Variational inequality solutions and finite stopping time for a class of shear-thinning flows

Abstract

The aim of this paper is to study the existence of a finite stopping time for solutions in the form of variational inequality to fluid flows following a power law (or Ostwald–DeWaele law) in dimension $N \in \{2,3\}$. We first establish the existence of solutions for generalized Newtonian flows, valid for viscous stress tensors associated with the usual laws such as Ostwald–DeWaele, Carreau–Yasuda, Herschel–Bulkley and Bingham, but also for cases where the viscosity coefficient satisfies a more atypical (logarithmic) form. To demonstrate the existence of such solutions, we proceed by applying a nonlinear Galerkin method with a double regularization on the viscosity coefficient. We then establish the existence of a finite stopping time for threshold fluids or shear-thinning power-law fluids, i.e. formally such that the viscous stress tensor is represented by a p-Laplacian for the symmetrized gradient for $p \in [1,2)$.

Appendices

Some examples of viscosity coefficients

In this section, we give some examples of functions F satisfying the conditions (C1)–(C4), most of which correspond to models of non-Newtonian coherent flows in the physical sense. This is the case for quasi-Newtonian fluids such as blood, threshold fluids such as mayonnaise, or more generally in the case of polymeric liquids.

• 1. Firstly, in order to describe power-law fluids (also known as Ostwald–DeWaele flows), we can consider functions $(F_{p})_{1< p < 2}$ given by: $\begin{array}{l|l} &{}(0,+\infty ) \rightarrow (0,+\infty )\\ F_{p} :&{}\\ &{}t \longmapsto t^{p-2}. \end{array}$

• 2. Considering functions $(F_{\mu ,p})_{\mu > 0, p \in [1,2)}$ of the form $\begin{array}{l|l} &{}(0,+\infty ) \rightarrow (0,+\infty )\\ F_{\mu ,p} :&{}\\ &{}t \longmapsto (\mu + t^2)^{\frac{p-2}{2}} \end{array}$ leads to Carreau flows.

• 3. Cross fluids are obtained by choosing function $(F_{\gamma ,p})_{\gamma > 0, p \in [1,2)}$ given by: $\begin{array}{l|l} &{}(0,+\infty ) \rightarrow (0,+\infty )\\ F_{\gamma ,p} :&{}\\ &{}t \longmapsto (\gamma + t^{2-p})^{-1}. \end{array}$

• 4. Another possible choice is to take functions $(F_{p,\beta ,\gamma })$ given $\begin{array}{l|l} &{}(0,+\infty ) \rightarrow (0,+\infty )\\ F_{p,\beta ,\gamma } :&{}\\ &{}t \longmapsto \left\{ \begin{array}{ll}t^{p-2}\text {log}(1+t)^{-\beta }&{}\quad \text {if} \; t \in (0,\gamma ] \\ {} &{} \\ \text {log}(1+\gamma )^{-\beta }t^{p-2}&{}\quad \text {if}\; t \in (\gamma ,+\infty )\end{array}\right. \end{array}$ for $1< p < 2$ and some $\beta ,\gamma >0$ with $\gamma$ small enough.

Useful lemmas and energy estimates

For the sake of clarity, in this “Appendix”, we state and prove some useful results employed for the proof of Theorem 3.1. We begin this “Appendix” with some technical lemmas and, in its second part, we give a proof for Proposition 4.1.

1.1 Technical lemmas

Lemma B.1

Let X be a Banach space, and $\gamma \ge \frac{1}{2}$. Then, the following inequality holds:

$$\begin{aligned} \forall (u,v) \in X^2,\; \Vert u + v \Vert ^{\gamma }_X \le 2^{\left( \gamma - \frac{1}{2}\right) }\left( \Vert u \Vert ^{\gamma }_X + \Vert v \Vert ^{\gamma }_X\right) . \end{aligned}$$

Proof

Using the convexity of $t \mapsto t^{2(2-p)}$ and triangle’s inequality of the norm, we get:

$$\begin{aligned} \Vert u + v \Vert ^{2\gamma }_X = 2^{2\gamma }\left\Vert \frac{u+v}{2} \right\Vert ^{2\gamma }_X \le 2^{2\gamma -1}\left( \Vert u \Vert ^{2\gamma }_X + \Vert v \Vert ^{2\gamma }_X\right) . \end{aligned}$$

Applying now the well-known inequality: $\forall (a,b) \in [0,+\infty )^2,\; \sqrt{a+b} \le \sqrt{a} + \sqrt{b}$, we get the result. $\square$

Lemma B.2

Consider that $\varphi \in L^2_{\textrm{loc}}(\mathbb {R}_+,H_0^1(\Omega ))$, then there exists a constant $C(\varepsilon ,\varphi ) > 0$ which goes to zero as $\varepsilon$ does, such that the following inequality holds:

$$\begin{aligned} j_{\varepsilon }(\varphi ) + C(\varepsilon ,\varphi ) \ge j(\varphi ), \end{aligned}$$

(B.1)

where $j_\varepsilon$ and j are defined by (2.2).

Proof

Recalling that the assumption (C3) states that $t \mapsto tF(t)$ is increasing, we get:

$$\begin{aligned} j(\varphi )&{:}{=} \int _{\Omega }\int _0^{|D(\varphi )|} sF(s)\; ds\, dx\\&\le \int _{\Omega }\int _0^{\sqrt{\varepsilon }} sF(s)\; ds\, dx + \int _{\Omega }\int _{\sqrt{\varepsilon }}^{\sqrt{\varepsilon } + |D(\varphi )|} sF(s)\; ds\, dx\\&\le \varepsilon \sqrt{\varepsilon }F(\varepsilon )|\Omega |+ \int _{\Omega }\int _0^{\sqrt{2|D(\varphi ) |\sqrt{\varepsilon } + |D(\varphi )|^2}}sF(\sqrt{\varepsilon + s^2})\; ds\, dx\\&\le \underbrace{\varepsilon \sqrt{\varepsilon }F(\varepsilon )|\Omega |+ \int _{\Omega }\int _{|D(\varphi ) |}^{2^{\frac{1}{2}}\varepsilon ^{\frac{1}{4}}|D(\varphi ) |^{\frac{1}{2}} + |D(\varphi ) |}sF(\sqrt{\varepsilon + s^2})\; ds \,dx}_{{:}{=} C(\varepsilon ,\varphi )} + j_{\varepsilon }(\varphi ), \end{aligned}$$

which is the wished result. $\square$

Lemma B.3

Consider $\Omega$ an open bounded subset of ${\mathbb {R}}^N$ with Lipschitz boundary, and a sequence $(w_n)_{n \in {\mathbb {N}}}$ such that there exists a positive constant $C > 0$ satisfying $\Vert w_n \Vert _{L^2_{\textrm{loc}}(\mathbb {R}_+,H_{0,\sigma }^1(\Omega )} \le C$. Then, for every fixed $T > 0$ and for almost all $(t,x) \in (0,T) \times \Omega$, the following inequality holds:

$$\begin{aligned} \underset{n \rightarrow +\infty }{\underline{\textrm{lim}}}\; |D(w_n)(t,x) |\ge |D(w)(t,x) |. \end{aligned}$$

Proof

Firstly, let us recall that Eberlein-S̆mulyan theorem leads up to an extraction to $w_n \rightharpoonup w$ in $L^2_{\textrm{loc}}(\mathbb {R}_+,H_0^1(\Omega ))$ then, for every fixed $T > 0$ and all Lebesgue points $t_0 \in (0,T)$ and $x_0 \in \Omega$, for all $\delta >$ and $R > 0$ small enough, we have $w_n \rightharpoonup w$ in $L^2((t_0 - \delta , t_0 + \delta ),H^1(B(x_0,R))$. Indeed, we have for all test function $\varphi$:

$$\begin{aligned} \int _0^T\int _{\Omega }\nabla w_n \cdot \nabla \varphi \;dt\,dx \underset{n \rightarrow +\infty }{\longrightarrow } \int _0^T\int _{\Omega }\nabla w \cdot \nabla \varphi \;dt\,dx. \end{aligned}$$

