
are imposed, where \(\Omega _{\varepsilon }\) is a family of domains with “rough” boundaries depending on the scaling parameter ɛ. More precisely, the boundaries \(\partial \Omega _{\varepsilon }\) differ from a limit shape by a family of small but still smooth asperities approximating the limit boundary in a similar way as the sequence of functions \(\varepsilon \sin \left (x/\varepsilon \right )\) approaches zero. In particular, as the fluid is viscous, such oscillating boundaries force the fluid velocity to vanish, meaning to satisfy the no-slip boundary condition
in the asymptotic limit ɛ → 0. Thus the scenario predicted by Conjecture II remains valid and we expect to recover strong convergence of the velocity fields.
7.1 Problem Formulation
Motivated by the previous discussion, we examine the low Mach number limit for the Navier-Stokes-Fourier system supplemented with either the no-slip boundary condition, or, alternatively, with the complete slip boundary conditions imposed on a family of domains with “oscillating” boundaries. In both cases, the fact that the fluid adheres completely (at least asymptotically in the latter case) to the wall of the physical space imposes additional restrictions on the propagation of acoustic waves. Our goal is to identify the geometrical properties of the domain, for which this implies strong convergence of the velocity field in the asymptotic limit.
7.1.1 Field Equations
We consider the same scaling of the field equations as in Chap. 5 Specifically, we set
obtaining
■ Scaled Navier-Stokes-Fourier system:
where
System (7.5)–(7.8) is supplemented, exactly as in Chap. 5, with the constitutive relations:
and
where
The reader will have noticed that the bulk viscosity has been neglected in (7.10) for the sake of simplicity.
As always in this book, Eqs. (7.5)–(7.8) are interpreted in the weak sense specified in Chap. 1 (see Sect. 7.2 below). We recall that the technical restrictions imposed on the constitutive functions are dictated by the existence theory developed in Chap. 3 and could be relaxed, to a certain extent, as far as the singular limit passage is concerned.
7.1.2 Physical Domain and Boundary Conditions
As indicated in the introductory part, the geometry of the physical domain plays a crucial role in the study of propagation of the acoustic waves. As already pointed out, the existence of an effective mechanism of dissipation of the acoustic waves is intimately linked to solvability of the (overdetermined) system (7.4) that can be written in a more concise form as
The problem of existence of a non-trivial, meaning non-constant, solution to (7.16) is directly related to the so-called Pompeiu property of the domain \(\Omega\). A remarkable result of Williams [273] asserts that if (7.16) possesses a non-constant solution in a domain in \(\mathbb{R}^{N}\) whose boundary is homeomorphic to the unit sphere, then, necessarily, \(\partial \Omega\) must admit a description by a system of charts that are real analytic. The celebrated Schiffer’s conjecture claims that (7.16) admits a non-trivial solution in the aforementioned class of domains only if \(\Omega\) is a ball.
In order to avoid the unsurmountable difficulties mentioned above, we restrict ourselves to a very simple geometry of the physical space. Similarly to Chap. 6, we assume the motion of the fluid is 2π-periodic in the horizontal variables (x 1, x 2), and the domain \(\Omega\) is an infinite slab determined by the graphs of two given functions B bottom, B top,
where \(\mathcal{T}^{\,2}\) denotes the flat torus,
Although the specific length of the period is not essential, this convention simplifies considerably the notation used in the remaining part of this chapter.
In the simple geometry described by (7.17), it is easy to see that problem (7.16) admits a non-trivial solution, namely r = cos(x 3) as soon as the boundary is flat, more precisely, if B bottom = −π, B top = 0. On the other hand, we claim that problem (7.16) possesses only the trivial solution in domains with variable bottoms as stated in the following assertion.
Proposition 7.1
Let \(\Omega\) be given through ( 7.17 ), with
Assume there is a function r ≠ const solving the eigenvalue problem ( 7.16 ) for a certain \(\Lambda\) .
Then h ≡ constant.
Proof
Since r is constant on the top part, specifically r(x 1, x 2, 0) = r 0, the function
satisfies
Accordingly, the function V extended to be zero in the upper half plane {x 3 > 0} solves the eigenvalue problem (7.16) in \(\Omega \cup \{ x_{3} \geq 0\}\). Consequently, by virtue of the unique continuation property of the elliptic operator \(\Delta _{x} + \Lambda I\) (analyticity of solutions to elliptic problems discussed in Sect. 11.3.1 in Appendix), we get V ≡ 0, in other words,
However, as r must be constant on the bottom part {x 3 = −π − h(x 1, x 2)}, we conclude that h ≡ const.
□
Our future considerations will be therefore concerned with fluids confined to domains described through (7.17), with flat “tops” and variables “bottoms” as in (7.18) with h ≢ const.
7.2 Main Result: The No-Slip Boundary Conditions
We start by imposing the no-slip boundary conditions for the velocity field
together with the no-flux boundary condition for the temperature
Accordingly, the system is energetically insulated in agreement with (7.8).
As a matter of fact, the approach delineated in this section applies to any bounded and sufficiently smooth spatial domain \(\Omega \subset \mathbb{R}^{3}\), on which the overdetermined problem (7.16) admits only the trivial (constant) solution r. In particular, the arguments in the proof of Proposition 7.1 can be used provided a part of the boundary is flat and the latter is connected.
7.2.1 Preliminaries: Global Existence
Exactly as in Chap. 5, we consider the initial data in the form
where
with positive constants \(\overline{\varrho }\), \(\overline{\vartheta }\).
For reader’s convenience, we recall the list of hypotheses, under which system (7.5)–(7.15), supplemented with the boundary conditions (7.19), (7.20), and the initial conditions (7.21), possesses a weak solution defined on an arbitrary time interval (0, T). To begin, we need the hypothesis of thermodynamic stability (1.44) expressed in terms of the function P as
together with the coercivity assumption
Similarly to Chap. 5, the transport coefficients μ, η, and κ are assumed to be continuously differentiable functions of the temperature ϑ satisfying the growth restrictions
and
where \(\underline{\mu }\), \(\overline{\mu }\), \(\underline{\kappa }\), and \(\overline{\kappa }\) are positive constants.
Now, as a direct consequence of the abstract existence result established in Theorem 3.1, we claim that for any ɛ > 0, the scaled Navier-Stokes-Fourier system (7.5)–(7.9), supplemented with the boundary conditions (7.19)–(7.20), and the initial conditions (7.21), possesses a weak solution {ϱ ɛ , u ɛ , ϑ ɛ } ɛ > 0 on the set \((0,T) \times \Omega\) such that
More specifically, we have:
-
(i)
Renormalized equation of continuity:
$$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\varrho _{\varepsilon }B(\varrho _{\varepsilon })\Big(\partial _{t}\varphi + \mathbf{u}_{\varepsilon } \cdot \nabla _{x}\varphi \Big)\ \mathrm{d}x\ \mathrm{d}t }$$(7.28)$$\displaystyle{=\int _{ 0}^{T}\int _{ \Omega }b(\varrho _{\varepsilon })\mathrm{div}_{x}\mathbf{u}_{\varepsilon }\varphi \ \mathrm{d}x\ \mathrm{d}t -\int _{\Omega }\varrho _{0,\varepsilon }B(\varrho _{0,\varepsilon })\varphi (0,\cdot )\ \mathrm{d}x}$$for any b as in (2.3) and any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\);
-
(ii)
Momentum equation:
$$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\Big(\varrho _{\varepsilon }\mathbf{u}_{\varepsilon } \cdot \partial _{t}\boldsymbol{\varphi } + \varrho _{\varepsilon }[\mathbf{u}_{\varepsilon } \otimes \mathbf{u}_{\varepsilon }]: \nabla _{x}\boldsymbol{\varphi } + \frac{1} {\varepsilon ^{2}} p(\varrho _{\varepsilon },\vartheta _{\varepsilon })\mathrm{div}_{x}\boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t }$$(7.29)$$\displaystyle{=\int _{ 0}^{T}\int _{ \Omega }\Big(\mathbb{S}_{\varepsilon }: \nabla _{x}\boldsymbol{\varphi } -\frac{1} {\varepsilon } \varrho _{\varepsilon }\nabla _{x}F\cdot \boldsymbol{\varphi }\Big)\ \mathrm{d}x\ \mathrm{d}t -\int _{\Omega }(\varrho _{0,\varepsilon }\mathbf{u}_{0,\varepsilon })\cdot \boldsymbol{\varphi }\ \mathrm{d}x}$$for any test function
$$\displaystyle{\boldsymbol{\varphi }\in C_{c}^{\infty }([0,T) \times \Omega; \mathbb{R}^{3});}$$ -
(iii)
Total energy balance:
$$\displaystyle{ \int _{\Omega }\Big(\frac{\varepsilon ^{2}} {2}\varrho _{\varepsilon }\vert \mathbf{u}_{\varepsilon }\vert ^{2} + \varrho _{\varepsilon }e(\varrho _{\varepsilon },\vartheta _{\varepsilon }) -\varepsilon \varrho _{\varepsilon }F\Big)(t)\ \mathrm{d}x }$$(7.30)$$\displaystyle{= \int _{\Omega }\Big(\frac{\varepsilon ^{2}} {2}\varrho _{0,\varepsilon }\vert \mathbf{u}_{0,\varepsilon }\vert ^{2} + \varrho _{ 0,\varepsilon }e(\varrho _{0,\varepsilon },\vartheta _{0,\varepsilon }) -\varepsilon \varrho _{\varepsilon }F\Big)\ \mathrm{d}x\ \mbox{ for a.a.}\ t \in (0,T);}$$ -
(iv)
Entropy balance:
$$\displaystyle{ \int _{0}^{T}\int _{ \Omega }\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })\Big(\partial _{t}\varphi + \mathbf{u}_{\varepsilon } \cdot \nabla _{x}\varphi \Big)\ \mathrm{d}x\ \mathrm{d}t +\int _{ 0}^{T}\int _{ \Omega }\frac{\mathbf{q}_{\varepsilon }} {\vartheta _{\varepsilon }} \cdot \nabla _{x}\varphi \ \mathrm{d}x\ \mathrm{d}t }$$(7.31)$$\displaystyle{+ <\sigma _{\varepsilon };\varphi> _{[\mathcal{M};C]([0,T]\times \overline{\Omega })} = -\int _{\Omega }\varrho _{0,\varepsilon }s(\varrho _{0,\varepsilon },\vartheta _{0,\varepsilon })\varphi (0,\cdot )\ \mathrm{d}x}$$for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\), where \(\sigma _{\varepsilon } \in \mathcal{M}^{+}([0,T] \times \overline{\Omega })\) satisfies (7.9).
