May  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:
$\begin{equation} \nonumber\frac{\partial u}{\partial t}+(f(u))_x +ν Λ^{α} u = 0, t ≥ 0,\ \ \ \ {\bf{x}} ∈ {\mathbb{T}}^d = ({\mathbb{R}}/{\mathbb{Z}})^d.\end{equation}$
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. Google Scholar

[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. Google Scholar

[3]

N. AlibaudJ. 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. Google Scholar

[4]

E. AurellU. FrischJ. 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. Google Scholar

[5]

C. BardosU. FrischW. PaulsS. 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. Google Scholar

[6]

J. Bec and K. Khanin, Burgers turbulence, Physics Reports, 447 (2007), 1-66. doi: 10.1016/j.physrep.2007.04.002. Google Scholar

[7]

P. BilerT. Funaki and W. Woyczynski, Fractal Burgers equations, Journal of Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

A. Boritchev, Erratum to: Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 344 (2016), 369-370. doi: 10.1007/s00220-016-2621-z. Google Scholar

[14]

A. Boritchev, Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 342 (2016), 441-489. doi: 10.1007/s00220-015-2521-7. Google Scholar

[15]

Z. BrzezniakL. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, Global and Stochastic Analysis, 1 (2011), 145-174. Google Scholar

[16]

J. M. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems, Reidel, 1974. doi: 10.1007/978-94-010-1745-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

A. Chorin, Lectures on Turbulence Theory, volume 5 of Mathematics Lecture Series, Publish or Perish, 1975. Google Scholar

[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. Google Scholar

[22]

P. ConstantinA. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, volume 229 of London Mathematical Society Lecture Notes. Cambridge University Press, 1996. Google Scholar

[27]

M. DabkowskiA. KiselevL. 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. Google Scholar

[28]

C. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics. Cambridge University Press, 1995. Google Scholar

[29]

H. J. DongD. 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. Google Scholar

[30]

J. DroniouT. 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. Google Scholar

[31]

L. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. Google Scholar

[32]

J. D. Fournier and U. Frisch, L'équation de Burgers déterministe et stastistique, Journal de Mécanique Théorique et Appliquée, 2 (1983), 699-750. Google Scholar

[33]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. Google Scholar

[34]

B. JourdainS. 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. Google Scholar

[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. Google Scholar

[36]

S. Kida, Asymptotic properties of Burgers turbulence, Journal of Fluid Mechanics, 93 (1979), 337-377. doi: 10.1017/S0022112079001932. Google Scholar

[37]

A. KiselevF. 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. Google Scholar

[38]

R. H. Kraichnan, Lagrangian-history statistical theory for Burgers' equation, Physics of Fluids, 11 (1968), 265-277. doi: 10.1063/1.1691900. Google Scholar

[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. Google Scholar

[40]

S. Kuksin, On turbulence in nonlinear Schrödinger equations, Geometric and Functional Analysis, 7 (1997), 783-822. doi: 10.1007/s000390050026. Google Scholar

[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. Google Scholar

[42]

A. Polyakov, Turbulence without pressure, Physical Review E, 52 (1995), 6183-6188. doi: 10.1103/PhysRevE.52.6183. Google Scholar

[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 Google Scholar

[44]

M. Taylor, Partial Differential Equations Ⅰ: Basic Theory, volume 115 of Applied Mathematical Sciences. Springer, 1996. Google Scholar

[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. Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, 1975. Google Scholar

[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. Google Scholar

[3]

N. AlibaudJ. 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. Google Scholar

[4]

E. AurellU. FrischJ. 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. Google Scholar

[5]

C. BardosU. FrischW. PaulsS. 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. Google Scholar

[6]

J. Bec and K. Khanin, Burgers turbulence, Physics Reports, 447 (2007), 1-66. doi: 10.1016/j.physrep.2007.04.002. Google Scholar

[7]

P. BilerT. Funaki and W. Woyczynski, Fractal Burgers equations, Journal of Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

A. Boritchev, Erratum to: Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 344 (2016), 369-370. doi: 10.1007/s00220-016-2621-z. Google Scholar

[14]

A. Boritchev, Multidimensional potential Burgers turbulence, Communications in Mathematical Physics, 342 (2016), 441-489. doi: 10.1007/s00220-015-2521-7. Google Scholar

[15]

Z. BrzezniakL. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, Global and Stochastic Analysis, 1 (2011), 145-174. Google Scholar

[16]

J. M. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems, Reidel, 1974. doi: 10.1007/978-94-010-1745-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

A. Chorin, Lectures on Turbulence Theory, volume 5 of Mathematics Lecture Series, Publish or Perish, 1975. Google Scholar

[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. Google Scholar

[22]

P. ConstantinA. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, volume 229 of London Mathematical Society Lecture Notes. Cambridge University Press, 1996. Google Scholar

[27]

M. DabkowskiA. KiselevL. 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. Google Scholar

[28]

C. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics. Cambridge University Press, 1995. Google Scholar

[29]

H. J. DongD. 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. Google Scholar

[30]

J. DroniouT. 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. Google Scholar

[31]

L. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. Google Scholar

[32]

J. D. Fournier and U. Frisch, L'équation de Burgers déterministe et stastistique, Journal de Mécanique Théorique et Appliquée, 2 (1983), 699-750. Google Scholar

[33]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. Google Scholar

[34]

B. JourdainS. 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. Google Scholar

[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. Google Scholar

[36]

S. Kida, Asymptotic properties of Burgers turbulence, Journal of Fluid Mechanics, 93 (1979), 337-377. doi: 10.1017/S0022112079001932. Google Scholar

[37]

A. KiselevF. 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. Google Scholar

[38]

R. H. Kraichnan, Lagrangian-history statistical theory for Burgers' equation, Physics of Fluids, 11 (1968), 265-277. doi: 10.1063/1.1691900. Google Scholar

[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. Google Scholar

[40]

S. Kuksin, On turbulence in nonlinear Schrödinger equations, Geometric and Functional Analysis, 7 (1997), 783-822. doi: 10.1007/s000390050026. Google Scholar

[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. Google Scholar

[42]

A. Polyakov, Turbulence without pressure, Physical Review E, 52 (1995), 6183-6188. doi: 10.1103/PhysRevE.52.6183. Google Scholar

[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 Google Scholar

[44]

M. Taylor, Partial Differential Equations Ⅰ: Basic Theory, volume 115 of Applied Mathematical Sciences. Springer, 1996. Google Scholar

[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. Google Scholar

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