Hence, we can take $\varphi$, which belongs to ${\mathcal {C}}^{\infty }_0((t_0 - \delta ,t_0 + \delta ) \times B(x_0,R))$ (up to arguing by density thereafter), satisfying:

$$\begin{aligned} \nabla \varphi = \left\{ \begin{array}{ll} \nabla \psi &{} on (t_0 - \delta , t_0 + \delta ) \times B(x_0,R)\\ 0&{} on (0,T) \times \Omega \backslash (t_0 - \delta , t_0 + \delta ) \times B(x_0,R) \end{array}\right. \end{aligned}$$

and so this leads to:

$$\begin{aligned} \int _{t_0- \delta }^{t_0 + \delta }\int _{B(x_0,R)}\nabla w_n \cdot \nabla \psi \;dt\,dx \underset{n \rightarrow +\infty }{\longrightarrow }\ \int _{t_0 - \delta }^{t_0 + \delta }\int _{B(x_0,R)}\nabla w \cdot \nabla \psi \;dt\,dx. \end{aligned}$$

That is $w_n\rightharpoonup w$ in $L^2((t_0 - \delta , t_0 + \delta ),H^1(B(x_0,R)))$. Also, from Korn’s $L^2$ equality and Lebesgue’s differentiation theorem over $(\delta ,R)$ after dividing by $2\delta |B(x_0,R)|$, one gets that for every Lebesgue point $(t_0,x_0) \in (0,T) \times \Omega$:

$$\begin{aligned} |D(w_n(t_0,x_0)) |^2 \le C \end{aligned}$$

Following the same line of arguments, we find that:

$$\begin{aligned} \underset{n \rightarrow +\infty }{\underline{\textrm{lim}}}\;\int _{t_0 - \delta }^{t_0 + \delta }\int _{B(x_0,R)} |D(w_n) |^2\;dx \,dt \ge \int _{t_0 - \delta }^{t_0 + \delta }\int _{B(x_0,R)} |D(w) |^2\;dx \,dt. \end{aligned}$$

Dividing each side by $2\delta |B(x_0,R) |$, we get:

then letting $(\delta ,R) \rightarrow (0,0)$ leads to the result, after applying a dominated convergence theorem. $\square$

1.2 Proof of Proposition 4.1

We now prove the energy estimates used for the convergence of the nonlinear Galerkin method appearing in the proof of Theorem 3.1.

Proof of Proposition 4.1

• 1. Setting $\varphi = u_{m,\varepsilon }$ in the weak formulation, we get:

  $$\begin{aligned} \frac{1}{2} \frac{d}{dt}\Vert u_{m,\varepsilon } \Vert _{L^2}^2{} & {} + \int _{\Omega } |D(u_{m,\varepsilon }) |^2\; dx + \underbrace{\langle j_{\varepsilon }'(u_{m,\varepsilon }),u_{m,\varepsilon } \rangle }_{\ge 0}\\{} & {} - \underbrace{\int _{\Omega }(u_{m,\varepsilon }\cdot \nabla u_{m,\varepsilon })\cdot u_{m,\varepsilon }\; dx}_{=0} = \langle f,u_{m,\varepsilon } \rangle . \end{aligned}$$

  Using the Korn’s $L^2$ equality for divergence free vectors fields, we get

  $$\begin{aligned} \frac{d}{dt}\Vert u_{m,\varepsilon }(t) \Vert _{L^2}^2 + \Vert u_{m,\varepsilon }(t) \Vert _{H_0^1}^2 \le 2\left\langle f(t),u_{m,\varepsilon }(t) \right\rangle \le 2\Vert f(t) \Vert _{H^{-1}}^2 + \frac{1}{2}\Vert u_{m,\varepsilon }(t) \Vert _{H_0^1}^2. \end{aligned}$$

