American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system
  • EECT Home
  • This Issue
  • Next Article
    Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement
June  2013, 2(2): 365-378. doi: 10.3934/eect.2013.2.365

Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces

Stefan Meyer 1, and  Mathias Wilke 1,

1. 

Martin Luther University Halle-Wittenberg, NWF II - Institute of Mathematics, D - 06099 Halle (Saale), Germany, Germany

Received  November 2012 Revised  February 2013 Published  March 2013

We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal $L_p$-regularity for parabolic equations and the implicit function theorem.
Keywords: quasilinear parabolic system, Kuznetsov's equation, global well-posedness, optimal regularity, exponential decay..
Mathematics Subject Classification: Primary: 35K59; Secondary: 35K51, 35Q35, 35B30, 35B35, 35B40, 35B45, 35B6.
Citation: Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365
References:
[1]

Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.  Google Scholar

[3]

Herbert Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[4]

Sigurd B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107. doi: 10.1017/S0308210500024598.  Google Scholar

[5]

Klaus Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.  Google Scholar

[6]

Robert Denk, Jürgen Saal and Jörg Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russ. J. Math. Phys., 15 (2008), 171-191. doi: 10.1134/S1061920808020040.  Google Scholar

[7]

Robert Denk, Matthias Hieber and Jan Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003).  Google Scholar

[8]

Robert Denk, Matthias Hieber and Jan Prüss, Optimal $L^p$- $L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[9]

Gabriella Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl. (4), 138 (1984), 55-104. doi: 10.1007/BF01762539.  Google Scholar

[10]

Barbara Kaltenbacher and Irena Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321. doi: 10.1002/mana.201000007.  Google Scholar

[11]

Manfred Kaltenbacher, "Numerical Simulation of Mechatronic Sensors and Actuators," Springer, 2007. Available from: http://dx.doi.org/10.1007/978-3-540-71360-9. Google Scholar

[12]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470. Google Scholar

[13]

Yuri Latushkin, Jan Prüss and Roland Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576. doi: 10.1007/s00028-006-0272-9.  Google Scholar

[14]

Alessandra Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[15]

Stefan Meyer and Mathias Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.  Google Scholar

[16]

Martin Meyries and Roland Schnaubelt, Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights, J. Funct. Anal., 262 (2012), 1200-1229. doi: 10.1016/j.jfa.2011.11.001.  Google Scholar

[17]

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

[18]

Hans Triebel, "Interpolation Theory, Function Spaces, Differential Operators," $2^{nd}$ edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[19]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

show all references

References:
[1]

Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.  Google Scholar

[3]

Herbert Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[4]

Sigurd B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107. doi: 10.1017/S0308210500024598.  Google Scholar

[5]

Klaus Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.  Google Scholar

[6]

Robert Denk, Jürgen Saal and Jörg Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russ. J. Math. Phys., 15 (2008), 171-191. doi: 10.1134/S1061920808020040.  Google Scholar

[7]

Robert Denk, Matthias Hieber and Jan Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003).  Google Scholar

[8]

Robert Denk, Matthias Hieber and Jan Prüss, Optimal $L^p$- $L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[9]

Gabriella Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl. (4), 138 (1984), 55-104. doi: 10.1007/BF01762539.  Google Scholar

[10]

Barbara Kaltenbacher and Irena Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321. doi: 10.1002/mana.201000007.  Google Scholar

[11]

Manfred Kaltenbacher, "Numerical Simulation of Mechatronic Sensors and Actuators," Springer, 2007. Available from: http://dx.doi.org/10.1007/978-3-540-71360-9. Google Scholar

[12]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470. Google Scholar

[13]

Yuri Latushkin, Jan Prüss and Roland Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576. doi: 10.1007/s00028-006-0272-9.  Google Scholar

[14]

Alessandra Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[15]

Stefan Meyer and Mathias Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.  Google Scholar

[16]

Martin Meyries and Roland Schnaubelt, Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights, J. Funct. Anal., 262 (2012), 1200-1229. doi: 10.1016/j.jfa.2011.11.001.  Google Scholar

[17]

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

[18]

Hans Triebel, "Interpolation Theory, Function Spaces, Differential Operators," $2^{nd}$ edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[19]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[1]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075
[2]

Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019
[3]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087
[4]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[5]

Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
[6]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
[7]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[8]

Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021147
[9]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[10]

Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781
[11]

E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156
[12]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
[13]

Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[14]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[15]

Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483
[16]

Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713
[17]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273
[18]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[19]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081
[20]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences