# American Institute of Mathematical Sciences

2018, 38(5): 2229-2249. doi: 10.3934/dcds.2018092

## Decaying turbulence for the fractional subcritical Burgers equation

 University of Lyon, CNRS UMR 5208, University Claude Bernard Lyon 1, Institut Camille Jordan, 43 Blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Received  December 2016 Revised  July 2017 Published  March 2018

We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting:
 $$$\nonumber\frac{\partial u}{\partial t}+(f(u))_x +ν Λ^{α} u = 0, t ≥ 0,\ \ \ \ {\bf{x}} ∈ {\mathbb{T}}^d = ({\mathbb{R}}/{\mathbb{Z}})^d.$$$
Here
 $f$
is strongly convex and satisfies a growth condition,
 $Λ = \sqrt{-Δ}, \ ν$
is small and positive, while
 $α ∈ (1,\ 2)$
is a constant in the subcritical range.
For solutions
 $u$
of this equation, we generalise the results obtained for the case
 $α = 2$
(i.e. when
 $-Λ^{α}$
is the Laplacian) in [12]. We obtain sharp estimates for the time-averaged Sobolev norms of
 $u$
as a function of
 $ν$
. These results yield sharp
 $ν$
-independent estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation
 $\ell$
in the physical space and the wavenumber
 $\bf{k}$
in the Fourier space.
The form of all estimates is the same as in the case
 $α = 2$
; the only thing which changes is that
 $ν$
is replaced by
 $ν^{1/(α-1)}$
.
Citation: Alexandre Boritchev. Decaying turbulence for the fractional subcritical Burgers equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2229-2249. doi: 10.3934/dcds.2018092
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975. [2] M. Alfaro and J. Droniou, General fractal conservation laws arising from a model of detonations in gases, Applied Mathematics Research eXpress, 2012 (2012), 127-151. [3] N. Alibaud, J. Droniou and J. Vovelle, Occurence and non-appearance of shocks in fractal Burgers equations, Journal of Hyperbolic Differential Equations, 4 (2007), 479-499. doi: 10.1142/S0219891607001227. [4] E. Aurell, U. Frisch, J. Lutsko and M. Vergassola, On the multifractal properties of the energy dissipation derived from turbulence data, Journal of Fluid Mechanics, 238 (1992), 467-486. doi: 10.1017/S0022112092001782. [5] C. Bardos, U. Frisch, W. Pauls, S. S. Ray and E. S. Titi, Entire solutions of hydrodynamical equations with exponential dissipation, Communications in Mathematical Physics, 293 (2010), 519-543. doi: 10.1007/s00220-009-0916-z. [6] J. Bec and K. Khanin, Burgers turbulence, Physics Reports, 447 (2007), 1-66. doi: 10.1016/j.physrep.2007.04.002. [7] P. Biler, T. Funaki and W. Woyczynski, Fractal Burgers equations, Journal of Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458. [8] A. Biryuk, Spectral properties of solutions of the Burgers equation with small dissipation, Functional Analysis and its Applications, 35 (2001), 1-12. doi: 10.1023/A:1004143415090. [9] A. Boritchev, Generalised Burgers Equation with Random Force and Small Viscosity, PhD thesis, Ecole Polytechnique, 2012. http://math.univ-lyon1.fr/homes-www/boritchev/Thesis.pdf. [10] A. Boritchev, Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation, Proceedings of the Royal Society of Edinburgh A, 143 (2013), 253-268. doi: 10.1017/S0308210511000989. [11] A. Boritchev, Sharp estimates for turbulence in white-forced generalised Burgers equation, Geometric and Functional Analysis, 23 (2013), 1730-1771. doi: 10.1007/s00039-013-0245-4. [12] A. Boritchev, Decaying turbulence in generalised Burgers equation, Archive for Rational Mechanics and Analysis, 214 (2014), 331-357. doi: 10.1007/s00205-014-0766-5. [13] A. Boritchev, Erratum to: Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 344 (2016), 369-370. doi: 10.1007/s00220-016-2621-z. [14] A. Boritchev, Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 342 (2016), 441-489. doi: 10.1007/s00220-015-2521-7. [15] Z. Brzezniak, L. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, Global and Stochastic Analysis, 1 (2011), 145-174. [16] J. M. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems, Reidel, 1974. doi: 10.1007/978-94-010-1745-9. [17] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations Ⅰ: Local theory, Inventiones Mathematicae, 173 (2008), 449-475. doi: 10.1007/s00222-008-0124-z. [18] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [19] C. H. Chan and M. Czubak, Regularity of solutions for the critical N-dimensional Burgers' equation, Annales de l'Institut Henri Poincare: Analyse non Lineaire, 27 (2010), 471-501. doi: 10.1016/j.anihpc.2009.11.008. [20] A. Chorin, Lectures on Turbulence Theory, volume 5 of Mathematics Lecture Series, Publish or Perish, 1975. [21] P. Clavin and B. Denet, Diamond patterns in the cellular front of an overdriven detonation Physical Review Letters, 88 (2002), 044502. doi: 10.1103/PhysRevLett.88.044502. [22] P. Constantin, A. Tarfulea and V. Vicol, Long time dynamics of forced critical SQG, Communications in Mathematical Physics, 335 (2015), 93-141. doi: 10.1007/s00220-014-2129-3. [23] P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geometric and Functional Analysis, 22 (2012), 1289-1321. doi: 10.1007/s00039-012-0172-9. [24] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Communications in Mathematical Physics, 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. [25] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 45 of Encyclopaedia of Mathematics and its Applications. Cambridge University Press, 1992. [26] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, volume 229 of London Mathematical Society Lecture Notes. Cambridge University Press, 1996. [27] M. Dabkowski, A. Kiselev, L. Silvestre and V. Vicol, Global well-posedness of slightly supercritical active scalar equations, Analysis & PDE, 7 (2014), 43-72. doi: 10.2140/apde.2014.7.43. [28] C. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics. Cambridge University Press, 1995. [29] H. J. Dong, D. Du and D. Li, Finite-time singularities and Global well-posedness for fractal Burgers' equations, Indiana University Mathematics Journal, 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505. [30] J. Droniou, T. Gallouët and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, Journal of Evolution Equations, 3 (2003), 499-521. doi: 10.1007/s00028-003-0503-1. [31] L. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. [32] J. D. Fournier and U. Frisch, L'équation de Burgers déterministe et stastistique, Journal de Mécanique Théorique et Appliquée, 2 (1983), 699-750. [33] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. [34] B. Jourdain, S. Méléard and W. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli, 11 (2005), 689-714. doi: 10.3150/bj/1126126765. [35] B. Jourdain and R. Roux, Convergence of a stochastic particle approximation for fractional scalar conservation laws, Stochastic Processes and their Applications, 121 (2011), 957-988. doi: 10.1016/j.spa.2011.01.012. [36] S. Kida, Asymptotic properties of Burgers turbulence, Journal of Fluid Mechanics, 93 (1979), 337-377. doi: 10.1017/S0022112079001932. [37] A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equations, Dynamics of PDE, 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2. [38] R. H. Kraichnan, Lagrangian-history statistical theory for Burgers' equation, Physics of Fluids, 11 (1968), 265-277. doi: 10.1063/1.1691900. [39] S.N. Kruzhkov, The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the first-order with several variables, Soviet Math. Doklady, 5 (1964), 493-496. [40] S. Kuksin, On turbulence in nonlinear Schrödinger equations, Geometric and Functional Analysis, 7 (1997), 783-822. doi: 10.1007/s000390050026. [41] S. Kuksin, Spectral properties of solutions for nonlinear PDEs in the turbulent regime, Geometric and Functional Analysis, 9 (1999), 141-184. doi: 10.1007/s000390050083. [42] A. Polyakov, Turbulence without pressure, Physical Review E, 52 (1995), 6183-6188. doi: 10.1103/PhysRevE.52.6183. [43] B. Protas and D. Yun, Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation, J. Nonlinear Sci., 28 (2018), 395-422, arXiv: 1610.09578 [44] M. Taylor, Partial Differential Equations Ⅰ: Basic Theory, volume 115 of Applied Mathematical Sciences. Springer, 1996. [45] A. Truman and J.-L. Wu, On a stochastic nonlinear equation arising from 1d integro-differential scalar conservation laws, Journal of Functional Analysis, 238 (2006), 612-635. doi: 10.1016/j.jfa.2006.01.012.