  Then, integrating on (0, t) we get

  $$\begin{aligned} \Vert u_{m,\varepsilon }(t) \Vert ^2_{L^2} + \frac{1}{2}\int _0^t\Vert u_{m,\varepsilon } \Vert ^2_{H_0^1}\; dt \le 2\int _0^t \Vert f \Vert ^2_{H^{-1}}\; dt + \Vert u_0 \Vert _{L^2}^2. \end{aligned}$$

  (B.2)

  Indeed, we recall that $(P_m(u_0),w_i)_{L^2} = (u_0,P_mw_i)_{L^2} = (u_0,w_i)_{L^2}$, and the conclusion follows. From now on, we will omit to detail this last part which is usual.

• 2. We have, using Cauchy-Schwarz’s inequality and Korn’s equality in the divergence free $L^2$ setting:

  $$\begin{aligned} \left\langle j_{\varepsilon }'(u_{m,\varepsilon }),\varphi \right\rangle&= \int _{\Omega }F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) D(u_{m,\varepsilon }):D(\varphi )\; dx \nonumber \\&\le \frac{1}{\sqrt{2}}\left( \int _{\Omega }F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) ^2|D(u_{m,\varepsilon }) |^2\; dx\right) ^{\frac{1}{2}} \Vert \varphi \Vert _{H_0^1}. \end{aligned}$$

  (B.3)

  From hypothesis (C4), setting $A = \Omega \cap \{ |D(u_{m,\varepsilon }) |\le t_0 \}$ and B its complement in $\Omega$, we obtain

  $$\begin{aligned} \int _{\Omega }F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) ^2|D(u_{m,\varepsilon }) |^2\; dx&= \int _{A}F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) ^2|D(u_{m,\varepsilon }) |^2\; dx\\&\quad + \int _{B}F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) ^2|D(u_{m,\varepsilon }) |^2\; dx. \end{aligned}$$

  Let’s estimate these two integrals independently. By assumption (C3), we have that the application $t \mapsto t^2F\left( \sqrt{\varepsilon + t^2}\right) ^2$ is non-decreasing, and we obtain directly:

  $$\begin{aligned} \int _{A}F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) ^2|D(u_{m,\varepsilon }) |^2\; dx&\le F\left( \sqrt{\varepsilon + {t_0}^2}\right) ^2{t_0}^2|A |\\&\le F\left( \sqrt{\varepsilon + {t_0}^2}\right) ^2{t_0}^2|\Omega |\\&\le F\left( \sqrt{1+ {t_0}^2}\right) ^2\sqrt{1+ {t_0}^2}|\Omega |\\&\le C. \end{aligned}$$

  Then we have, using again (C4):

  $$\begin{aligned} \int _{B}F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) ^2|D(u_{m,\varepsilon }) |^2\; dx&\le K\int _{B}\frac{|D(u_{m,\varepsilon }) |^2}{\left( \varepsilon + |D(u_{m,\varepsilon }) |^2\right) ^{2-p}}\; dx\\&\le K\int _{B}|D(u_{m,\varepsilon }) |^{2(p-1)}\; dx\\&\le K\int _B|\nabla u_{m,\varepsilon } |^{2(p-1)}\; dx\\&\le C\Vert u_{m,\varepsilon } \Vert ^{2(p-1)}_{H_0^1}, \end{aligned}$$

  where we used Jensen’s inequality in the concave setting with $t \mapsto t^{p-1}$ in the last line. So, we obtain:

  $$\begin{aligned} \left( \int _{\Omega }F\left( \sqrt{\varepsilon + |D(u_{m,\varepsilon }) |^2}\right) ^2|D(u_{m,\varepsilon }) |^2\; dx\right) ^{\frac{1}{2}} \le \left( C + C\Vert u_{m,\varepsilon } \Vert ^{2(p-1)}_{H_0^1}\right) ^{\frac{1}{2}}. \end{aligned}$$

