Skip to main content
SpringerLink
Log in

Gradient Blow-Up for Dispersive and Dissipative Perturbations of the Burgers Equation

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order \(\alpha \), and the fractal Burgers equation of order \(\beta \), where \(\alpha , \beta \in [0,1)\), and the Whitham equation. For all \(\alpha , \beta \in [0,1)\), we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a “shock-like” singularity. Moreover, we provide an asymptotic description of the blow-up. To the best of our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range \(\alpha \in [2/3, 1)\), as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range \(\beta \in [2/3, 1)\). Our construction is based on modulation theory, where the well-known smooth self-similar solutions to the inviscid Burgers equation are used as profiles. A somewhat amusing point is that the profiles that are less stable under initial data perturbations (in that the number of unstable directions is larger) are more stable under perturbations of the equation (in that higher order dispersive and/or dissipative terms are allowed) due to their slower rates of concentration. Another innovation of this article, which may be of independent interest, is the development of a streamlined weighted \(L^{2}\)-based approach (in lieu of the characteristic method) for establishing the sharp spatial behavior of the solution in self-similar variables, which leads to the sharp Hölder regularity of the solution up to the blow-up time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during this study.

Notes

  1. Here, (s, y, U) are the self-similar variables for solutions defined for negatives times that blow up at \((t, x) = (0, 0)\).

  2. At this point, we note the interesting work of Collot–Ghoul–Masmoudi [10], which considered a two-dimensional partially dissipative perturbation of (Burgers) and constructed blow-up solutions with both ground and excited states as blow-up profiles. We refer to Sect. 1.2 for further discussion.

  3. It is important to distinguish the perturbations of the equations discussed here, which are terms of the form Lu for some linear operator L, with an external forcing term f, which is independent of u. The effect of an external forcing term with compact support in spacetime should resemble that of a compactly supported initial data perturbation.

  4. This property is equivalent to the requirement that the Fourier multiplier \(\Gamma \partial _{x} + \Upsilon \) maps real-valued functions to real-valued functions.

  5. More precisely, we will have boundedness of the gradient of these terms, and controlled growth for the terms themselves.

  6. For this purpose, we need to ensure that the coefficient \((2k)! + w_{2k+1}\) is uniformly bounded away from zero; this assertion will be one of the bootstrap assumptions below.

  7. Local well-posedness of the equation considered holds, for instance, in the space \(H^{2k+3}(\mathbb {R})\): if \(u_0 \in H^{2k+3}(\mathbb {R})\), there exists a local-in-time classical solution \(u(t,x)\in C^1([0,T], H^{2k+3}(\mathbb {R}))\) which solves the Eq. (1), and such that \(u(0,x) = u_0(x)\).

  8. The reason why we separate out the case is \(\max \{\alpha , \beta \} = \frac{1}{2k+1}\) is entirely technical; see Lemma 5.1 below. We note that \(\frac{2k-\frac{3}{2}}{2k}\) can be replaced by any positive number strictly less than \(\frac{2k-1}{2k}\).

  9. Recall that \(P_{\le 0}\) is the Fourier multiplier operator with symbol \(P_{\le 0}(\xi )\), where \(P_{\le 0}(\xi )\) is a nonnegative smooth function supported in \([-2, 2]\) which equals 1 on \([-1, 1]\). Moreover, \(P_{> 0}(\xi ) = 1 - P_{\le 0}(\xi )\).

  10. Recall that \({\tilde{P}}_k\) is a function whose support properties and bounds are the same as \(P_k\).

  11. Recall that \(\vert {\tilde{K}_{k}(y)}\vert \lesssim _{N} \frac{2^{k}}{\langle {2^{k} y}\rangle ^{N}}\).

  12. Note that, for the nonlinear ODE \(\dot{x} = -x - x^2\), the equilibrium point \(x = 0\) attracts all orbits originating in \((-1, \infty )\).

  13. This shows a lower bound. The upper bound comes from (68).

  14. Note that this choice can be made independently of \(\sigma _1\).

  15. This follows from bound (64) on \(E^{(1)}\).

  16. Recall display (25), where D and M are defined, and moreover recall that \({\mathcal {N}}(\vec {w}(s))\) is a vector with quadratic entries as functions of the entries of \(\vec {w}\), and \(\vec {f}\) is the vector \(((1+e^{s} \tau _{s}) F^{(2)}(s,0), \ldots , (1+e^{s} \tau _{s}) F^{(2k-1)}(s, 0))\).

