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On the local and global existence of solutions to 1d transport equations with nonlocal velocity
1. | Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan, Republic of Korea |
2. | Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain |
3. | Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, Sevilla, Spain |
$ \begin{equation*} \label{intro eq} \begin{split} &\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta = 0, \\ & u = \mathcal{N}(\theta), \end{split} \end{equation*} $ |
$ \mathcal{N} $ |
$ \Lambda^{\gamma} $ |
$ \widehat{\Lambda^{\gamma} f}(\xi) = |\xi|^{\gamma}\widehat{f}(\xi) $ |
References:
[1] |
H. Bae, D. Chae and H. Okamoto,
On the well-posedness of various one-dimensional model equations for fluid motion, Nonlinear Anal., 160 (2017), 25-43.
doi: 10.1016/j.na.2017.05.002. |
[2] |
H. Bae and R. Granero-Belinchón,
Global existence for some transport equations with nonlocal velocity, Adv. Math., 269 (2015), 197-219.
doi: 10.1016/j.aim.2014.10.016. |
[3] |
H. Bae, R. Granero-Belinchón and O. Lazar, Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018) 1484–1515.
doi: 10.1088/1361-6544/aaa2e0. |
[4] |
H. Bahouri, J-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, 2011.
doi: 10.1007/978-3-642-16830-7. |
[5] |
G. R. Baker, X. Li and A. C. Morlet,
Analytic structure of 1D transport equations with nonlocal fluxes, Physica D., 91 (1996), 349-375.
doi: 10.1016/0167-2789(95)00271-5. |
[6] |
J. A. Carrillo, L. C. F. Ferreira and J. C. Precioso,
A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231 (2012), 306-327.
doi: 10.1016/j.aim.2012.03.036. |
[7] |
A. Castro and D. Córdoba,
Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219 (2008), 1916-1936.
doi: 10.1016/j.aim.2008.07.015. |
[8] |
A. Castro and D. Córdoba,
Self-similar solutions for a transport equation with non-local flux, Chinese Annals of Mathematics, Series B, 30 (2009), 505-512.
doi: 10.1007/s11401-009-0180-8. |
[9] |
D. Chae, A. Cordoba, D. Cordoba and M. Fontelos,
Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194 (2005), 203-223.
doi: 10.1016/j.aim.2004.06.004. |
[10] |
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. |
[11] |
A. Córdoba, D. Córdoba and M. Fontelos,
Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1-13.
doi: 10.4007/annals.2005.162.1377. |
[12] |
M. Cotlar,
A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana, 1 (1955), 105-167.
|
[13] |
S. De Gregorio,
On a one-dimensional model for the 3D vorticity equation, J. Statist. Phys., 59 (1990), 1251-1263.
doi: 10.1007/BF01334750. |
[14] |
H. Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, (2008), 3070–3097.
doi: 10.1016/j.jfa.2008.08.005. |
[15] |
H. Dong, On a multi-dimensional transport equation with nonlocal velocity, Adv. Math., 264 (2014), 747–761.
doi: 10.1016/j.aim.2014.07.028. |
[16] |
H. Dong and D. Li, On a one-dimensional $\alpha$-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc., 366 (2014), 2041–2061.
doi: 10.1090/S0002-9947-2013-06075-8. |
[17] |
J. Duoandikoetxea, Fourier Analysis, Translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics, 29, American Mathematical Society, 2000. |
[18] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[19] |
A. Kiselev,
Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.
doi: 10.1051/mmnp/20105410. |
[20] |
O. Lazar,
On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion, Journal of Diff. Eq., 261 (2016), 4974-4996.
doi: 10.1016/j.jde.2016.07.009. |
[21] |
O. Lazar and P.-G. Lemarié-Rieusset, Infinite energy solutions for a 1D transport equation with nonlocal velocity, Dynamics of PDEs, 13 (2016), 107-131.
doi: 10.4310/DPDE.2016.v13.n2.a2. |
[22] |
D. Li, On Kato-Ponce and fractional Leibniz, arXiv: 1609.01780.
doi: 10.4171/rmi/1049. |
[23] |
D. Li and J. Rodrigo,
Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math., 217 (2008), 2563-2568.