  (B.4)

  Thus, combining the inequality (B.3)–(B.4), using Lemma B.1 with $\gamma = \frac{2}{N}$, and integrating in time leads to:

  $$\begin{aligned} \Vert j_{\varepsilon }'(u_{m,\varepsilon }) \Vert ^{\frac{4}{N}}_{L^{\frac{4}{N}}\left( (0,T),H^{-1}\right) } \le C + C\Vert u_{m,\varepsilon } \Vert ^{\frac{4(p-1)}{N}}_{L^{\frac{4(p-1)}{N}}((0,T),H_0^1)}. \end{aligned}$$

  Then, since $0<\frac{4(p-1)}{N} \le 2$, we get, using the embedding $L^2 \hookrightarrow L^{\frac{4(p-1)}{N}}$ and Lemma B.1 with $X := H_0^1$, $q = \frac{4(p-1)}{N}$ and $p=2$ on $\Vert u_{m,\varepsilon } \Vert _{L^{\frac{4(p-1)}{N}}((0,T),H_0^1)}$:

  $$\begin{aligned} \Vert j_{\varepsilon }'(u_{m,\varepsilon }) \Vert ^{\frac{4}{N}}_{L^{\frac{4}{N}}\left( (0,T),H^{-1}\right) } \le C + C\Vert u_{m,\varepsilon } \Vert ^{\frac{4(p-1)}{N}}_{L^{2}((0,T),H_0^1)}. \end{aligned}$$

  Using the first point of the proposition for $t=T$, and since $\frac{4(p-1)}{N} \ge 0$, we get:

  $$\begin{aligned} \Vert j_{\varepsilon }'(u_{m,\varepsilon }) \Vert ^{\frac{4}{N}}_{L^{\frac{4}{N}}\left( (0,T),H^{-1}\right) } \le C + C(\Vert f \Vert _{L^{2}((0,T),H^{-1})} + \Vert u_0 \Vert _{L^2})^{\frac{4(p-1)}{N}}. \end{aligned}$$

  Then, using the exponent $\frac{N}{4}$ on both sides and applying once again Lemma B.1 with $\gamma = \frac{N}{4}$ on the right-hand side in the inequality above leads us to:

  $$\begin{aligned} \Vert j_{\varepsilon }'(u_{m,\varepsilon }) \Vert _{L^{\frac{4}{N}}\left( (0,T),H^{-1}\right) } \le C +C(\Vert f \Vert _{L^{2}((0,T),H^{-1})} + \Vert u_0 \Vert _{L^2})^{p-1}. \end{aligned}$$

  This is the wished result.

• 3. From the weak formulation (4.4) we get

  $$\begin{aligned} \langle \partial _tu_{m,\varepsilon },\varphi \rangle{} & {} = - \int _{\Omega } D(u_{m,\varepsilon }):D(\varphi )\; dx - \langle j_{\varepsilon }'(u_{m,\varepsilon }),\varphi \rangle \nonumber \\{} & {} \quad \ + \int _{\Omega }(u_{m,\varepsilon } \cdot \nabla u_{m,\varepsilon }) \cdot \varphi \; dx + \langle f,\varphi \rangle . \end{aligned}$$

  (B.5)

  Let us point out that

  $$\begin{aligned} \int _{\Omega }D(u_{m,\varepsilon }):D(\varphi )\; dx = \frac{1}{2}\int _{\Omega }\nabla u_{m,\varepsilon }\cdot \nabla \varphi \; dx \le \frac{1}{2}\Vert u_{m,\varepsilon } \Vert _{H_0^1}\Vert \varphi \Vert _{H_0^1}. \end{aligned}$$

  (B.6)

