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Local Existence of Solutions to a Navier–Stokes-Nonlinear-Schrödinger Model of Superfluidity

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Abstract

In Pitaevskii (Sov Phys JETP 35(8):282–287, 1959), a micro-scale model of superfluidity was derived from first principles, to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model couples two of the most fundamental PDEs in mathematics: the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations (NSE). In this article, we show the local existence of solutions—strong in wavefunction and velocity, weak in density—to this system in a smooth bounded domain in 3D, by deriving the required a priori estimates. (We will also establish an energy inequality obeyed by the weak solutions constructed in Kim (SIAM J Math Anal 18(1):89–96, 1987) for the incompressible, inhomogeneous NSE.) To the best of our knowledge, this is the first rigorous mathematical analysis of a bidirectionally coupled system of the NLS and NSE.

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Notes

  1. Parallels between superfluidity and superconductivity had been drawn for quite some time: the quantized vortex filaments in the former were analogous to the quantized magnetic flux tubes in the latter, and both phenomena were characterized as order-disorder transitions. Furthermore, following the success of the BCS theory of superconductivity, it became clear that the same explanation (Cooper pairing) can be extended to the superfluidity of the fermionic He-3.

  2. Interestingly, this formulation is used in David Bohm’s pilot wave theory, a deterministic yet complicated interpretation of quantum mechanics. This posits that a pilot wave (whose dynamics are governed by QHD) guides quantum particles in a classical manner, at odds with other descriptions, like the inherently random Copenhagen interpretation or the fantastical multiverse theories.

  3. For a justification of the exclusion of \(t=0\) in the boundary conditions for the wavefunction, see Remark 2.5.

  4. \(\mu >0\) (resp. \(\mu <0\)) is called the defocusing (resp. focusing) NLS.

  5. There is also the cubic nonlinearity term, which is to say that the relaxation to equilibrium also depends on the potential energy of the superfluid.

  6. See Remark .

  7. Of course, the local existence time depends on the choice of \(\varepsilon \) and should ideally be written as \(T_{\varepsilon }\). However, we will assume that the value of \(\varepsilon \) is fixed throughout this article, and for brevity, drop the subscript.

  8. Both the normal and tangential derivatives of \(\psi \) are zero on the boundary, the latter because \(\psi \) is zero on a smooth boundary.

  9. This trick will be used again for deriving the higher-order a priori estimates.

  10. Recall that \(\gamma = \mu ^2 \left( \Lambda + \frac{1}{\Lambda } \right) \).

  11. \(\left\Vert fg\right\Vert _{H^r}\lesssim \left\Vert f\right\Vert _{H^r}\left\Vert g\right\Vert _{H^r}\) for \(r>\frac{d}{2}\) in d dimensions.

  12. Compared to the Pitaevskii model, there is one extra vanishing derivative on the boundary for these eigenfunctions. This is to ensure some that the a priori estimates work out. See the handling of the \(B\psi \) term in Sect. 3.4.

  13. This is the same as the local existence time defined earlier, due to the a priori estimates. The latter guarantee that as long as the density is bounded below, the energy of the system is bounded above, implying that the coefficients of time dependence are bounded.

  14. Refer to Sect. 1.1 for the notation used in the case of Sobolev spaces of the x-variable.

  15. It should be mentioned here that T is finite, depending only on the initial data and size of the domain.

  16. It is worth noting that the obstacle to an energy equality in the work by Kim was the lack of strong convergence of the dissipative term; yet again, this boils down to the fact that there is no uniform bound on \(\partial _t u\) (since the density is not bounded below).

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Acknowledgements

The authors appreciate the discussions with Yan Guo that helped correct some minor errors. Both authors would like to thank the anonymous referees for their suggestions which helped improve the original manuscript. P.C.J. was partially supported by the Ann Wylie Fellowship at UMD. Both P.C.J. and K.T. gratefully acknowledge the support of the National Science Foundation under the awards DMS-1614964 and DMS-2008568.

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  1. Department of Physics, University of Maryland, College Park, MD, 20742, USA

    Pranava Chaitanya Jayanti

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Konstantina Trivisa

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Correspondence to Pranava Chaitanya Jayanti.

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Jayanti, P.C., Trivisa, K. Local Existence of Solutions to a Navier–Stokes-Nonlinear-Schrödinger Model of Superfluidity. J. Math. Fluid Mech. 24, 46 (2022). https://doi.org/10.1007/s00021-022-00681-8

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