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Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions

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Abstract

Existence and uniqueness of a solution to the nonstationary Navier–Stokes equations having a prescribed flow rate (flux) in the infinite cylinder \(\Pi =\{x=(x^\prime , x_n)\in {{\mathbb {R}}}^n:\; x^\prime \in \sigma \subset {{\mathbb {R}}}^{n-1},\; -\infty<x_n<\infty ,\, n=2,3\}\) are proved. It is assumed that the flow rate \(F\in L^2(0, T)\) and the initial data \(\mathbf{u}_0=\big (0,\ldots ,0, u_{0n}\big )\in L^2(\sigma )\). The nonstationary Poiseuille solution has the form \(\mathbf{u}(x,t)=\big (0,\ldots ,0, U(x^\prime , t)\big ), \; p(x,t)=-q(t)x_n+p_0(t)\), where \((U(x',t), q(t))\) is a solution of an inverse problem for the heat equation with a specific over-determination condition.

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Notes

  1. Since, by the incompressibility of the fluid, the total flux in a system of pipes has to be zero, even if we know the pressure slope in some of the pipes and can compute the corresponding fluxes, in at least one pipe we have to solve the problem with prescribed flow rate in order to satisfy zero flux condition.

  2. Since \(U\in L^2(0, T; L^2(\sigma ))\), \(S_U\) is a primitive of U, i.e., \((S_U)_t=U\), and we have the following inclusion \((S_U)_t\in L^2(0, T; L^2(\sigma ))\).

  3. The condition that \(U|_{\partial \sigma }=0\) is understood in the usual sense of traces (see [1], [10]). If \(S_U\in L^2(0,T; \mathring{W}^{1,2}(\sigma ))\), then \(S_U=\intop \limits _0^tU(\cdot ,\tau )\mathrm{d}\tau \in \mathring{W}^{1,2}(\sigma )\) for a.a. \(t\in (0,T)\) and \(\intop \limits _0^tU(x',\tau )\mathrm{d}\tau |_{\partial \sigma }=0\) in the sense of traces for such t. But then also \(U(x',t)_{\partial \sigma }=0\) in the sense of traces for a.a. \(t\in (0,T)\).

  4. These consideration does not prove the continuity of \(\Vert U(x,t)\Vert _{L^2(\sigma )}\) at \(t=0\). To have this property, we need F to be an element of \(W^{\varepsilon ,2}(0,T)\) with arbitrary \(\varepsilon >0\) (see Remark 5.1).

  5. Recall that for \(U\in L^2(0,T; L^2(\sigma ))\) the function \(S_U\) coincides with the primitive \(\intop \limits _0^t U(x', \tau )\mathrm{d}\tau \).

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Acknowledgements

The authors would like to thank referees for their comments which are helpful and constructive. The research of K. Pileckas was funded by the Grant No. S-MIP-17-68 from the Research Council of Lithuania.

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  1. Institute of Applied Mathematics, Vilnius University, Naugarduko Str. 24, Vilnius, Lithuania

    K. Pileckas

  2. Vilnius Gediminas Technical University, Saulėtekio av. 11, Vilnius, Lithuania

    R. Čiegis

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  1. K. Pileckas
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  2. R. Čiegis
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Pileckas, K., Čiegis, R. Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions. Z. Angew. Math. Phys. 71, 192 (2020). https://doi.org/10.1007/s00033-020-01422-5

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