Note that the satisfaction of the no-slip boundary conditions is ensured by the fact that the velocity field u ɛ (t, ⋅ ) belongs to the Sobolev space \(W_{0}^{1,2}(\Omega; \mathbb{R}^{3})\) defined as a completion of \(C_{c}^{\infty }(\Omega; \mathbb{R}^{3})\) with respect to the W 1,2-norm. Accordingly, the test functions in the momentum equation (7.29) must be compactly supported in \(\Omega\), in particular, the Helmholtz projection \(\mathbf{H}[\boldsymbol{\varphi }]\) is no longer an admissible test function in (7.29).
7.2.2 Compactness of the Family of Velocities
In order to avoid confusion, let us point out that the principal result to be shown in this part is pointwise compactness of the family of velocity fields {u ɛ } ɛ > 0. Then following step by step the analysis presented in Chap. 5 we could show that the limit system obtained by letting ɛ → 0 is the same as in Theorem 5.2, specifically, the Oberbeck-Boussinesq approximation (5.161)–(5.166).
■ Compactness of Velocities on Domains with Variable Bottoms:
Theorem 7.1
Let \(\Omega\) be the infinite slab introduced in ( 7.17 ), ( 7.18 ), where the “bottom” part of the boundary is given by a function h satisfying
Assume that \(\mathbb{S}_{\varepsilon }\) , q ɛ as well as the thermodynamic functions p, e, and s are given by ( 7.10 )–( 7.15 ), where P meets the structural hypotheses ( 7.23 )–( 7.25 ), while the transport coefficients μ and κ satisfy ( 7.26 ), ( 7.27 ). Finally, let {ϱ ɛ , u ɛ , ϑ ɛ } ɛ > 0 be a family of weak solutions to the Navier-Stokes-Fourier system satisfying ( 7.28 )–( 7.31 ), where the initial data are given by ( 7.21 ), ( 7.22 ).
Then, at least for a suitable subsequence,
where \(\mathbf{U} \in L^{2}(0,T;W_{0}^{1,2}(\Omega; \mathbb{R}^{3}))\) , div x U = 0.
The bulk of the remaining part of this chapter, specifically Sects. 7.3–7.5, is devoted to the proof of Theorem 7.1 which is tedious and rather technical. It is based on careful analysis of the singular eigenvalue problem (7.2), (7.3) in a boundary layer by means of the abstract method proposed by Vishik and Ljusternik [267] and later adapted to the low Mach number limit problems in the context of isentropic fluid flows by Desjardins et al. [81]. In contrast with [81], we “save” one degree of approximation—a fact that simplifies considerably the analysis and makes the proof relatively transparent and easily applicable to other choices of boundary conditions (see [118]).
7.3 Uniform Estimates
We begin the proof of Theorem 7.1 by recalling the uniform estimates that can be obtained exactly as in Chap. 5 Thus we focus only the principal ideas referring to the corresponding parts of Sect. 5.2 for all technical details.
As the initial distribution of the density is a zero mean perturbation of the constant state \(\overline{\varrho }\), we have
in particular,
To obtain further estimates, we combine (7.30), (7.31) to deduce the dissipation balance equality in the form
where \(H_{\overline{\vartheta }}\) is the Helmholtz function introduced in (2.48).
As we have observed in (2.49), (2.50), the hypothesis of thermodynamic stability ∂ ϱ p > 0, ∂ ϑ e > 0, expressed in terms of (7.23), (7.24), implies that
while
Consequently, subtracting a suitable affine function of ϱ from both sides of (7.35), and using the coercivity properties of \(H_{\overline{\vartheta }}\) stated in Lemma 5.1 we deduce the following list of uniform estimates:
-
Energy estimates:
$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\|\sqrt{\varrho _{\varepsilon }}\mathbf{u}_{\varepsilon }\|_{L^{2}(\Omega;\mathbb{R}^{3})} \leq c\ \mbox{ [ cf. (5.49) ]}, }$$(7.36)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\varrho _{\varepsilon } -\overline{\varrho }} {\varepsilon } \Big]_{\mathrm{ess}}\Big\|_{L^{2}(\Omega )} \leq c\ \ \mbox{ [ cf. (5.46) ]}, }$$(7.37)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\varrho _{\varepsilon } -\overline{\varrho }} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{ L^{\frac{5} {3} }(\Omega )} \leq \varepsilon ^{\frac{1} {5} }c\ \ \mbox{ [ cf. (5.45), (5.48) ]}, }$$(7.38)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\vartheta _{\varepsilon } -\overline{\vartheta }} {\varepsilon } \Big]_{\mathrm{ess}}\Big\|_{L^{2}(\Omega )} \leq c\ \ \mbox{ [ cf. (5.47) ]}, }$$(7.39)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|[\vartheta _{\varepsilon }]_{\mathrm{res}}\Big\|_{L^{4}(\Omega )} \leq \varepsilon ^{\frac{1} {2} }c\ \ \mbox{ [ cf. (5.48) ]}, }$$(7.40)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{p(\varrho _{\varepsilon },\vartheta _{\varepsilon }) - p(\overline{\varrho },\overline{\vartheta })} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{L^{1}(\Omega )} \leq \varepsilon c\ \ \mbox{ [ cf. (5.45), (5.100) ]}. }$$(7.41) -
Estimates based on energy dissipation:
$$\displaystyle{ \|\sigma _{\varepsilon }\|_{\mathcal{M}^{+}([0,T]\times \overline{\Omega })} \leq \varepsilon ^{2}c\ \ \mbox{ [ cf. (5.50) ]}, }$$(7.42)$$\displaystyle{ \int _{0}^{T}\|\mathbf{u}_{\varepsilon }\|_{ W_{0}^{1,2}(\Omega;\mathbb{R}^{3})}^{2}\ \mathrm{d}t \leq c\ \ \mbox{ [ cf. (5.51) ]}, }$$(7.43)$$\displaystyle{ \int _{0}^{T}\Big\|\frac{\vartheta _{\varepsilon } -\overline{\vartheta }} {\varepsilon } \Big\|_{W^{1,2}(\Omega )}^{2}\ \mathrm{d}t \leq c\ \ \mbox{ [ cf. (5.52) ] }, }$$(7.44)$$\displaystyle{ \int _{0}^{T}\Big\|\frac{\log (\vartheta _{\varepsilon }) -\log (\overline{\vartheta })} {\varepsilon } \Big\|_{W^{1,2}(\Omega )}^{2}\ \mathrm{d}t \leq c\ \ \mbox{ [ cf. (5.53) ] }. }$$(7.45) -
Entropy estimates:
$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{L^{1}(\Omega )}\mathrm{d}t \leq \varepsilon c\ \ \mbox{ [ cf. (5.44) ]}, }$$(7.46)$$\displaystyle{ \int _{0}^{T}\Big\|\Big[\frac{\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{L^{q}(\Omega )}^{q}\ \mathrm{d}t \leq c\ \mbox{ for a certain}\ q> 1\ \ \mbox{ [ cf. (5.54) ]}, }$$(7.47)$$\displaystyle{ \int _{0}^{T}\Big\|\Big[\frac{\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \mathbf{u}_{\varepsilon }\Big]_{\mathrm{res}}\Big\|_{L^{q}(\Omega;\mathbb{R}^{3})}^{q}\ \mathrm{d}t \leq c\ \mbox{ for a certain}\ q> 1\ \ \mbox{ [ cf. (5.55) ]}, }$$(7.48)$$\displaystyle{ \int _{0}^{T}\Big\|\Big[\frac{\kappa (\vartheta _{\varepsilon })} {\vartheta _{\varepsilon }} \Big]_{\mathrm{res}}\frac{\nabla _{x}\vartheta _{\varepsilon }} {\varepsilon } \Big\|_{L^{q}(\Omega;\mathbb{R}^{3})}^{q}\ \mathrm{d}t \rightarrow 0\ \mbox{ for a certain}\ q> 1\ \ \mbox{ [ cf. (5.56) ]}. }$$(7.49)
Let us recall that the “essential” component [h]ess of a function h and its “residual” counterpart [h]res have been introduced in (4.44), (4.45).
