December  2014, 3(4): 595-626. doi: 10.3934/eect.2014.3.595

Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling

1. 

Institut für Mathematik, Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt am Wörthersee

2. 

Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588

Received  December 2013 Revised  March 2014 Published  October 2014

In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying $L_\infty$-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.
Citation: Rainer Brunnhuber, Barbara Kaltenbacher, Petronela Radu. