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Gradient regularity for a singular parabolic equation in non-divergence form
Filtering the $ L^2- $critical focusing Schrödinger equation
Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud Ⅺ, CNRS, Université Paris-Saclay, F-91405 Orsay, France |
$ L^2- $ |
$ \begin{equation*} i\partial_t u + \partial_x^2 u + \Pi(|u|^4 u) = 0, \quad (t, x)\in \mathbb{R}\times \mathbb{R}, \qquad u(0, \cdot) = u_0. \end{equation*} $ |
$ P(u) = \langle -i\partial_x u, u\rangle_{L^2} $ |
$ \dot{H}^{\frac{1}{2}}- $ |
$ H^1_+ = \Pi(H^1(\mathbb{R})) $ |
$ u_0 $ |
$ L^2- $ |
$ u_0 $ |
$ u_{\omega, c}(t, x) = e^{i\omega t}Q(x+ct) $ |
$ \omega, c>0 $ |
$ Q $ |
$ \begin{equation*} I^{(\gamma)}(f) = \frac{\|\partial_x f\|_{L^2}^2\|f\|_{L^2}^{4}+\gamma \langle -i \partial_x f , f\rangle_{L^2}^2 \|f\|_{L^2}^2}{\|f\|_{L^{6}}^{6}}, \qquad \forall f\in H^1_+ \backslash \{0\}, \end{equation*} $ |
$ \gamma\geq 0 $ |
$ \gamma = 2 $ |
$ I^{(0)} $ |
References:
[1] |
C. Amick and J. F. Toland,
Uniqueness and related analytic properties for the Benjamin-Ono equation – a nonlinear Neumann problem in the plane, Acta Math., 167 (1991), 107-126.
doi: 10.1007/BF02392447. |
[2] |
H. Bahouri and P. Gérard,
High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[3] |
T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, Vol. 26, Instituto de Matemática UFRJ, 1996. |
[4] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York University, Courant Institute of Mathematical Sciences, AMS, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[5] |
T. Cazenave and P.-L. Lions,
Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[6] |
T. Cazenave and F. Weissler,
The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta. Math., 61 (1988), 477-494.
doi: 10.1007/BF01258601. |
[7] |
T. Cazenave and F. Weissler,
The Cauchy problem for the nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[8] |
B. Dodson,
Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.
doi: 10.1016/j.aim.2015.04.030. |
[9] |
D. Foschi,
Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS), 9 (2007), 739-774.
doi: 10.4171/JEMS/95. |
[10] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[11] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[12] |
P. Gérard,
Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.
doi: 10.1051/cocv:1998107. |
[13] |
P. Gérard and S. Grellier,
The cubic Szegő equation, Ann. Sci. l'Éc. Norm. Supér., 43 (2010), 761-810.
doi: 10.24033/asens.2133. |
[14] |
P. Gérard and S. Grellier, The cubic Szegő equation and Hankel operators, in Astérisque Vol. 389, 2017. |
[15] |
P. Gérard, E. Lenzmann, O. Pocovnicu and P. Raphaël, A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line, Ann. PDE, 4 (2018), 166 pp.
doi: 10.1007/s40818-017-0043-7. |
[16] |
R. Glassey,
On the blowing up of solutions to the Cauchy problem for non linear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1063/1.523491. |
[17] |
T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., (2005), No. 46, 2815–2828.
doi: 10.1155/IMRN.2005.2815. |
[18] |
T. Hmidi and S. Keraani,
Remarks on the blow-up for the $L^2-$critical nonlinear Schrödinger equations, SIAM J. Math. Anal., 38 (2006), 1035-1047.
doi: 10.1137/050624054. |
[19] |
C. Kenig and F. Merle,
Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[20] |
C. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[21] |
R. Killip and M. Vişan,
Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885.
doi: 10.2140/apde.2012.5.855. |
[22] |
R. Killip, M. Vişan and X. Zhang,
Energy-critical NLS with quadratic potentials, Comm. PDE., 34 (2009), 1531-1565.
doi: 10.1080/03605300903328109. |
[23] |
J. Krieger and J. Lührmann, Concentration compactness for the critical Maxwell-Klein-Gordon equation, Ann. PDE, 1 (2015), no. 1, Art. 5,208 pp.
doi: 10.1007/s40818-015-0004-y. |
[24] |
J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, Series of Lectures in Mathematics, European Mathematical Society, Zúrich, 2012.
doi: 10.4171/106. |
[25] |
E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, preprint, 2018, arXiv: 1805.06294.
doi: 10.1093/imrn/rnz274. |
[26] |
P.-L. Lions,
The concentration-compactness principle in calculus of variations. The locally compact case. Part 1, Ann. Inst. Henri Poincaré, Analyse non linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[27] |
P.-L. Lions,
The concentration-compactness principle in calculus of variations. The locally compact case. Part 2, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 1 (1984), 223-283.
|
[28] |
F. Merle and P. Raphaël,
On universality of blow-up profile for $L^2-$critical non linear Schrödinger equation, Invent. Math., 156 (2004), 565-672.