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975. [2] M. Alfaro and J. Droniou, General fractal conservation laws arising from a model of detonations in gases, Applied Mathematics Research eXpress, 2012 (2012), 127-151. [3] N. Alibaud, J. Droniou and J. Vovelle, Occurence and non-appearance of shocks in fractal Burgers equations, Journal of Hyperbolic Differential Equations, 4 (2007), 479-499. doi: 10.1142/S0219891607001227. [4] E. Aurell, U. Frisch, J. Lutsko and M. Vergassola, On the multifractal properties of the energy dissipation derived from turbulence data, Journal of Fluid Mechanics, 238 (1992), 467-486. doi: 10.1017/S0022112092001782. [5] C. Bardos, U. Frisch, W. Pauls, S. S. Ray and E. S. Titi, Entire solutions of hydrodynamical equations with exponential dissipation, Communications in Mathematical Physics, 293 (2010), 519-543. doi: 10.1007/s00220-009-0916-z. [6] J. Bec and K. Khanin, Burgers turbulence, Physics Reports, 447 (2007), 1-66. doi: 10.1016/j.physrep.2007.04.002. [7] P. Biler, T. Funaki and W. Woyczynski, Fractal Burgers equations, Journal of Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458. [8] A. Biryuk, Spectral properties of solutions of the Burgers equation with small dissipation, Functional Analysis and its Applications, 35 (2001), 1-12. doi: 10.1023/A:1004143415090. [9] A. Boritchev, Generalised Burgers Equation with Random Force and Small Viscosity, PhD thesis, Ecole Polytechnique, 2012. http://math.univ-lyon1.fr/homes-www/boritchev/Thesis.pdf. [10] A. Boritchev, Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation, Proceedings of the Royal Society of Edinburgh A, 143 (2013), 253-268. doi: 10.1017/S0308210511000989. [11] A. Boritchev, Sharp estimates for turbulence in white-forced generalised Burgers equation, Geometric and Functional Analysis, 23 (2013), 1730-1771. doi: 10.1007/s00039-013-0245-4. [12] A. Boritchev, Decaying turbulence in generalised Burgers equation, Archive for Rational Mechanics and Analysis, 214 (2014), 331-357. doi: 10.1007/s00205-014-0766-5. [13] A. Boritchev, Erratum to: Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 344 (2016), 369-370. doi: 10.1007/s00220-016-2621-z. [14] A. Boritchev, Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 342 (2016), 441-489. doi: 10.1007/s00220-015-2521-7. [15] Z. Brzezniak, L. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, Global and Stochastic Analysis, 1 (2011), 145-174. [16] J. M. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems, Reidel, 1974. doi: 10.1007/978-94-010-1745-9. [17] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations Ⅰ: Local theory, Inventiones Mathematicae, 173 (2008), 449-475. doi: 10.1007/s00222-008-0124-z. [18] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [19] C. H. Chan and M. Czubak, Regularity of solutions for the critical N-dimensional Burgers' equation, Annales de l'Institut Henri Poincare: Analyse non Lineaire, 27 (2010), 471-501. doi: 10.1016/j.anihpc.2009.11.008. [20] A. Chorin, Lectures on Turbulence Theory, volume 5 of Mathematics Lecture Series, Publish or Perish, 1975. [21] P. Clavin and B. Denet, Diamond patterns in the cellular front of an overdriven detonation Physical Review Letters, 88 (2002), 044502. doi: 10.1103/PhysRevLett.88.044502. [22] P. Constantin, A. Tarfulea and V. Vicol, Long time dynamics of forced critical SQG, Communications in Mathematical Physics, 335 (2015), 93-141. doi: 10.1007/s00220-014-2129-3. [23] P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geometric and Functional Analysis, 22 (2012), 1289-1321. doi: 10.1007/s00039-012-0172-9. [24] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Communications in Mathematical Physics, 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. [25] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 45 of Encyclopaedia of Mathematics and its Applications. Cambridge University Press, 1992. [26] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, volume 229 of London Mathematical Society Lecture Notes. Cambridge University Press, 1996. [27] M. Dabkowski, A. Kiselev, L. Silvestre and V. Vicol, Global well-posedness of slightly supercritical active scalar equations, Analysis & PDE, 7 (2014), 43-72. doi: 10.2140/apde.2014.7.43. [28] C. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics. Cambridge University Press, 1995. [29] H. J. Dong, D. Du and D. Li, Finite-time singularities and Global well-posedness for fractal Burgers' equations, Indiana University Mathematics Journal, 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505. [30] J. Droniou, T. Gallouët and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, Journal of Evolution Equations, 3 (2003), 499-521. doi: 10.1007/s00028-003-0503-1. [31] L. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. [32] J. D. Fournier and U. Frisch, L'équation de Burgers déterministe et stastistique, Journal de Mécanique Théorique et Appliquée, 2 (1983), 699-750. [33] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. [34] B. Jourdain, S. Méléard and W. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli, 11 (2005), 689-714. doi: 10.3150/bj/1126126765. [35] B. Jourdain and R. Roux, Convergence of a stochastic particle approximation for fractional scalar conservation laws, Stochastic Processes and their Applications, 121 (2011), 957-988. doi: 10.1016/j.spa.2011.01.012. [36] S. Kida, Asymptotic properties of Burgers turbulence, Journal of Fluid Mechanics, 93 (1979), 337-377. doi: 10.1017/S0022112079001932. [37] A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equations, Dynamics of PDE, 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2. [38] R. H. Kraichnan, Lagrangian-history statistical theory for Burgers' equation, Physics of Fluids, 11 (1968), 265-277. doi: 10.1063/1.1691900. [39] S.N. Kruzhkov, The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the first-order with several variables, Soviet Math. Doklady, 5 (1964), 493-496. [40] S. Kuksin, On turbulence in nonlinear Schrödinger equations, Geometric and Functional Analysis, 7 (1997), 783-822. doi: 10.1007/s000390050026. [41] S. Kuksin, Spectral properties of solutions for nonlinear PDEs in the turbulent regime, Geometric and Functional Analysis, 9 (1999), 141-184. doi: 10.1007/s000390050083. [42] A. Polyakov, Turbulence without pressure, Physical Review E, 52 (1995), 6183-6188. doi: 10.1103/PhysRevE.52.6183. [43] B. Protas and D. Yun, Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation, J. Nonlinear Sci., 28 (2018), 395-422, arXiv: 1610.09578 [44] M. Taylor, Partial Differential Equations Ⅰ: Basic Theory, volume 115 of Applied Mathematical Sciences. Springer, 1996. [45] A. Truman and J.-L. Wu, On a stochastic nonlinear equation arising from 1d integro-differential scalar conservation laws, Journal of Functional Analysis, 238 (2006), 612-635. doi: 10.1016/j.jfa.2006.01.012.
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