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On singularities in the quaternionic Burgers equation

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Abstract

We consider the equation \(q_t+qq_x=q_{xx}\) for \(q:{{\mathbf {R}}}\times (0,\infty )\rightarrow {\mathbf {H}}\) (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. Similar results are also established for the same equation in \(\mathbf{S}^1\times (0,\infty )\), where \(\mathbf{S}^1\) is the standard one-dimensional circle.

Résumé

L’équation \(q_t+qq_x=q_{xx}\) pour \(q:{{\mathbf {R}}}\times (0,\infty )\rightarrow {\mathbf {H}}\) (les quaternions) est considérée. Nous montrons que bien que des singularités peuvent se développer en temps fini à partir de données initiales lisses à support compact, cette situation n’est pas générique. Les singularités disparaissent après une perturbation générique lisse arbitrairement petite de la donnée initiale. Des résultats similaires sont également établis pour la même équation dans \(S^1\times (0,\infty )\), où \(S^1\) est le cercle unidimensionnel.

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Notes

  1. The model certainly loses predictive power at the level of low-regularity weak solutions. In the inviscid case this was first shown by Scheffer [17]. Schnirelman [18] studied this phenomenon from a new angle. In recent years, applications of convex integration lead to further important developments, such as [2, 5, 8].

  2. One of the many options is to use (a subset of) techniques developed in a 1934 paper by Leray [12] for the Navier–Stokes equation.

  3. In this case we can take for example \(X=L^1(\mathbf{S}^1,{\mathbf {H}})\), or the smooth functions on \(\mathbf{S}^1\).

  4. Our method for establishing this will use the special structure of the equation, it will not be based on finding a good functional-analytic setup for applying a standard fixed-point theorem by establishing a contraction property of the Picard iteration. It might be interesting to see what is the lowest regularity threshold for a proof via Picard interation.

  5. This will again be seen from the special structure of the equation and not via more general methods by which such statements are usually proved.

  6. See, for example, [14].

  7. Of course, for a given smooth function some non-effective bounds are implied by the fact that a continuous function on a compact set is bounded.

  8. Here and below we will slightly abuse notation by writing \(L^1({{\mathbf {R}}})\) also for vector valued functions (such as \(L^1({{\mathbf {R}}},{\mathbf {H}})\), for example.

  9. The corresponding linearized equation at a constant steady state \(\alpha \in {\mathbf {H}}\) is given by \(q_t+\alpha q_x=q_{xx}\). The operator \(q\rightarrow q_{xx}-\alpha q_x\) always has an eigenvalue \(\lambda =0\) (with the eigenvectors being constant functions), corresponding to the shift of the constant solution to another constant \(\alpha '\). By a simple change of variables \(q\rightarrow \beta q\beta ^{-1}\) we can assume that \(\alpha \) is complex. The stability condition that—with the exception of the modes accounted for by the constant solutions—the spectrum of the linearized operator is in \(\{\lambda \,,\,{\text {Re}}\lambda < 0\}\) is easily seen to be \(|\alpha -{{\bar{\alpha }}}|<2\). This is satisfied for the constant \(-2\beta ^{-1}(ik_0+a)\beta \) in (4.24) in the generic case.

  10. The linearized operator has a non-trivial kernel, corresponding to the tangent space to the manifold of the constant steady states. The rest of the spectrul lies in \(\{\lambda , {\text {Re}}\lambda <0\}\).

References

  1. van den Berg, J., B., Williams, J. F., (In-)stability of singular equivariant solutions to the Landau-Lifshitz-Gilbert equation. European J. Appl. Math. 24 (2013), no. 6, 921–948.

  2. Buckmaster, T., Vicol, V., Convex integration constructions in hydrodynamics. Bull. Amer. Math. Soc. 58 (2021), no. 1, 1–44.

  3. Cole, J., On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics. 9 (3): 225–236.

  4. Caffarelli, L., Kohn, R., Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831.

  5. De Lellis, C., Szekelyhidi, L. Jr., The h-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 3, 347–375.

  6. Guillod, J., Sverak, V., Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces, arXiv:1704.00560

  7. Hopf, E., The partial differential equation \(u_t+uu_x=\mu u_{xx}\). Comm. Pure Appl. Math. 3 (1950), 201–230.

  8. Isett, P., A proof of Onsager’s conjecture. Ann. of Math. (2) 188 (2018), no. 3, 871–963.

  9. Jia, H., Sverak, V., Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. Math. 196 (2014), no. 1, 233–265.

  10. Jia, H., Sverak, V., Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space? J. Funct. Anal. 268 (2015), no. 12, 3734–3766.

  11. Ladyzhenskaya, O. A., The sixth millennium problem: Navier-Stokes equations, existence and smoothness. Uspekhi Mat. Nauk 58 (2003), no. 2(350), 45–78; translation in Russian Math. Surveys 58 (2003), no. 2, 251–286.

  12. Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248.

  13. Lin, F., A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51 (1998), no. 3, 241–257.

  14. Plechac, P., Sverak, V., On self-similar singular solutions of the complex Ginzburg-Landau equation. Comm. Pure Appl. Math. 54 (2001), no. 10, 1215–1242.

  15. Polacik, P., Sverak, V., Zeros of complex caloric functions and singularities of complex viscous Burgers equation. J. Reine Angew. Math. 616 (2008), 205–217.

  16. Scheffer, V., Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66 (1976), no. 2, 535–552.

  17. Scheffer, V., An inviscid flow with compact support in space-time. J. Geom. Anal. 3 (1993), no. 4, 343–401.

  18. Shnirelman, A., On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. 50 (1997), no. 12, 1261–1286.

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Acknowledgements

The research of the author was supported in part by grant DMS 1956092 from the National Science Foundation.

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  1. University of Minnesota, Minneapolis, USA

    Vladimir Sverak

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Correspondence to Vladimir Sverak.

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Dedicated to Professor Alexander Shnirelman on the occasion of his 75th birthday.

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Sverak, V. On singularities in the quaternionic Burgers equation. Ann. Math. Québec 46, 41–54 (2022). https://doi.org/10.1007/s40316-021-00175-5

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