We conclude with the estimate on the “measure of the residual set” established in (5.46), specifically,
with \(\mathcal{M}_{\mathrm{res}}^{\varepsilon }[t] \subset \Omega\) introduced in (4.43).
7.4 Analysis of Acoustic Waves
7.4.1 Acoustic Equation
The acoustic equation governing the time oscillations of the gradient part of the velocity field is essentially the same as in Chap. 5 However, a refined analysis to be performed below requires a more elaborate description of the “small” terms as well as the knowledge of the precise rate of convergence of these quantities toward zero.
We start rewriting the equation of continuity (7.5) in the form
for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\).
Similarly, the momentum equation (7.29) can be written as
for any \(\boldsymbol{\varphi }\in C_{c}^{\infty }([0,T) \times \Omega; \mathbb{R}^{3})\), where we have set
and
It is important to notice that validity of (7.52) can be extended to the class of test functions satisfying
by means of a simple density argument. Indeed, in accordance with the integrability properties of the weak solutions established in Theorem 3.2, it is enough to use the density of \(C_{c}^{\infty }(\Omega )\) in \(W_{0}^{1,p}(\Omega )\) for any finite p.
Since \(\mathbf{u}_{\varepsilon } \in L^{2}(0,T;W_{0}^{1,2}(\Omega; \mathbb{R}^{3}))\), in particular, the trace of u ɛ vanishes on the boundary, we are allowed to use the Gauss-Green theorem to obtain
for any \(\boldsymbol{\varphi }\) as in (7.56), where we have introduced the notation
In a similar fashion, the entropy balance (7.31) can be rewritten as
for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\).
Summing up relations (7.51)–(7.58) we obtain, exactly as in Sect. 5.4.3, a linear hyperbolic equation describing the propagation of acoustic waves.
■ Acoustic Equation:
for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega })\),
for any \(\boldsymbol{\varphi }\in C_{c }^{\infty } ([0, T) \times \mathbb{R}^{3 }; \mathbb{R}^{3 } )\), \(\varphi \vert _{\partial \Omega } = 0\),
where we have set
with
Note that the integral identities (7.59), (7.60) represent a weak formulation of Eq. (7.1), where the “small” terms read as follows:
and
7.4.2 Spectral Analysis of the Acoustic Operator
In this part, we are concerned with the spectral analysis of the linear operator associated to problem (7.59), (7.60), namely we examine the differential operator
with
that can be viewed as the formal adjoint of the generator in (7.59), (7.60). In accordance with (7.19), we impose the homogeneous Dirichlet boundary condition for w,
Let us start with the limit eigenvalue problem
which can be equivalently reformulated as
where the boundary condition (7.70) transforms to \(\nabla _{x}v\vert _{\partial \Omega } = 0\), in particular,
Note that the null space (kernel) of \(\mathcal{A}\) is
As is well-known, the Neumann problem (7.72), (7.73) admits a countable set of real eigenvalues \(\{\Lambda _{n}\}_{n=0}^{\infty }\),
where the associated family of real eigenfunctions \(\{v_{n,m}\}_{n=0,m=1}^{\infty,m_{n}}\) forms an orthonormal basis of the Hilbert space \(L^{2}(\Omega )\). Moreover, we denote
the eigenspace corresponding to the eigenvalue \(\Lambda _{n}\) of multiplicity m n . In particular, m 0 = 1, E 0 = span{1} (see Theorem 11.9 in Appendix).
Under hypothesis (7.32), Proposition 7.1 implies that \(v_{0} = 1/\sqrt{\vert \Omega \vert }\) is the only eigenfunction that satisfies the supplementary boundary condition \(\nabla _{x}v_{0}\vert _{\partial \Omega } = 0\). Thus the term \(\varepsilon \mathcal{B}\), together with (7.70), may be viewed as a singular perturbation of the operator \(\mathcal{A}\).
Accordingly, the eigenvalue problem (7.71), (7.73) admits a system of eigenvalues
lying on the imaginary axis. The associated eigenspaces are
Here and hereafter, we fix n > 0 and set
together with
In order to match the incompatibility of the boundary conditions (7.70), (7.73), we look for “approximate” eigenfunctions of the perturbed problem (7.80), (7.82) in the form
where we set
The functions v ɛ , w ɛ are determined as solutions to the following approximate problem.
■ Approximate Eigenvalue Problem:
meaning,
where
supplemented with the homogeneous Dirichlet boundary condition
There is a vast amount of literature, in particular in applied mathematics, devoted to formal asymptotic analysis of singularly perturbed problems based on the so-called WKB (Wentzel-Kramers-Brilbuin) expansions for boundary layers similar to (7.77), (7.78). An excellent introduction to the mathematical aspects of the theory is the book by Métivier [211]. The “interior” functions v int,k = v int,k(x), w int,k = w int,k(x) depend only on \(x \in \Omega\), while the “boundary layer” functions v bl,k(x, Z) = v bl,k(x, Z), w bl,k = w bl,k(x, Z) depend on x and the fast variable \(Z = d(x)/\sqrt{\varepsilon }\), where d is a generalized distance function to \(\partial \Omega\),
Note that the distance function enjoys the same regularity as the boundary \(\partial \Omega\), namely as the function h appearing in hypothesis (7.32).
The rest of this section is devoted to identifying all terms in the asymptotic expansions (7.77), (7.78), the remainders s ɛ 1, s ɛ 2, and the value of λ 1. In accordance with the heuristic arguments in the introductory part of this chapter, we expect to recover λ 1 ≠ 0, specifically, Re[λ 1] < 0 yielding the desired exponential decay rate of order \(\sqrt{\varepsilon }\) (no contradiction with the sign of Re[λ ɛ ] in the introductory section as the elliptic part of problem (7.80)–(7.82) has negative spectrum!). This rather tedious task is accomplished in several steps.
Differential Operators Applied to the Boundary Layer Correction Functions To avoid confusion, we shall write \(\nabla _{x}\mathbf{w}^{\mathrm{bl},k}(x,d(x)/\sqrt{\varepsilon })\) for the gradient of the composed function \(x\mapsto \mathbf{w}^{\mathrm{bl},k}(x,d(x)/\sqrt{\varepsilon })\), while ∇ x w bl,k(x, Z), ∂ Z w bl,k(x, Z) stand for the differential operators applied to a function of two variables x and Z. It is a routine matter to compute:
Similarly, we get
and
for k = 0, 1, where Z stands for \(d(x)/\sqrt{\varepsilon }\).
Consequently, substituting ansatz (7.77), (7.78) in (7.80), (7.81), we arrive at the following system of equations:
and
Moreover, the remainders s ɛ 1, s ɛ 2 are determined by means of (7.80) as
where \(Z = d(x)/\sqrt{\varepsilon }\).
Finally, in agreement with (7.82), we require
Determining the Zeroth Order Terms System (7.84)–(7.89) consists of six equations for the unknowns v bl,0, w bl,0, v int,1, w int,1, and v bl,1, w bl,1. Note that, in agreement with (7.79),
Moreover, since the matrix \(\{\int _{\partial \Omega }\nabla _{x}v_{(i)} \cdot \nabla _{x}\mathbf{v}_{(\,j)}\ \mathrm{d}S_{x}\}_{i,j=1}^{m}\) is diagonalizable, the basis {v (1), …, v (m)} of the eigenspace E introduced in (7.75), (7.76) may be chosen in such a way that
where v int,0 = v (1).
Since there are no boundary conditions imposed on the component v, we can take
in particular, Eq. (7.88) holds.
Furthermore, Eq. (7.86) requires the quantity w bl,0(x, Z) ⋅ ∇ x d(x) to be independent of Z. On the other hand, by virtue of (7.73), (7.92), the function \(x\mapsto \mathbf{w}^{\mathrm{bl},0}(x,d(x)/\sqrt{\varepsilon })\) must have zero normal trace on \(\partial \Omega\). Since d(x) = 0, ∇ x d(x) = −n(x) for any \(x \in \partial \Omega\), we have to take
Consequently, Eq. (7.89) reduces to
For a fixed \(x \in \overline{\Omega }\), relation (7.97) represents an ordinary differential equation of second order in Z, for which the initial conditions w bl,0(x, 0) are uniquely determined by (7.92), namely
It is easy to check that problem (7.97), (7.98) admits a unique solution that decays to zero for Z → ∞, specifically,
where χ ∈ C ∞[0, ∞),
and
It seems worth-noting that formula (7.99) is compatible with (7.96) as for \(x \in \Omega\) the point x −∇ x d(x)∕d(x) is the nearest to x on \(\partial \Omega\) as soon as d(x) coincides with \(\mathrm{dist}[x,\partial \Omega ]\).
First Order Terms Equation (7.87), together with the ansatz made in (7.95), give rise to
In view of (7.99), Eq. (7.102) admits a unique solution with exponential decay for Z → ∞ for any fixed \(x \in \overline{\Omega }\), namely
Thus we can set
for a function H such that
to be determined below. Note that, in accordance with formula (7.99), \(\vert \nabla _{x}d(x)\vert = \vert \nabla _{x}\mathrm{dist}[x,\partial \Omega ]\vert = 1\) on the set where w bl,0 ≠ 0.
Determining λ 1 Our ultimate goal is to identify v int,1, w int,1, and, in particular λ 1, by help of equations of (7.84), (7.85). In accordance with (7.92), the normal trace of the quantity w int,1(x) + w bl,1(x, 0) must vanish for \(x \in \partial \Omega\); whence, by virtue of (7.103),
for any \(x \in \partial \Omega\).
As a consequence of (7.93), system (7.84), (7.85) can be rewritten as a second order elliptic equation
where \(\Lambda = -(\lambda ^{0})^{2}/\omega\). Problem (7.106) is supplemented with the non-homogeneous Neumann boundary condition determined by means of (7.93), (7.85), and (7.105), namely
According to the standard Fredholm alternative for elliptic problems (see Sect. 11.3.2 in Appendix), system (7.106), (7.107) is solvable as long as
where {v (1), …, v (m)} is the system of eigenvectors introduced in (7.94). In accordance with our agreement v (1) = v int,0, therefore we set
and verify that
To this end, use (7.93), (7.99) to compute
whenever \(\mathrm{dist}[x,\partial \Omega ] <\delta /2\). Consequently,
where the symbol \(\Delta _{S}\) denotes the Laplace-Beltrami operator on the (compact) Riemannian manifold \(\partial \Omega\). Indeed expression \(\Big[\nabla _{x}^{2}v^{\mathrm{int},0}: (\mathbf{n} \otimes \mathbf{n} - \mathbb{I})\Big]\) represents the standard “flat” Laplacian of the function v int,0 with respect to the tangent plane at each point of \(\partial \Omega\) that coincides (up to a sign) with the associated Laplace-Beltrami operator on the manifold \(\partial \Omega\) applied to the restriction of \(v^{\mathrm{int},0}\vert _{\partial \Omega }\) provided ∇ x v int,0 ⋅ n = 0 on \(\partial \Omega\) (see Gilbarg and Trudinger [136, Chap. 16]).
In accordance with (7.94), we infer that
In particular, we get (7.109), and, using (7.72), (7.101),
Seeing that \(\Lambda> 0\), and, by virtue of (7.101), \(\mathrm{Re}[\Gamma ]> 0\), we utilize hypothesis (7.32) together with Proposition 7.1 to deduce the desired conclusion
This is the crucial point of the proof of Theorem 7.1.
Having identified v int,1 by means of (7.106), (7.107) we use (7.85) to compute
Finally, in order to meet the boundary conditions (7.92), we set
in (7.103), with χ given by (7.100).
Conclusion By a direct inspection of (7.90), (7.91), where all quantities are evaluated by means (7.95), (7.99), (7.103), we infer that
in particular s ɛ 1, s ɛ 2 are uniformly bounded in \(\overline{\Omega }\) and tend to zero uniformly on any compact \(K \subset \Omega\).
The results obtained in this section are summarized in the following assertion.
Proposition 7.2
Let \(\Omega\) be given through ( 7.17 ), with
Assume that v int, 0 , w int, 0 , and λ 0 ≠ 0 solve the eigenvalue problem ( 7.71 ), ( 7.73 ).
Then the approximate eigenvalue problem ( 7.80 )–( 7.82 ) admits a solution in the form ( 7.77 ), ( 7.78 ), where
-
the functions v int, 1 = v int, 1 (x), w int, 1 = w int, 1 (x) belong to the class \(C^{2}(\overline{\Omega })\) ;
-
the boundary layer corrector functions v bl, 0 = v bl, 1 = 0, w bl, 0 = w bl, 0 (x, Z), w bl, 1 = w bl, 1 (x, Z) are all of the form \(\mathbf{h}(x)\exp (-\Gamma Z)\) , where \(\mathbf{h} \in C^{2}(\overline{\Omega }; \mathbb{R}^{3})\) , and \(\mathrm{Re}[\Gamma ]> 0\) ;
-
the approximate eigenvalue λ ɛ is given by ( 7.81 ), where
$$\displaystyle{ \mathrm{Re}[\lambda ^{1}] <0; }$$(7.111) -
the remainders s ɛ 1 , s ɛ 2 satisfy
$$\displaystyle{ s_{\varepsilon }^{1} \rightarrow 0\ \mathit{\mbox{ in}}\ L^{q}(\Omega ),\ \mathbf{s}_{\varepsilon }^{2} \rightarrow 0\ \mathit{\mbox{ in}}\ L^{q}(\Omega; \mathbb{R}^{3})\ \mathit{\mbox{ as}}\ \varepsilon \rightarrow 0\ \mathit{\mbox{ for any}}\ 1 \leq q <\infty. }$$(7.112)
7.5 Strong Convergence of the Velocity Field
We are now in a position to establish the main result of this chapter stated in Theorem 7.1, namely
We recall that, in accordance with (7.43),
at least for a suitable subsequence. Moreover, exactly as in Sect. 5.3.1, we have
Consequently, it remains to control possible oscillations of the velocity field in time. To this end, similarly to Chap. 5, the problem is reduced to a finite number of acoustic modes that can be treated by means of the spectral theory developed in the preceding section.
7.5.1 Compactness of the Solenoidal Component
It follows from the uniform estimates (7.36)–(7.38) that
Using quantities
as test functions in the momentum equation (7.29) we deduce, by means of the standard Arzelà-Ascoli theorem, that the scalar functions
Note that
as \(\boldsymbol{\phi }\) is a divergenceless vector field.
Consequently, by help of (7.115) and a simple density argument, we infer that the family
for any \(\boldsymbol{\phi }\in C_{c}^{\infty }(\Omega; \mathbb{R}^{3})\), where H denotes the Helmholtz projection introduced in Sect. 5.4.1. In other words,
Let us point out that H[ϕ] is not an admissible test function in (7.29), however, it can be approximated in \(L^{p}(\Omega; \mathbb{R}^{3})\) by smooth solenoidal functions with compact support for finite p (see Sect. 11.7 in Appendix).