  17. We use Lipschitz continuity here as it is easier to observe for (1), which is quasilinear. The minor price we have to pay is that we cannot work with the highest order topology \(H^{2k+3}\), but rather with the lower order topology \(H^{2k+2}\).

References

  1. Alibaud, N., Droniou, J., Vovelle, J.: Occurrence and non-appearance of shocks in fractal Burgers equations. J. Hyperbol. Differ. Equ. 4(3), 479–499, 2007

    Article  MathSciNet  Google Scholar 

  2. Buckmaster, T., Iyer, S.: Formation of unstable shocks for 2d isentropic compressible Euler. ArXiv preprint: arXiv:2007.15519 (2020)

  3. Buckmaster, T., Shkoller, S., Vicol, V.: Formation of point shocks for 3d compressible Euler. ArXiv preprint: arXiv:1912.04429 (2019)

  4. Buckmaster, T., Shkoller, S., Vicol, V.: Formation of shocks for 2d isentropic compressible Euler. ArXiv preprint: arXiv:1907.03784 (2019)

  5. Buckmaster, T., Shkoller, S., Vicol, V.: Shock formation and vorticity creation for 3d Euler. ArXiv preprint: arXiv:2006.14789 (2020)

  6. Castro, A., Córdoba, D., Gancedo, F.: Singularity formations for a surface wave model. Nonlinearity 23(11), 2835–2847, 2010

    Article  ADS  MathSciNet  Google Scholar 

  7. Chickering, K. R., Moreno-Vasquez, R. C., Pandya, G.: Asymptotically self-similar shock formation for 1d fractal Burgers equation. ArXiV Preprint: arXiv:2105.15128 (2021)

  8. Christodoulou, D.: The formation of shocks in 3-dimensional fluids. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2007)

  9. Christodoulou, D.: The shock development problem. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2019)

  10. Collot, C., Ghoul, T.-E., Masmoudi, N.: Singularity formation for burgers equation with transverse viscosity. ArXiv preprint: arXiv:1803.07826 (2018)

  11. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243, 1998

    Article  MathSciNet  Google Scholar 

  12. Constantin, P., Wu, J.: Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(6), 1103–1110, 2008

    Article  ADS  MathSciNet  Google Scholar 

  13. Côte, R., Martel, Y., Merle, F.: Construction of multi-soliton solutions for the \(L^2\)-supercritical gKdV and NLS equations. Rev. Mat. Iberoam. 27(1), 273–302, 2011

    Article  MathSciNet  Google Scholar 

  14. Dong, H., Du, D., Li, D.: Finite time singularities and global well-posedness for fractal Burgers equations. Indiana Univer. Math. J. 58(2), 807–821, 2009

    Article  MathSciNet  Google Scholar 

  15. Eggers, J., Fontelos, M.A.: Singularities: formation, structure, and propagation. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2015)

    Book  Google Scholar 

  16. Elgindi, T.: Finite-time singularity formation for \({\cal{C}}^{1,\alpha }\) solutions to the incompressible Euler equations on \({\mathbb{R}}^3\). ArXiv preprint: arXiv:1904.04795 (2019)

  17. Hur, V.M.: Wave breaking in the Whitham equation. Adv. Math. 317, 410–437, 2017

    Article  MathSciNet  Google Scholar 

  18. Hur, V.M., Tao, L.: Wave breaking for the Whitham equation with fractional dispersion. Nonlinearity 27(12), 2937–2949, 2014

    Article  ADS  MathSciNet  Google Scholar 

  19. Ibdah, H.: Lipschitz continuity of solutions to drift-diffusion equations in the presence of nonlocal terms. ArXiV Preprint: arXiv:2006.01859 (2020)

  20. Kiselev, A., Nazarov, F., Shterenberg, R.: Blow up and regularity for fractal Burgers equation. Dyn. Part. Differ. Equ. 5(3), 211–240, 2008

    Article  MathSciNet  Google Scholar 

  21. Klein, C., Linares, F., Pilod, D., Saut, J.-C.: On Whitham and related equations. Stud. Appl. Math. 140(2), 133–177, 2018

    Article  MathSciNet  Google Scholar 

  22. Klein, C., Saut, J.-C.: A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation. Phys. D 295(296), 46–65, 2015

    Article  MathSciNet  Google Scholar 

  23. Klein, C., Saut, J.-C., Wang, Y.: On the modified fractional Korteweg-de Vries and related equations. ArXiv preprint: arXiv:2010.05081 (2020)

  24. Linares, F., Pilod, D., Saut, J.-C.: Dispersive perturbations of Burgers and hyperbolic equations I: local theory. SIAM J. Math. Anal. 46(2), 1505–1537, 2014

    Article  MathSciNet  Google Scholar 

  25. Luk, J., Speck, J.: Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity. Invent. Math. 214(1), 1–169, 2018

    Article  ADS  MathSciNet  Google Scholar 

  26. Merle, F., Raphaël, P., Rodnianski, I., Szeftel, J.: On blow up for the energy super critical defocusing non linear Schrödinger equations. ArXiv preprint: arXiv:1912.11005 (2019)

  27. Merle, F., Raphaël, P., Rodnianski, I., Szeftel, J.: On smooth self similar solutions to the compressible Euler equations. ArXiv preprint: arXiv:1912.10998 (2019)

  28. Merle, F., Raphaël, P., Rodnianski, I., Szeftel, J.: On the implosion of a three dimensional compressible fluid. ArXiv preprint: arXiv:1912.11009 (2019)

  29. Molinet, L., Pilod, D., Vento, S.: On well-posedness for some dispersive perturbations of Burgers’ equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(7), 1719–1756, 2018

    Article  ADS  MathSciNet  Google Scholar 

  30. Naumkin, P. I., Shishmarëv, I. A.: Nonlinear Nonlocal Equations in the Theory of Waves, Volume 133 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1994). Translated from the Russian manuscript by Boris Gommerstadt.

  31. Rimah Said, A.: On the Cauchy problem of dispersive burgers type equations. ArXiv preprint: arXiv:2006.03803 (2021)

  32. Saut, J.-C., Wang, Y.: The wave breaking for Whitham-type equations revisited. ArXiv preprint: arXiv:2103.03588 (2020)

  33. Saut, J.-C., Wang, Y.: Global dynamics of small solutions to the modified fractional Korteweg–de Vries and fractional cubic nonlinear Schrödinger equations. Commun. Part. Differ. Equ. 46(10), 1851–1891, 2021

    Article  Google Scholar 

  34. Seliger, R.L.: A note on the breaking of waves. Proc. R. Soc. Lond. A 303(1475), 493–496, 1968

    Article  ADS  Google Scholar 

  35. Silvestre, L.: Hölder estimates for advection fractional-diffusion equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(4), 843–855, 2012

    MathSciNet  Google Scholar 

  36. Silvestre, L.: On the differentiability of the solution to an equation with drift and fractional diffusion. Indiana Univer. Math. J. 61(2), 557–584, 2012

    Article  MathSciNet  Google Scholar 

  37. Whitham, G.B.: Variational methods and applications to water waves. Proc. R. Soc. Lond. A 299(1456), 6–25, 1967

    Article  ADS  Google Scholar 

  38. Whitham, G. B.: Linear and nonlinear waves. Pure and Applied Mathematics (New York). Wiley, New York (1999). Reprint of the 1974 original, A Wiley-Interscience Publication.

  39. Yang, R.: Shock formation for the Burgers–Hilbert equation. ArXiv preprint: arXiv:2006.05568 (2020)

Download references

Acknowledgements

F. Pasqualotto would like to acknowledge Tristan Buckmaster and Javier Gómez-Serrano for insightful discussions on blow-up constructions. This material is based partially upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2021 semester. S.-J. Oh was partially supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1702-02, a Sloan Research Fellowship and a National Science Foundation CAREER Grant under NSF-DMS-1945615.

Author information

Authors and Affiliations

  1. Department of Mathematics, UC Berkeley and School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea

    Sung-Jin Oh

  2. Department of Mathematics, Duke University, Durham, NC, USA

    Federico Pasqualotto

Authors
  1. Sung-Jin Oh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federico Pasqualotto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Federico Pasqualotto.

Additional information

Communicated by N. Masmoudi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oh, SJ., Pasqualotto, F. Gradient Blow-Up for Dispersive and Dissipative Perturbations of the Burgers Equation. Arch Rational Mech Anal 248, 54 (2024). https://doi.org/10.1007/s00205-024-01985-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00205-024-01985-x

Search

Navigation