doi: 10.1007/s00222-003-0346-z. |
[29] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1-$solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[30] |
G. Perelman,
On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2 (2001), 605-673.
doi: 10.1007/PL00001048. |
[31] |
O. Pocovnicu,
Traveling waves for the cubic Szegő equation on the real line, Anal. PDE, 4 (2011), 379-404.
doi: 10.2140/apde.2011.4.379. |
[32] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. |
[33] |
R. Strichartz,
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[34] |
R. Sun,
Long time behavior of the NLS-Szegő equation, Dyn. Partial. Differ. Equ., 16 (2019), 325-357.
doi: 10.4310/DPDE.2019.v16.n4.a2. |
[35] |
R. Sun, Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds, preprint, arXiv: 2004.10007. |
[36] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/cbms/106. |
[37] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
|
show all references
References:
[1] |
C. Amick and J. F. Toland,
Uniqueness and related analytic properties for the Benjamin-Ono equation – a nonlinear Neumann problem in the plane, Acta Math., 167 (1991), 107-126.
doi: 10.1007/BF02392447. |
[2] |
H. Bahouri and P. Gérard,
High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[3] |
T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, Vol. 26, Instituto de Matemática UFRJ, 1996. |
[4] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York University, Courant Institute of Mathematical Sciences, AMS, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[5] |
T. Cazenave and P.-L. Lions,
Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[6] |
T. Cazenave and F. Weissler,
The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta. Math., 61 (1988), 477-494.
doi: 10.1007/BF01258601. |
[7] |
T. Cazenave and F. Weissler,
The Cauchy problem for the nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[8] |
B. Dodson,
Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.
doi: 10.1016/j.aim.2015.04.030. |
[9] |
D. Foschi,
Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS), 9 (2007), 739-774.
doi: 10.4171/JEMS/95. |
[10] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[11] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[12] |
P. Gérard,
Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.
doi: 10.1051/cocv:1998107. |
[13] |
P. Gérard and S. Grellier,
The cubic Szegő equation, Ann. Sci. l'Éc. Norm. Supér., 43 (2010), 761-810.
doi: 10.24033/asens.2133. |
[14] |
P. Gérard and S. Grellier, The cubic Szegő equation and Hankel operators, in Astérisque Vol. 389, 2017. |
[15] |
P. Gérard, E. Lenzmann, O. Pocovnicu and P. Raphaël, A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line, Ann. PDE, 4 (2018), 166 pp.
doi: 10.1007/s40818-017-0043-7. |
[16] |
R. Glassey,
On the blowing up of solutions to the Cauchy problem for non linear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1063/1.523491. |
[17] |
T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., (2005), No. 46, 2815–2828.
doi: 10.1155/IMRN.2005.2815. |
[18] |
T. Hmidi and S. Keraani,
Remarks on the blow-up for the $L^2-$critical nonlinear Schrödinger equations, SIAM J. Math. Anal., 38 (2006), 1035-1047.
doi: 10.1137/050624054. |
[19] |
C. Kenig and F. Merle,
Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[20] |
C. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[21] |
R. Killip and M. Vişan,
Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885.
doi: 10.2140/apde.2012.5.855. |
[22] |
R. Killip, M. Vişan and X. Zhang,
Energy-critical NLS with quadratic potentials, Comm. PDE., 34 (2009), 1531-1565.
doi: 10.1080/03605300903328109. |
[23] |
J. Krieger and J. Lührmann, Concentration compactness for the critical Maxwell-Klein-Gordon equation, Ann. PDE, 1 (2015), no. 1, Art. 5,208 pp.
doi: 10.1007/s40818-015-0004-y. |
[24] |
J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, Series of Lectures in Mathematics, European Mathematical Society, Zúrich, 2012.
doi: 10.4171/106. |
[25] |
E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, preprint, 2018, arXiv: 1805.06294.
doi: 10.1093/imrn/rnz274. |
[26] |
P.-L. Lions,
The concentration-compactness principle in calculus of variations. The locally compact case. Part 1, Ann. Inst. Henri Poincaré, Analyse non linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[27] |
P.-L. Lions,
The concentration-compactness principle in calculus of variations. The locally compact case. Part 2, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 1 (1984), 223-283.
|
[28] |
F. Merle and P. Raphaël,
On universality of blow-up profile for $L^2-$critical non linear Schrödinger equation, Invent. Math., 156 (2004), 565-672.
doi: 10.1007/s00222-003-0346-z. |
[29] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1-$solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[30] |
G. Perelman,
On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2 (2001), 605-673.
doi: 10.1007/PL00001048. |
[31] |
O. Pocovnicu,
Traveling waves for the cubic Szegő equation on the real line, Anal. PDE, 4 (2011), 379-404.
doi: 10.2140/apde.2011.4.379. |
[32] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. |
[33] |
R. Strichartz,
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[34] |
R. Sun,
Long time behavior of the NLS-Szegő equation, Dyn. Partial. Differ. Equ., 16 (2019), 325-357.
doi: 10.4310/DPDE.2019.v16.n4.a2. |
[35] |
R. Sun, Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds, preprint, arXiv: 2004.10007. |
[36] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/cbms/106. |
[37] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
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