Thus, combining relations (7.114), (7.116), we infer
which, together with estimates (7.37), (7.38), gives rise to
yielding, finally, the desired conclusion
7.5.2 Reduction to a Finite Number of Modes
Exactly as in (5.146), we decompose the space L 2 as a sum of the subspace of solenoidal vector fields L σ 2 and gradients L g 2:
Since we already know that the solenoidal components of the velocity field u ɛ are precompact in L 2, the proof of (7.113) reduces to showing
Moreover, since the embedding \(W_{0}^{1,2}(\Omega; \mathbb{R}^{3})\hookrightarrow L^{2}(\Omega; \mathbb{R}^{3})\) is compact, it is enough to show
for any fixed \(\mathbf{w} = \frac{1} {\lambda } \nabla _{x}v\), where v, w, λ ≠ 0 solve the eigenvalue problem (7.71), (7.73) (cf. Sect. 5.4.6).
In view of (7.37), (7.38), relation (7.118) follows as soon as we show
where the latter quantity can be expressed by means of the acoustic equation (7.59), (7.60). In addition, since the solutions of the eigenvalue problem (7.71), (7.73) come in pairs [v, w, λ], [v, −w, −λ], it is enough to show
for any solution v, w of (7.71), (7.73) associated to an eigenvalue λ ≠ 0, where r ɛ , V ɛ are given by (7.61).
Finally, in order to exploit the information on the spectrum of the perturbed acoustic operator, we claim that (7.119) can be replaced by
where v ɛ , w ɛ are the solutions of the approximate eigenvalue problem (7.80), (7.82) constructed in the previous section. Indeed, by virtue of Proposition 7.2, we have
Accordingly, the proof of Theorem 7.1 reduces to showing (7.120). This will be done in the following section.
7.5.3 Strong Convergence
In order to complete the proof of Theorem 7.1, our ultimate goal consists in showing (7.120). To this end, we make use of the specific form of the acoustic equation (7.59), (7.60), together with the associated spectral problem (7.80), (7.82). Taking the quantities ψ(t)v ɛ (x), ψ(t)w ɛ (x), ψ ∈ C c ∞(0, T), as test functions in (7.59), (7.60), respectively, we obtain
where we have set
and the symbols I m ɛ stand for the “small” terms:
and
where G 8 ɛ is given by (7.67).
Our aim is to show that each of the integrals can be written in the form
where
This rather tedious task, to be achieved by means of Proposition 7.2 combined with the uniform estimates listed in Sect. 7.3, consists in several steps as follows:
-
(i)
By virtue of Hölder’s inequality, we have
$$\displaystyle{ \Big\vert \int _{\Omega }\Big[\frac{\kappa (\vartheta _{\varepsilon })} {\vartheta _{\varepsilon }} \nabla _{x}\vartheta _{\varepsilon } \cdot \nabla _{x}v_{\varepsilon }\ \mathrm{d}x\Big\vert }$$(7.122)$$\displaystyle\begin{array}{rcl} & \leq \varepsilon \|v_{\varepsilon }\|_{W^{1,\infty }(\Omega )}\Big\vert \int _{\Omega }\Big[\frac{\kappa (\vartheta _{\varepsilon })} {\vartheta _{\varepsilon }} \Big]_{\mathrm{ess}}\Big\vert \frac{\nabla _{x}\vartheta _{\varepsilon }} {\varepsilon } \Big\vert \ \mathrm{d}x\Big\vert +\Big \vert \int _{\Omega }\Big[\frac{\kappa (\vartheta _{\varepsilon })} {\vartheta _{\varepsilon }} \Big]_{\mathrm{res}}\Big\vert \frac{\nabla _{x}\vartheta _{\varepsilon }} {\varepsilon } \Big\vert \ \mathrm{d}x\Big\vert & {}\\ & = \varepsilon \gamma _{1,1}^{\varepsilon },\ \mbox{ with}\ \{\gamma _{1}^{\varepsilon }\}_{\varepsilon>1}\ \mbox{ bounded in}\ L^{q}(0,T) & \ \mbox{ for a certain}\ q> 1, {}\\ \end{array}$$where we have used estimates (7.44) and (7.49). Note that, in accordance with Proposition 7.2, both correction terms v bl,0, v bl,1 vanish identically, in particular,
$$\displaystyle{ \|v_{\varepsilon }\|_{W^{1,\infty }(\Omega )} \leq c\ \mbox{ uniformly in}\ \varepsilon. }$$(7.123)In a similar way,
$$\displaystyle{ \Big\vert \int _{\Omega }\Big(\varrho _{\varepsilon }s(\overline{\varrho },\overline{\vartheta }) -\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })\Big)\mathbf{u}_{\varepsilon } \cdot \nabla _{x}v_{\varepsilon }\ \mathrm{d}x\Big\vert }$$(7.124)$$\displaystyle\begin{array}{rcl} & \leq \varepsilon \|v_{\varepsilon }\|_{W^{1,\infty }(\Omega )}\left [\int _{\Omega }\Big\vert \ \Big[\frac{\varrho _{\varepsilon }s(\overline{\varrho },\overline{\vartheta })-\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \Big]_{\mathrm{ess}}\ \Big\vert \vert \mathbf{u}_{\varepsilon }\vert \ \mathrm{d}x\right. & {}\\ & \left.+\int _{\Omega }\Big\vert \ \Big[\frac{\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \Big]_{\mathrm{res}}\mathbf{u}_{\varepsilon }\Big\vert \ \mathrm{d}x\ \mathrm{d}t + \vert s(\overline{\varrho },\overline{\vartheta })\vert \int _{\Omega }\Big[\frac{\varrho _{\varepsilon }} {\varepsilon }\Big]_{\mathrm{res}}\vert \mathbf{u}_{\varepsilon }\vert \ \mathrm{d}x\right ] = \varepsilon \gamma _{1,2}^{\varepsilon }.& {}\\ \end{array}$$Thus we can use Proposition 5.2, together with estimates (7.37)–(7.39), (7.43), (7.48), (7.50), in order to conclude that
$$\displaystyle{\{\gamma _{1,2}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ is bounded in}\ L^{q}(0,T)\ \mbox{ for a certain}\ q> 1.}$$Summing up (7.122), (7.124) we infer that
$$\displaystyle{ I_{1}^{\varepsilon } = \varepsilon \int _{ 0}^{T}\psi (t)\gamma _{ 1}^{\varepsilon }(t)\ \mathrm{d}t,\ \mbox{ with}\ \{\gamma _{ 1}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ bounded in}\ L^{q}(0,T)\ \mbox{ for a certain}\ q> 1. }$$(7.125) -
(ii)
As a straightforward consequence of estimate (7.42) we obtain
$$\displaystyle{ I_{2}^{\varepsilon } = \varepsilon ^{2} <\Gamma _{ 2}^{\varepsilon };\psi> _{ [\mathcal{M};C][0,T]},\ \mbox{ where}\ \{\Gamma _{2}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ is bounded in}\ \mathcal{M}^{+}[0,T]. }$$(7.126) -
(iii)
Taking advantage of the form of w bl,0, w bl,1 specified in Proposition 7.2, we obtain
$$\displaystyle{\|\varepsilon \mathrm{div}_{x}[[\nabla _{x}\mathbf{w}_{\varepsilon }]]\ \|_{L^{\infty }(\Omega;\mathbb{R}^{3})} \leq c}$$uniformly for ɛ → 0. This fact, combined with the uniform bounds established in (7.37), (7.38), (7.43), and the standard embedding \(W^{1,2}(\Omega )\hookrightarrow L^{6}(\Omega )\), gives rise to
$$\displaystyle{ I_{3}^{\varepsilon } = \varepsilon \int _{ 0}^{T}\psi (t)\gamma _{ 3}^{\varepsilon }(t)\ \mathrm{d}t, }$$(7.127)where
$$\displaystyle{\{\gamma _{3}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ is bounded in}\ L^{2}(0,T).}$$ -
