doi: 10.3934/dcdsb.2019215

Existence and blow up of solutions to the $ 2D $ Burgers equation with supercritical dissipation

1. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematical Sciences and Institute of Natural Science, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: binbinshi@sjtu.edu.cn(Binbin Shi)

Received  December 2018 Revised  May 2019 Published  September 2019

This paper is concerned with the Cauchy problem for a fractal Burgers equation in two dimensions. When $ \alpha\in (0, 1) $, the same problem has been studied in one dimensions, we can refer to [1, 17, 24]. In this paper, we study well-posedness of solutions to the Burgers equation with supercritical dissipation. We prove the local existence with large initial data and global existence with small initial data in critical Besov space by energy method. Furthermore, we show that solutions can blow up in finite time if initial data is not small by contradiction method.

Citation: Binbin Shi, Weike Wang. Existence and blow up of solutions to the $ 2D $ Burgers equation with supercritical dissipation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019215
References:
[1]

N. AlibaudJ. Droniou and J. Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ., 4 (2007), 479-499. doi: 10.1142/S0219891607001227. Google Scholar

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

[3]

J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. Google Scholar

[4]

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

[5]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0. Google Scholar

[6]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935. doi: 10.1088/0951-7715/23/4/009. Google Scholar

[7]

L. Brandolese and G. Karch, Far field asymptotics of solutions to convection equation with anomalous diffusion, J. Evol. Equ., 8 (2008), 307-326. doi: 10.1007/s00028-008-0356-9. Google Scholar

[8]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903–1930. doi: 10.4007/annals.2010.171.1903. Google Scholar

[9]

C. H. Chan and M. Czubak, Regularity of solutions for the critical $N$-dimensional Burgers' equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 471-501. doi: 10.1016/j.anihpc.2009.11.008. Google Scholar

[10]

J.-Y. Chemin, Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Google Scholar

[11]

Q. ChenC. Miao and Z. Zhang, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271 (2007), 821-838. doi: 10.1007/s00220-007-0193-7. Google Scholar

[12]

P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), 97–108, https://doi.org/10.1512/iumj.2008.57.3629, Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). doi: 10.1512/iumj.2001.50.2153. Google Scholar

[13]

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321. doi: 10.1007/s00039-012-0172-9. Google Scholar

[14]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333. Google Scholar

[15]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. Google Scholar

[16]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X. Google Scholar

[17]

H. DongD. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J., 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505. Google Scholar

[18]

J. Droniou, T. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499–521, Dedicated to Philippe Bénilan. doi: 10.1007/s00028-003-0503-1. Google Scholar

[19]

R. Granero-Belinchón, On a drift-diffusion system for semiconductor devices, Ann. Henri Poincaré, 17 (2016), 3473-3498. doi: 10.1007/s00023-016-0493-6. Google Scholar

[20]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161-181. doi: 10.1007/s00220-004-1256-7. Google Scholar

[21]

G. Karch, Nonlinear evolution equations with anomalous diffusion, in Qualitative Properties of Solutions to Partial Differential Equations, vol. 5 of Jindřich Nečas Cent. Math. Model. Lect. Notes, Matfyzpress, Prague, 2009, 25–68. Google Scholar

[22]

N. H. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z. Google Scholar

[23]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3. Google Scholar

[24]

A. KiselevF. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ., 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2. Google Scholar

[25]

D. LiJ. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoam., 26 (2010), 295-332. doi: 10.4171/RMI/602. Google Scholar

[26]

F. Li and F. Rong, Decay of solutions to fractal parabolic conservation laws with large initial data, Commun. Pure Appl. Anal., 12 (2013), 973-984. doi: 10.3934/cpaa.2013.12.973. Google Scholar

[27]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3. Google Scholar

[28]

C. Miao and G. Wu, Global well-posedness of the critical Burgers equation in critical Besov spaces, J. Differential Equations, 247 (2009), 1673-1693. doi: 10.1016/j.jde.2009.03.028. Google Scholar

[29]

R. Shi and W. Wang, Nonlinear stability of large perturbation around the viscous shock wave for the 2D scalar viscous conservation law, Indiana Univ. Math. J., 65 (2016), 1137-1182. doi: 10.1512/iumj.2016.65.5850. Google Scholar

[30]

Y. SugiyamaM. Yamamoto and K. Kato, Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space, J. Differential Equations, 258 (2015), 2983-3010. doi: 10.1016/j.jde.2014.12.033. Google Scholar

[31]

H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar

[32]

L. Wang and W. Wang, Large-time behavior of periodic solutions to fractal Burgers equation with large initial data, Chin. Ann. Math. Ser. B, 33 (2012), 405-418. doi: 10.1007/s11401-012-0710-7. Google Scholar

[33]

W. Wang and W. Wang, Blow up and global existence of solutions for a model system of the radiating gas, Nonlinear Anal., 81 (2013), 12-30. doi: 10.1016/j.na.2012.12.010. Google Scholar

[34]

J. Wu, The quasi-geostrophic equation and its two regularizations, Comm. Partial Differential Equations, 27 (2002), 1161-1181. doi: 10.1081/PDE-120004898. Google Scholar

