Existence of a local strong solution to the beam–polymeric fluid interaction system

Definition 2.2. (Weak solution)Let $$(\mathbf {f}, g, \eta _0, \eta _\star, \mathbf {u}_0, \widehat{f}_0)$$ be a dataset such that

• (a) the shell displacement $$\eta$$ satisfies $$\Vert \eta \Vert _{L^\infty (I \times \omega)} <L$$ and
  $$$$\begin{align*} \eta \in W^{1,\infty } {\left(I; L^{2}(\omega) \right)}\cap L^\infty {\left(I; W^{2,2}(\omega) \right)} \cap W^{1,2}{\left(I;W^{1,2}(\omega) \right)}; \end{align*}$$$$
• (b) the velocity $$\mathbf {u}$$ is such that $$\mathbf {u}\circ \bm {\varphi }_{\eta } =(\partial _t\eta)\mathbf {n}$$ on $$I\times \omega$$ and
  $$$$\begin{align*} \mathbf {u}\in L^{\infty } {\left(I; L^2_{\mathrm{div}_{\mathbf {x}}}(\Omega _{\eta (t)}) \right)}\cap L^2 {\left(I; W^{1,2}(\Omega _{\eta (t)} \right)}; \end{align*}$$$$
• (c) the probability density function $$ \widehat{f}$$ satisfies
  $$$$\begin{align*} \widehat{f} \in L^\infty {\left(I; L^2(\Omega _{\eta (t)}; L^2_M(B))\right)} \cap L^2 {\left(I; W^{1,2}(\Omega _{\eta (t)}; L^2_M(B))\right)} \cap L^2 {\left(I; L^2(\Omega _{\eta (t)}; H^1_M(B))\right)}; \end{align*}$$$$
• (d) for all $$(\phi, \bm {\phi }, \varphi) \in C^\infty (\overline{I}\times \omega) \times C^\infty (\overline{I}; C^\infty _{\mathrm{div}_{\mathbf {x}}}(\mathbb {R}^3)) \times C^\infty (\overline{I}\times \mathbb {R}^3 \times \overline{B})$$ with $$\phi (T,\cdot)=0$$, $$\bm {\phi }(T,\cdot)=0$$ and $$\bm {\phi }\circ \bm {\varphi }_{\eta }= \phi \mathbf {n}$$, we have
  $$$$\begin{align*} & \int _I \frac{d}{\, {d}t}{\left(\int _\omega \partial _t \eta \, \phi \, {d}\mathbf {y}+ \int _{\Omega _{\eta (t)}} \mathbf {u}\cdot \bm {\phi }\, {d} \mathbf {x}+ \int _{\Omega _{\eta (t)} \times B}M \widehat{f} \, \varphi \, {d} \mathbf {q}\, {d} \mathbf {x}\right)}\, {d}t\\ &\quad = \int _I \int _\omega {\left(\partial _t \eta \, \partial _t\phi - \partial _t \nabla _{\mathbf {y}}\eta \cdot \nabla _{\mathbf {y}}\phi + g\, \phi - \Delta _{\mathbf {y}}\eta \, \Delta _{\mathbf {y}}\phi \right)}\, {d}\mathbf {y}\, {d}t\\ &\qquad + \int _I \int _{\Omega _{\eta (t)}}{\left(\mathbf {u}\cdot \partial _t \bm {\phi } + \mathbf {u}\otimes \mathbf {u}: \nabla _{\mathbf {x}}\bm {\phi } \right)} \, {d} \mathbf {x}\, {d}t\\ &\qquad -\int _I \int _{\Omega _{\eta (t)}}{\left(\nabla _{\mathbf {x}}\mathbf {u}:\nabla _{\mathbf {x}}\bm {\phi } + \mathbb {S}_\mathbf {q}(\widehat{f}) :\nabla _{\mathbf {x}}\bm {\phi }-\mathbf {f}\cdot \bm {\phi } \right)} \, {d} \mathbf {x}\, {d}t\\ &\qquad +\int _I\int _{\Omega _{\eta (t)} \times B}{\left(M \widehat{f} \,\partial _t \varphi + M\mathbf {u}\widehat{f} \cdot \nabla _{\mathbf {x}}\varphi \right)} \, {d} \mathbf {q}\, {d} \mathbf {x}\, {d}t\\ &\qquad + \int _I\int _{ \Omega _{\eta (t)} \times B} {\left(M(\nabla _{\mathbf {x}}\mathbf {u}) \mathbf {q}\widehat{f}- M \nabla _{\mathbf {q}}\widehat{f} \right)} \cdot \nabla _{\mathbf {q}}\varphi \, {d} \mathbf {q}\, {d} \mathbf {x}\, {d}t, \end{align*}$$$$
  with $$\eta (0)=\eta _0$$ and $$\partial _t\eta =\eta _\star$$ a.e. in $$\omega$$, $$\mathbf {u}(0)=\mathbf {u}_0$$ a.e. in $$\Omega _{\eta _0}$$ as well as $$\widehat{f}(0)=\widehat{f}_0$$ a.e. in $$\Omega _{\eta _0}\times B$$.

Theorem 2.4.Let $$(\mathbf {f}, g, \eta _0, \eta _\star, \mathbf {u}_0, \widehat{f}_0)$$ be a dataset satisfying (2.7)–(2.10). There is a time $$T_*>0$$ such that a unique strong solution $$(\eta, \mathbf {u}, \pi, \widehat{f})$$ of (1.1)–(1.4), in the sense of Definition 2.3, exists.

Remark 2.5.We note that the Fokker–Planck equation is conservative and its solution is an actual probability density function (meaning that $$\widehat{f}\ge 0$$) within the flexible geometry under consideration. More precisely, we note that if we integrate (1.4) over $$\Omega _{\eta (t)}\times B$$ and use (1.12)–(1.14) together with Reynold's transport theorem, we obtain

Remark 2.6.The reason for the choice of the topology in Definition 2.3 comes from the coupling of the Navier–Stokes equations and the Fokker–Planck equation with center-of-mass diffusion in a bounded domain: the velocity field must be Lipschitz in space, we cannot allow more than two spatial derivatives for the probability density function (this is related to (1.13) and (1.14)). Moreover, we need additional temporal regularity to close the fixed point argument for the fully coupled system in Section 5. As we have explained in Section 1.4 these difficulties are not related to the moving boundary and our result is even new for fixed boundaries (referring to $$\eta _0=\eta _\star =g=0$$ in (1.1) and (1.9)).

Remark 2.7.It remains open if a result similar to Theorem 2.4 holds if the Fokker–Planck equation without center-of-mass diffusion, that is $$\varepsilon =0$$ in (1.4), is considered. This is related to the nontrivial boundary conditions for the fluid as well as the moving domain. For the fixed point argument for the fully coupled system in Section 5, we need to prove a stability estimate for two different solutions of (1.4) being posed in two different moving domains. This requires to transform them to the reference domain, which eventually creates several boundary terms. They can only be controlled with the help of the additional regularity coming from the center-of-mass diffusion.