doi: 10.1016/j.aim.2007.11.002. |
[24] |
D. Li and J. Rodrigo,
On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal., 43 (2011), 507-526.
doi: 10.1137/100794924. |
[25] |
A. Morlet,
Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl., 221 (1998), 132-160.
doi: 10.1006/jmaa.1997.5801. |
[26] |
J. Simon,
Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
show all references
References:
[1] |
H. Bae, D. Chae and H. Okamoto,
On the well-posedness of various one-dimensional model equations for fluid motion, Nonlinear Anal., 160 (2017), 25-43.
doi: 10.1016/j.na.2017.05.002. |
[2] |
H. Bae and R. Granero-Belinchón,
Global existence for some transport equations with nonlocal velocity, Adv. Math., 269 (2015), 197-219.
doi: 10.1016/j.aim.2014.10.016. |
[3] |
H. Bae, R. Granero-Belinchón and O. Lazar, Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018) 1484–1515.
doi: 10.1088/1361-6544/aaa2e0. |
[4] |
H. Bahouri, J-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, 2011.
doi: 10.1007/978-3-642-16830-7. |
[5] |
G. R. Baker, X. Li and A. C. Morlet,
Analytic structure of 1D transport equations with nonlocal fluxes, Physica D., 91 (1996), 349-375.
doi: 10.1016/0167-2789(95)00271-5. |
[6] |
J. A. Carrillo, L. C. F. Ferreira and J. C. Precioso,
A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231 (2012), 306-327.
doi: 10.1016/j.aim.2012.03.036. |
[7] |
A. Castro and D. Córdoba,
Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219 (2008), 1916-1936.
doi: 10.1016/j.aim.2008.07.015. |
[8] |
A. Castro and D. Córdoba,
Self-similar solutions for a transport equation with non-local flux, Chinese Annals of Mathematics, Series B, 30 (2009), 505-512.
doi: 10.1007/s11401-009-0180-8. |
[9] |
D. Chae, A. Cordoba, D. Cordoba and M. Fontelos,
Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194 (2005), 203-223.
doi: 10.1016/j.aim.2004.06.004. |
[10] |
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. |
[11] |
A. Córdoba, D. Córdoba and M. Fontelos,
Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1-13.
doi: 10.4007/annals.2005.162.1377. |
[12] |
M. Cotlar,
A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana, 1 (1955), 105-167.
|
[13] |
S. De Gregorio,
On a one-dimensional model for the 3D vorticity equation, J. Statist. Phys., 59 (1990), 1251-1263.
doi: 10.1007/BF01334750. |
[14] |
H. Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, (2008), 3070–3097.
doi: 10.1016/j.jfa.2008.08.005. |
[15] |
H. Dong, On a multi-dimensional transport equation with nonlocal velocity, Adv. Math., 264 (2014), 747–761.
doi: 10.1016/j.aim.2014.07.028. |
[16] |
H. Dong and D. Li, On a one-dimensional $\alpha$-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc., 366 (2014), 2041–2061.
doi: 10.1090/S0002-9947-2013-06075-8. |
[17] |
J. Duoandikoetxea, Fourier Analysis, Translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics, 29, American Mathematical Society, 2000. |
[18] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[19] |
A. Kiselev,
Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.
doi: 10.1051/mmnp/20105410. |
[20] |
O. Lazar,
On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion, Journal of Diff. Eq., 261 (2016), 4974-4996.
doi: 10.1016/j.jde.2016.07.009. |
[21] |
O. Lazar and P.-G. Lemarié-Rieusset, Infinite energy solutions for a 1D transport equation with nonlocal velocity, Dynamics of PDEs, 13 (2016), 107-131.
doi: 10.4310/DPDE.2016.v13.n2.a2. |
[22] |
D. Li, On Kato-Ponce and fractional Leibniz, arXiv: 1609.01780.
doi: 10.4171/rmi/1049. |
[23] |
D. Li and J. Rodrigo,
Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math., 217 (2008), 2563-2568.
doi: 10.1016/j.aim.2007.11.002. |
[24] |
D. Li and J. Rodrigo,
On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal., 43 (2011), 507-526.
doi: 10.1137/100794924. |
[25] |
A. Morlet,
Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl., 221 (1998), 132-160.
doi: 10.1006/jmaa.1997.5801. |
[26] |
J. Simon,
Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
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