  Also, from Gagliardo-Nirenberg’s inequality, we get the existence of a positive constant C which only depends on N and $\Omega$ such that:

  $$\begin{aligned} \Vert u \Vert _{L^4}^2 \le C\Vert \nabla u\Vert _{L^2}^{\frac{N}{2}}\Vert u \Vert _{L^2}^{\frac{4-N}{2}}. \end{aligned}$$

  (B.7)

  The latter leads, as for the Navier–Stokes equations:

  $$\begin{aligned} \left|\int _{\Omega }(u_{m,\varepsilon }.\nabla u_{m,\varepsilon }).\varphi \; dx \right|\le C \Vert u_{m,\varepsilon } \Vert _{L^2}^{\frac{4-N}{2}}\Vert u_{m,\varepsilon } \Vert _{H_0^1}^{\frac{N}{2}}\Vert \varphi \Vert _{H_0^1}. \end{aligned}$$

  So, putting (B.6)–() and the second estimate of the Proposition 4.1 in (B.5), we obtain

  $$\begin{aligned} \langle \partial _tu_{m,\varepsilon },\varphi \rangle \le&\; \frac{1}{2}\Vert u_{m,\varepsilon } \Vert _{H_0^1}\Vert \varphi \Vert _{H_0^1} + \Vert j_{\varepsilon }'(u_{m,\varepsilon }) \Vert _{H^{-1}}\Vert \varphi \Vert _{H_0^1} +C \Vert u_{m,\varepsilon } \Vert _{L^2}^{\frac{4-N}{2}}\Vert u_{m,\varepsilon } \Vert _{H_0^1}^{\frac{N}{2}}\Vert \varphi \Vert _{H_0^1}\\&+ \Vert f \Vert _{H^{-1}}\Vert \varphi \Vert _{H_0^1}, \end{aligned}$$

  and therefore

  $$\begin{aligned} \Vert \partial _tu_{m,\varepsilon }(t) \Vert _{H^{-1}} \le \; \frac{1}{2}\Vert u_{m,\varepsilon } \Vert _{H_0^1} + \Vert j_{\varepsilon }'(u_{m,\varepsilon }) \Vert _{H^{-1}} + C \Vert u_{m,\varepsilon } \Vert _{L^2}^{\frac{4-N}{2}}\Vert u_{m,\varepsilon } \Vert _{H_0^1}^{\frac{N}{2}}+ \Vert f \Vert _{H^{-1}}. \end{aligned}$$

  Now, using the following convexity inequality

  $$\begin{aligned} \forall k \in {\mathbb {N}},\; \forall (x_i)_{1 \le i \le k} \in (0,+\infty )^k,\; \exists C > 0,\; \left( \sum _{i=1}^kx_i\right) ^{\frac{4}{N}} \le C\sum _{i=1}^k x_i^{\frac{4}{N}} \end{aligned}$$

  we get, after integrating in time an using the the embedding $L^2(\Omega ) \hookrightarrow L^{\frac{4}{N}}(\Omega )$ (which is valid since $N \in \{2,3\}$, so that we have $\frac{4}{N} \le 2$):

  $$\begin{aligned} \Vert \partial _tu_{m,\varepsilon } \Vert _{L^{\frac{4}{N}}((0,T),H^{-1})}^{\frac{4}{N}}&\; \le C\left( \Vert u_{m,\varepsilon } \Vert _{L^2((0,T),H_0^1)}^{\frac{4}{N}} + \Vert j_{\varepsilon }'(u_{m,\varepsilon }) \Vert _{L^{\frac{4}{N}}((0,T),H^{-1})}^{\frac{4}{N}}\right) \\&\quad + C \Vert u_{m,\varepsilon } \Vert _{L^{\infty }((0,T),L^2)}^{\frac{8-2N}{N}}\Vert u_{m,\varepsilon } \Vert _{L^2((0,T),H_0^1)}^{\frac{4}{N}}+ C\Vert f \Vert _{L^2((0,T),H^{-1})}^{\frac{4}{N}}. \end{aligned}$$

  Using the previously given convexity inequality and the first and second points of the proposition we obtain the desired result.

$\square$