(iv)
Similarly to the preceding step, we deduce
$$\displaystyle{ \|\sqrt{\varepsilon }\mathbf{w}_{\varepsilon }\|_{W^{1,\infty }(\Omega;\mathbb{R}^{3})} \leq c; }$$(7.128)whence, by virtue of (7.40), (7.43), and (7.44),
$$\displaystyle{ I_{4}^{\varepsilon } = \varepsilon ^{3/2}\int _{ 0}^{T}\psi (t)\Gamma _{ 4}^{\varepsilon }(t)\ \mathrm{d}t, }$$(7.129)where
$$\displaystyle{\{\Gamma _{4}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ is bounded in}\ L^{1}(0,T).}$$ -
(v)
Probably the most delicate issue is to handle the integrals in I 5 ɛ. To this end, we first write
$$\displaystyle\begin{array}{rcl} & \int _{0}^{T}\psi \int _{\Omega }\varepsilon \varrho _{\varepsilon }\mathbf{u}_{\varepsilon } \otimes \mathbf{u}_{\varepsilon }: \nabla _{x}\mathbf{w}_{\varepsilon }\ \mathrm{d}x\ \mathrm{d}t & {}\\ & =\int _{ 0}^{T}\psi \int _{\Omega }\varepsilon ^{2}\Big(\frac{\varrho _{\varepsilon }-\overline{\varrho }} {\varepsilon } \Big)\mathbf{u}_{\varepsilon } \otimes \mathbf{u}_{\varepsilon }: \nabla _{x}\mathbf{w}_{\varepsilon }\ \mathrm{d}x\ \mathrm{d}t + \overline{\varrho }\int _{0}^{T}\psi \int _{ \Omega }\varepsilon \mathbf{u}_{\varepsilon } \otimes \mathbf{u}_{\varepsilon }: \nabla _{x}\mathbf{w}_{\varepsilon }\ \mathrm{d}x\ \mathrm{d}t,& {}\\ \end{array}$$where, by virtue of (7.37), (7.38), (7.43), and the gradient estimate established in (7.128),
$$\displaystyle{ \int _{0}^{T}\psi \int _{ \Omega }\varepsilon ^{2}\Big(\frac{\varrho _{\varepsilon } -\overline{\varrho }} {\varepsilon } \Big)\mathbf{u}_{\varepsilon } \otimes \mathbf{u}_{\varepsilon }: \nabla _{x}\mathbf{w}_{\varepsilon }\ \mathrm{d}x\ \mathrm{d}t = \varepsilon ^{3/2}\int _{ 0}^{T}\psi (t)\Gamma _{ 5,1}^{\varepsilon }(t)\ \mathrm{d}t, }$$(7.130)with
$$\displaystyle{\{\Gamma _{5,1}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ bounded in}\ L^{1}(0,T).}$$On the other hand, a direct computation yields
$$\displaystyle{ \int _{\Omega }(\mathbf{u}_{\varepsilon } \otimes \mathbf{u}_{\varepsilon }): \nabla _{x}\mathbf{w}_{\varepsilon }\ \mathrm{d}x = -\int _{\Omega }\mathrm{div}_{x}\mathbf{u}_{\varepsilon }\mathbf{u}_{\varepsilon } \cdot \mathbf{w}_{\varepsilon }\ \mathrm{d}x -\int _{\Omega }(\nabla _{x}\mathbf{u}_{\varepsilon }\mathbf{u}_{\varepsilon }) \cdot \mathbf{w}_{\varepsilon }\ \mathrm{d}x. }$$(7.131)Now, we have
$$\displaystyle{\int _{\Omega }\mathrm{div}_{x}\mathbf{u}_{\varepsilon }\mathbf{u}_{\varepsilon } \cdot \mathbf{w}_{\varepsilon }\ \mathrm{d}x = \int _{\Omega }\mathrm{div}_{x}\mathbf{u}_{\varepsilon }[\mathbf{u}_{\varepsilon }]_{\mathrm{ess}} \cdot \mathbf{w}_{\varepsilon }\ \mathrm{d}x + \int _{\Omega }\mathrm{div}_{x}\mathbf{u}_{\varepsilon }[\mathbf{u}_{\varepsilon }]_{\mathrm{res}} \cdot \mathbf{w}_{\varepsilon }\ \mathrm{d}x,}$$where, in accordance with estimates (7.36), (7.43),
$$\displaystyle{\{\mathrm{div}_{x}\mathbf{u}_{\varepsilon }[\mathbf{u}_{\varepsilon }]_{\mathrm{ess}}\}_{\varepsilon>0}\ \mbox{ is bounded in}\ L^{2}(0,T;L^{1}(\Omega; \mathbb{R}^{3})),}$$while
$$\displaystyle\begin{array}{rcl} & \|\mathrm{div}_{x}\mathbf{u}_{\varepsilon }[\mathbf{u}_{\varepsilon }]_{\mathrm{res}}\|_{L^{1}(0,T;L^{1}(\Omega;\mathbb{R}^{3}))} & {}\\ & \leq c\varepsilon ^{2/3}\|\nabla _{x}\mathbf{u}_{\varepsilon }\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{3\times 3}))}\|\mathbf{u}_{\varepsilon }\|_{L^{2}(0,T;L^{6}(\Omega;\mathbb{R}^{3}))},& {}\\ \end{array}$$where we have used (7.43), the embedding \(W^{1,2}(\Omega )\hookrightarrow L^{6}(\Omega )\), and the bound on the measure of the “residual set” established in (7.50).
Applying the same treatment to the latter integral on the right-hand side of (7.131) and adding the result to (7.130) we conclude that
$$\displaystyle{ I_{5}^{\varepsilon } = \varepsilon ^{3/2}\int _{ 0}^{T}\psi (t)\Gamma _{ 5,1}^{\varepsilon }\ \mathrm{d}t + \varepsilon \int _{ 0}^{T}\psi (t)\gamma _{ 5}^{\varepsilon }(t)\ \mathrm{d}t + \varepsilon ^{5/3}\int _{ 0}^{T}\psi (t)\Gamma _{ 5,2}^{\varepsilon }\ \mathrm{d}t, }$$(7.132)where
$$\displaystyle{\{\gamma _{5}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ is bounded in}\ L^{2}(0,T),}$$and
$$\displaystyle{\{\Gamma _{5,1}^{\varepsilon }\}_{ \varepsilon>0},\ \{\Gamma _{5,2}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ are bounded in}\ L^{1}(0,T).}$$ -
(vi)
In view of estimates (7.37), (7.38), it is easy to check that
$$\displaystyle{ I_{6}^{\varepsilon } = \varepsilon \int _{ 0}^{T}\psi (t)\gamma _{ 6}^{\varepsilon }(t)\ \mathrm{d}t, }$$(7.133)with
$$\displaystyle{\{\gamma _{6}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ bounded in}\ L^{\infty }(0,T).}$$ -
(vii)
Finally, in accordance with the first equation in (7.80) and Proposition 7.2,
$$\displaystyle{\|\mathrm{div}_{x}\mathbf{w}_{\varepsilon }\|_{L^{\infty }(\Omega )} \leq c;}$$therefore relations (7.38)–(7.41), (7.46), together with Proposition 5.2, can be used in order to conclude that
$$\displaystyle{ I_{7}^{\varepsilon } = \varepsilon \int _{ 0}^{T}\psi (t)\gamma _{ 7}^{\varepsilon }(t)\ \mathrm{d}t, }$$(7.134)where
$$\displaystyle{\{\gamma _{7}^{\varepsilon }\}_{ \varepsilon>0}\ \mbox{ is bounded in}\ L^{\infty }(0,T).}$$
We are now in a position to use relation (7.121) in order to show (7.120). To begin, we focus on the integral
appearing on the left-hand side of (7.121), with r ɛ , V ɛ specified in (7.61). Writing
we can use the uniform estimates (7.36)–(7.41), together with pointwise convergence of the remainders established in (7.112), in order to deduce that
where
Next, we use a family of standard regularizing kernels
in order to handle the “measure-valued” term in (7.121). To this end, we take ζ δ as a test function in (7.121) to obtain
where we have set
for t ∈ (δ, T −δ).
In accordance with the uniform estimates (7.122)–(7.134), we have
uniformly for δ → 0, where we have used the standard properties of mollifiers recorded in Theorem 11.3 in Appendix. Similarly, by virtue of (7.135), (7.136),
Here all functions in (7.138), (7.139) have been extended to be zero outside (δ, T −δ).
The standard variation-of-constants formula yields
therefore letting first δ → 0 and then ɛ → 0 yields the desired conclusion (7.120). Note that, in accordance with (7.111),
in particular
uniformly for ɛ → 0. The proof of Theorem 7.1 is now complete.
7.6 Asymptotic Limit on Domains with Oscillatory Boundaries and Complete Slip Boundary Conditions
Although the no-slip boundary condition (7.19) is probably the most widely accepted for viscous fluids in contact with an impermeable boundary, it is sometimes more convenient to approximate a complicated topography of the real physical boundary by a smooth one endowed with a suitable wall law similar to the slip boundary condition (5.15) rather than (7.19) (see Jaeger and Mikelic [158], Mohammadi et al. [215], among others).
Similarly to the preceding part, we consider the infinite slab (7.17), (7.18), with flat top and variable bottom determined through a function
where \(h \in C^{2}(\mathcal{T}^{\,2})\) is the same as in (7.18), \(\omega \in C^{2}(\mathcal{T}^{\,2})\), and k(ɛ) is a sequence of positive integers, k(ɛ) → ∞ as ɛ → 0. Thus the functions ω ɛ are 2π∕k(ɛ)-periodic, with amplitude proportional to 1∕k(ɛ).
We set
and impose the following boundary conditions for the velocity:
The no-slip boundary condition is therefore prescribed only on the top part while complete slip boundary conditions, used in the preceding Chaps. 5 and 6, are required at the bottom part of \(\Omega _{\varepsilon }\). Our goal is to show that (7.142) provides the same effect as the no-slip boundary conditions provided the “oscillatory” part of the boundary here represented by ω ɛ is non-degenerate, meaning not constant in any direction. In particular, the velocities u ɛ in the asymptotic low Mach number limit will approach the limit profile u strongly with respect to the L 1-topology.
We claim the following variant of Theorem 7.1.
■ Compactness of Velocities on Domains with Variable Bottoms: The complete slip boundary conditions
Theorem 7.2
Let \(\Omega _{\varepsilon }\) be a family of domains determined through ( 7.140 ), ( 7.141 ), where the “bottom” part of the boundary is given by functions h, ω satisfying
and ω is non-degenerate, specifically, for any w = [w 1, w 2] ≠ 0 there is \((x_{1},x_{2}) \in \mathcal{T}^{\,2}\) such that
Let
Let \(F \in W^{1,\infty }(\mathbb{R}^{3})\) be given such that
where
Assume that \(\mathbb{S}\) , q as well as the thermodynamic functions p, e, and s are given by ( 7.10 )–( 7.15 ), where P meets the structural hypotheses ( 7.23 )–( 7.25 ), while the transport coefficients μ and κ satisfy ( 7.26 ), ( 7.27 ).
Finally, let {ϱ ɛ , u ɛ , ϑ ɛ } ɛ > 0 be a family of weak solutions to the Navier-Stokes-Fourier system satisfying ( 7.5 )–( 7.9 ) in \((0,T) \times \Omega _{\varepsilon }\) , with the boundary conditions ( 7.20 ), ( 7.142 ), with the initial data
where
and
Then, at least for a suitable subsequence,
where \(\mathbf{U} \in L^{2}(0,T;W_{0}^{1,2}(\Omega; \mathbb{R}^{3}))\) , div x U = 0.
Remark
It is worth noting that the limit velocity profile U satisfies the no-slip boundary condition on both the top and the bottom part of the boundary of the limit domain \(\Omega\). Similarly to the preceding part, we leave to the reader to show that the limit quantities satisfy the Oberbeck-Boussinesq system introduced in Sect. 5.
Remark
The weak solutions are defined exactly as in Sect. 7.2.1, with the obvious modifications
whereas the test functions φ in the momentum equation (7.29) are taken from the space
Remark
For the sake of simplicity, we have assumed that the initial data are defined on the whole physical space \(\mathbb{R}^{3}\). As \(\Omega \subset \Omega _{\varepsilon }\), the statement (7.147) makes sense.
The rest of this chapter is devoted to the proof of Theorem 7.2. The idea is that the rapidly oscillating boundary along with the effect of viscosity will force the fluid to be at rest on the boundary of the limit domain; whence the methods developed in Sects. 7.4, 7.5 can be applied.
7.7 Uniform Bounds
The uniform bounds on the sequence of solutions [ϱ ɛ , u ɛ ] are essentially the same as in Sect. 7.3. However, we should keep in mind that the underlying spatial domains \(\Omega _{\varepsilon }\) depend on the scaling parameter ɛ. Accordingly, the constants appearing in Korn’s and Poicaré’s inequality used in Sect. 7.3 may depend on ɛ. Fortunately, by virtue of hypotheses (7.140), (7.141), the family \(\Omega _{\varepsilon }\) is uniformly Lipschitz, and, consequently, the corresponding constants are the same for all \(\Omega _{\varepsilon }\), see Theorem 11.24 in Appendix. With this observation in mind, we report the following list of estimates.
-
Energy estimates:
$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\|\sqrt{\varrho _{\varepsilon }}\mathbf{u}_{\varepsilon }\|_{L^{2}(\Omega _{\varepsilon };\mathbb{R}^{3})} \leq c, }$$(7.148)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\varrho _{\varepsilon } -\overline{\varrho }} {\varepsilon } \Big]_{\mathrm{ess}}\Big\|_{L^{2}(\Omega _{\varepsilon })} \leq c, }$$(7.149)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\varrho _{\varepsilon } -\overline{\varrho }} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{ L^{\frac{5} {3} }(\Omega _{\varepsilon })} \leq \varepsilon ^{\frac{1} {5} }c, }$$(7.150)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\vartheta _{\varepsilon } -\overline{\vartheta }} {\varepsilon } \Big]_{\mathrm{ess}}\Big\|_{L^{2}(\Omega _{\varepsilon })} \leq c, }$$(7.151)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|[\vartheta _{\varepsilon }]_{\mathrm{res}}\Big\|_{L^{4}(\Omega _{\varepsilon })} \leq \varepsilon ^{\frac{1} {2} }c, }$$(7.152)$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{p(\varrho _{\varepsilon },\vartheta _{\varepsilon }) - p(\overline{\varrho },\overline{\vartheta })} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{L^{1}(\Omega _{\varepsilon })} \leq \varepsilon c. }$$(7.153) -
Estimates based on energy dissipation:
$$\displaystyle{ \|\sigma _{\varepsilon }\|_{\mathcal{M}^{+}([0,T]\times \overline{\Omega }_{\varepsilon })} \leq \varepsilon ^{2}c, }$$(7.154)$$\displaystyle{ \int _{0}^{T}\|\mathbf{u}_{\varepsilon }\|_{ W^{1,2}(\Omega _{\varepsilon };\mathbb{R}^{3})}^{2}\ \mathrm{d}t \leq c, }$$(7.155)$$\displaystyle{ \int _{0}^{T}\Big\|\frac{\vartheta _{\varepsilon } -\overline{\vartheta }} {\varepsilon } \Big\|_{W^{1,2}(\Omega _{\varepsilon })}^{2}\ \mathrm{d}t \leq c, }$$(7.156)$$\displaystyle{ \int _{0}^{T}\Big\|\frac{\log (\vartheta _{\varepsilon }) -\log (\overline{\vartheta })} {\varepsilon } \Big\|_{W^{1,2}(\Omega _{\varepsilon })}^{2}\ \mathrm{d}t \leq c. }$$(7.157) -
Entropy estimates:
$$\displaystyle{ \mathrm{ess}\sup _{t\in (0,T)}\Big\|\Big[\frac{\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{L^{1}(\Omega _{\varepsilon })}\mathrm{d}t \leq \varepsilon c, }$$(7.158)$$\displaystyle{ \int _{0}^{T}\Big\|\Big[\frac{\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \Big]_{\mathrm{res}}\Big\|_{L^{q}(\Omega _{\varepsilon })}^{q}\ \mathrm{d}t \leq c\ \mbox{ for a certain}\ q> 1, }$$(7.159)$$\displaystyle{ \int _{0}^{T}\Big\|\Big[\frac{\varrho _{\varepsilon }s(\varrho _{\varepsilon },\vartheta _{\varepsilon })} {\varepsilon } \mathbf{u}_{\varepsilon }\Big]_{\mathrm{res}}\Big\|_{L^{q}(\Omega _{\varepsilon };\mathbb{R}^{3})}^{q}\ \mathrm{d}t \leq c\ \mbox{ for a certain}\ q> 1, }$$(7.160)$$\displaystyle{ \int _{0}^{T}\Big\|\Big[\frac{\kappa (\vartheta _{\varepsilon })} {\vartheta _{\varepsilon }} \Big]_{\mathrm{res}}\frac{\nabla _{x}\vartheta _{\varepsilon }} {\varepsilon } \Big\|_{L^{q}(\Omega _{\varepsilon };\mathbb{R}^{3})}^{q}\ \mathrm{d}t \rightarrow 0\ \mbox{ for a certain}\ q> 1. }$$(7.161)
7.8 Convergence of the Velocity Trace on Oscillatory Boundary
Our goal is to show that the traces of the velocities {u ɛ } ɛ > 0 vanish on the boundary of the limit domain \(\Omega\) in the asymptotic limit ɛ → 0.