[35]

X. Xu, Local well-posedness and ill-posedness for the fractal Burgers equation in homogeneous Sobolev spaces, Math. Methods Appl. Sci., 32 (2009), 359-370. doi: 10.1002/mma.1046. Google Scholar

[36]

W. P. Ziemer, Weakly Differentiable Functions, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989, Sobolev spaces and functions of bounded variation. doi: 10.1007/978-1-4612-1015-3. Google Scholar

show all references

References:
[1]

N. AlibaudJ. Droniou and J. Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ., 4 (2007), 479-499. doi: 10.1142/S0219891607001227. Google Scholar

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

[3]

J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. Google Scholar

[4]

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

[5]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0. Google Scholar

[6]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935. doi: 10.1088/0951-7715/23/4/009. Google Scholar

[7]

L. Brandolese and G. Karch, Far field asymptotics of solutions to convection equation with anomalous diffusion, J. Evol. Equ., 8 (2008), 307-326. doi: 10.1007/s00028-008-0356-9. Google Scholar

[8]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903–1930. doi: 10.4007/annals.2010.171.1903. Google Scholar

[9]

C. H. Chan and M. Czubak, Regularity of solutions for the critical $N$-dimensional Burgers' equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 471-501. doi: 10.1016/j.anihpc.2009.11.008. Google Scholar

[10]

J.-Y. Chemin, Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Google Scholar

[11]

Q. ChenC. Miao and Z. Zhang, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271 (2007), 821-838. doi: 10.1007/s00220-007-0193-7. Google Scholar

[12]

P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), 97–108, https://doi.org/10.1512/iumj.2008.57.3629, Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). doi: 10.1512/iumj.2001.50.2153. Google Scholar

[13]

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321. doi: 10.1007/s00039-012-0172-9. Google Scholar

[14]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333. Google Scholar

[15]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. Google Scholar

[16]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X. Google Scholar

[17]

H. DongD. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J., 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505. Google Scholar

[18]

J. Droniou, T. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499–521, Dedicated to Philippe Bénilan. doi: 10.1007/s00028-003-0503-1. Google Scholar

[19]

R. Granero-Belinchón, On a drift-diffusion system for semiconductor devices, Ann. Henri Poincaré, 17 (2016), 3473-3498. doi: 10.1007/s00023-016-0493-6. Google Scholar

[20]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161-181. doi: 10.1007/s00220-004-1256-7. Google Scholar

[21]

G. Karch, Nonlinear evolution equations with anomalous diffusion, in Qualitative Properties of Solutions to Partial Differential Equations, vol. 5 of Jindřich Nečas Cent. Math. Model. Lect. Notes, Matfyzpress, Prague, 2009, 25–68. Google Scholar

[22]

N. H. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z. Google Scholar

[23]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3. Google Scholar

[24]

A. KiselevF. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ., 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2. Google Scholar

[25]

D. LiJ. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoam., 26 (2010), 295-332. doi: 10.4171/RMI/602. Google Scholar

[26]

F. Li and F. Rong, Decay of solutions to fractal parabolic conservation laws with large initial data, Commun. Pure Appl. Anal., 12 (2013), 973-984. doi: 10.3934/cpaa.2013.12.973. Google Scholar

[27]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3. Google Scholar

[28]

C. Miao and G. Wu, Global well-posedness of the critical Burgers equation in critical Besov spaces, J. Differential Equations, 247 (2009), 1673-1693. doi: 10.1016/j.jde.2009.03.028. Google Scholar

[29]

R. Shi and W. Wang, Nonlinear stability of large perturbation around the viscous shock wave for the 2D scalar viscous conservation law, Indiana Univ. Math. J., 65 (2016), 1137-1182. doi: 10.1512/iumj.2016.65.5850. Google Scholar

[30]

Y. SugiyamaM. Yamamoto and K. Kato, Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space, J. Differential Equations, 258 (2015), 2983-3010. doi: 10.1016/j.jde.2014.12.033. Google Scholar

[31]

H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar

[32]

L. Wang and W. Wang, Large-time behavior of periodic solutions to fractal Burgers equation with large initial data, Chin. Ann. Math. Ser. B, 33 (2012), 405-418. doi: 10.1007/s11401-012-0710-7. Google Scholar

[33]

W. Wang and W. Wang, Blow up and global existence of solutions for a model system of the radiating gas, Nonlinear Anal., 81 (2013), 12-30. doi: 10.1016/j.na.2012.12.010. Google Scholar

[34]

J. Wu, The quasi-geostrophic equation and its two regularizations, Comm. Partial Differential Equations, 27 (2002), 1161-1181. doi: 10.1081/PDE-120004898. Google Scholar

[35]

X. Xu, Local well-posedness and ill-posedness for the fractal Burgers equation in homogeneous Sobolev spaces, Math. Methods Appl. Sci., 32 (2009), 359-370. doi: 10.1002/mma.1046. Google Scholar

[36]

W. P. Ziemer, Weakly Differentiable Functions, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989, Sobolev spaces and functions of bounded variation. doi: 10.1007/978-1-4612-1015-3. Google Scholar

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