# zbMATH — the first resource for mathematics

On a one-dimensional $$\alpha$$-patch model with nonlocal drift and fractional dissipation. (English) Zbl 1302.35334
Authors’ abstract: We consider a one-dimensional nonlocal nonlinear equation of the form $$\partial_tu=(\Lambda^{-\alpha}u)\partial_xu-\nu\Lambda^\beta u$$, where $$\Lambda =(-\partial_{xx})^{\frac{1}{2}}$$ is the fractional Laplacian and $$\nu\geq 0$$ is the viscosity coefficient. We primarily consider the regime $$0<\alpha <1$$ and $$0\leq\beta\leq 2$$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $$\alpha$$-patch models. In the critical and subcritical range $$1-\alpha\leq\beta\leq 2$$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $$0\leq\beta <1-\alpha$$, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonlocal weighted inequality which can be of independent interest.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35B44 Blow-up in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35R11 Fractional partial differential equations
Full Text:
##### References:
 [1] Nathaël Alibaud, Jérôme Droniou, and Julien Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 479 – 499. · Zbl 1144.35038 · doi:10.1142/S0219891607001227 · doi.org [2] Piotr Biler, Tadahisa Funaki, and Wojbor A. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998), no. 1, 9 – 46. · Zbl 0911.35100 · doi:10.1006/jdeq.1998.3458 · doi.org [3] Angel Castro, Diego Córdoba, and Francisco Gancedo, Singularity formations for a surface wave model, Nonlinearity 23 (2010), no. 11, 2835 – 2847. · Zbl 1223.35016 · doi:10.1088/0951-7715/23/11/006 · doi.org [4] Dongho Chae, Peter Constantin, and Jiahong Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 35 – 62. · Zbl 1266.76010 · doi:10.1007/s00205-011-0411-5 · doi.org [5] Chi Hin Chan and Magdalena Czubak, Regularity of solutions for the critical \?-dimensional Burgers’ equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 2, 471 – 501 (English, with English and French summaries). · Zbl 1189.35354 · doi:10.1016/j.anihpc.2009.11.008 · doi.org [6] Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511 – 528. · Zbl 1309.76026 · doi:10.1007/s00220-004-1055-1 · doi.org [7] Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (2005), no. 3, 1377 – 1389. · Zbl 1101.35052 · doi:10.4007/annals.2005.162.1377 · doi.org [8] Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl. (9) 86 (2006), no. 6, 529 – 540 (English, with English and French summaries). · Zbl 1106.35059 · doi:10.1016/j.matpur.2006.08.002 · doi.org [9] Diego Córdoba, Marco A. Fontelos, Ana M. Mancho, and Jose L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005), no. 17, 5949 – 5952. · Zbl 1135.76315 · doi:10.1073/pnas.0501977102 · doi.org [10] Hongjie Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal. 255 (2008), no. 11, 3070 – 3097. · Zbl 1170.35004 · doi:10.1016/j.jfa.2008.08.005 · doi.org [11] Hongjie Dong and Dapeng Du, Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, Discrete Contin. Dyn. Syst. 21 (2008), no. 4, 1095 – 1101. · Zbl 1141.35436 · doi:10.3934/dcds.2008.21.1095 · doi.org [12] Hongjie Dong and Dong Li, Finite time singularities for a class of generalized surface quasi-geostrophic equations, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2555 – 2563. · Zbl 1143.35084 [13] Hongjie Dong, Dapeng Du, and Dong Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J. 58 (2009), no. 2, 807 – 821. · Zbl 1166.35030 · doi:10.1512/iumj.2009.58.3505 · doi.org [14] Francisco Gancedo, Existence for the \?-patch model and the QG sharp front in Sobolev spaces, Adv. Math. 217 (2008), no. 6, 2569 – 2598. · Zbl 1148.35099 · doi:10.1016/j.aim.2007.10.010 · doi.org [15] A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), no. 3, 445 – 453. · Zbl 1121.35115 · doi:10.1007/s00222-006-0020-3 · doi.org [16] Alexander Kiselev, Fedor Nazarov, and Roman Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008), no. 3, 211 – 240. · Zbl 1186.35020 · doi:10.4310/DPDE.2008.v5.n3.a2 · doi.org [17] Alexander Kiselev, Nonlocal maximum principles for active scalars, Adv. Math. 227 (2011), no. 5, 1806 – 1826. · Zbl 1244.35022 · doi:10.1016/j.aim.2011.03.019 · doi.org [18] Dong Li and Jose Rodrigo, Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math. 217 (2008), no. 6, 2563 – 2568. · Zbl 1138.35381 · doi:10.1016/j.aim.2007.11.002 · doi.org [19] Dong Li and Jose Rodrigo, Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation, Comm. Math. Phys. 286 (2009), no. 1, 111 – 124. · Zbl 1172.86301 · doi:10.1007/s00220-008-0585-3 · doi.org [20] Changxing Miao and Gang Wu, Global well-posedness of the critical Burgers equation in critical Besov spaces, J. Differential Equations 247 (2009), no. 6, 1673 – 1693. · Zbl 1184.35003 · doi:10.1016/j.jde.2009.03.028 · doi.org [21] Hisashi Okamoto, Takashi Sakajo, and Marcus Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity 21 (2008), no. 10, 2447 – 2461. · Zbl 1221.35300 · doi:10.1088/0951-7715/21/10/013 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.