Proposition 7.3
Let \(\Omega _{\varepsilon }\) be a family of domains satisfying the hypotheses of Theorem 7.2 .
Then
for any \(\mathbf{v} \in W^{1,2}(\Omega _{\varepsilon }; \mathbb{R}^{3})\) satisfying
where the constant is independent of ɛ → 0.
Proof
Obviously, we can restrict ourselves to the strip
containing the bottom part B of the boundary \(\partial \Omega\),
Next, writing
we observe that it is enough to show (7.162) on each S ɛ n, m.
Finally, after the scaling x ≈ k(ɛ)x and an obvious space shift, the problem reduces to proving
for any \(\mathbf{v} \in W^{1,2}(\mathbb{R}^{3}, \mathbb{R}^{3})\),
where χ ɛ → χ in C 1[0, 1]2 as ɛ → 0, and where χ is an affine function.
Arguing by contradiction, we obtain a sequence \(\mathbf{v}_{\varepsilon } \in W^{1,2}(\mathbb{R}^{3}, \mathbb{R}^{3})\),
where g(ɛ) → 0 as ɛ → 0. In addition, we may assume
and
Consequently, in view of the compact embedding \(W^{1,2}(K; \mathbb{R}^{3})\hookrightarrow L^{2}(K; \mathbb{R}^{3})\), \(K \subset \mathbb{R}^{3}\) compact, the limit function v satisfies
where
Finally, we claim that
Indeed seeing that
for any
we infer that
for any
which implies (7.165).
In view of (7.164), the limit v is constant in Q, which is incompatible with (7.165) as long as ω satisfies the non-degeneracy condition (7.144).
□
Thus Proposition 7.3, together with the uniform bound (7.155) yield
The remaining part of the proof is obvious although technically involved. We restrict ourselves to the limit domain \(\Omega\), where the boundary terms arising in by parts integration will be controlled by (7.166).
7.9 Strong Convergence of the Velocity Field Revisited
Our final goal is to establish the strong convergence of the velocities claimed in (7.147). As a consequence of (7.155), we may assume that
provided u ɛ were extended to the set where x 3 ≤ −π − h(x 1, x 2) −ω ɛ (x 1, x 2). Moreover, as a consequence of (7.166),
7.9.1 Solenoidal Component
The velocity fields, restricted to the target domain \(\Omega\), decompose as
where H denotes the Helmholtz projection defined on \(\Omega\). Using the uniform bounds obtained in Sect. 7.7 and repeating the arguments of Sect. 7.5.1 we deduce that the family of scalar functions
for any \(\phi \in C_{c}^{\infty }(\Omega; \mathbb{R}^{3}),\ \mathrm{div}_{x}\phi = 0\); whence, in accordance with (7.168),
Consequently, by virtue of (7.167) and compactness of the embedding \(W^{1,2}(\Omega )\hookrightarrow L^{5}(\Omega )\), we may infer that
yielding
7.9.2 Acoustic Waves
In view of (7.169), it remains to show
We use arguments similar to those in Sect. 7.4 starting with the acoustic equation (7.59), (7.60) for the unknowns
The equation of continuity together with the entropy balance give rise to
for any \(\varphi \in C_{c}^{\infty }([0,T) \times \overline{\Omega }_{\varepsilon })\), with
and
cf. (7.59).
Next, as \(\Omega \subset \Omega _{\varepsilon }\), we may consider
as a test function in (7.29) obtaining
with
and
cf. (7.52). It is easy to observe that validity of (7.172) can be extended to
Next, we may compute
for all
recalling that
Thus (7.172) finally reads as
for any
where
cf. (7.60). Unlike its counterpart (7.59), (7.60), however, Eqs. (7.171), (7.173) are considered on different spatial domains \(\Omega _{\varepsilon }\), \(\Omega\), respectively.
7.9.3 Strong Convergence of the Gradient Component
In view of exactly the same arguments as in Sect. 7.5.2, the proof of (7.170) can be reduced to showing
where r ɛ , V ɛ satisfy the acoustic system (7.171), (7.173), and v ɛ , w ɛ are the solutions of the approximate eigenvalue problem (7.80), (7.82).
A natural idea is to use v ɛ , w ɛ as test functions in (7.171), (7.173), respectively. Unfortunately, however, v ɛ is defined only on the set \(\Omega\) and therefore must be extended as \(\tilde{v}_{\varepsilon }\) to \(\Omega _{\varepsilon }\) in such a way that
where c is independent of ɛ. As the family \(\{\Omega _{\varepsilon }\}_{\varepsilon>0}\) is equi-Lipschitz, such an extension exists (see Theorem 8). Accordingly, we will show
instead of (7.174).
Following the line of arguments in Sect. 7.5.3, we take the quantities \(\psi (t)\tilde{v}_{\varepsilon }(x)\), ψ(t)w ɛ (x), ψ ∈ C c ∞(0, T), as test functions in (7.171), (7.173) to obtain
where
and
and
Note that the integrals I 1 ɛ–I 7 ɛ are the same as their counterparts in (7.121). In particular, the same arguments as in Sect. 7.5.3 can be used to show that each of them can be written in the form
where
Now, we come to the crucial point of the proof using the bound (7.166) on the trace of the solution u ɛ on \(\partial \Omega\) to obtain
With help of (7.155) and Proposition 7.2 yielding the necessary bound for \(\sqrt{\varepsilon }\nabla _{x}\mathbf{w}_{\varepsilon }\) we conclude that
Next, as \(\vert \Omega _{\varepsilon }\setminus \Omega \vert \approx \varepsilon\), we get
Furthermore, by means of Hölder’s inequality,
therefore, by virtue of (7.155) and Proposition 7.2,
where
Finally, to control the integral I 10 ɛ, we write
where,
and, similarly,
Thus the uniform bounds (7.149)–(7.153) yield the desired conclusion
where
Having controlled all the integrals in (7.176) and seeing that m > 1, we are in the situation described in the last part of Sect. 7.5.3. Consequently, repeating step by step the arguments used therein, we can show (7.174) and therefore complete the proof of Theorem 7.2.
7.10 Concluding Remarks
We have shown that the no-slip boundary conditions and the complete slip boundary conditions considered on “oscillatory” boundary produce the same effect in the low Mach number limit, specifically, the acoustic waves are effectively damped as long as the boundary of the target domain is non-degenerate (non-flat). As a matter of fact, a proper choice of the boundary conditions for the velocity of a viscous fluid confined to a bounded physical space has been discussed by many prominent physicists and mathematicians over the last two centuries (see the survey paper by Priezjev and Troian [235]).
For a long time, the no-slip boundary conditions have been the most widely accepted for their tremendous success in reproducing the observed velocity profiles for macroscopic flows. Still the no-slip boundary condition is not intuitively obvious. Recently developed technologies of micro and nano-fluidics have shown the slip of the fluid on the boundary to be relevant when the system size approaches the nanoscale. The same argument applies in the case when the shear rate is sufficiently strong in comparison with the characteristic length scale as in some meteorological models (see Priezjev and Troian [235]). As a matter of fact, an alternative microscopic explanation of the no-slip condition argues that because most real surfaces are rough, the viscous dissipation as the fluid passes the surface irregularities brings it to rest regardless the character of the intermolecular forces acting between the fluid and the solid wall. A rigorous mathematical evidence of this hypothesis has been provided in a series of papers by Amirat et al. [9, 10], Casado-Díaz et al. [51] or, more recently, [52]. Thus the roughness argument, used also in this chapter, reconciles convincingly the ubiquitous success of the no-slip condition with the boundary behaviour of real fluids predicted by molecular dynamics (cf. Qian et al. [236]).
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Feireisl, E., Novotný, A. (2017). Interaction of Acoustic Waves with Boundary. In: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-63781-5_7
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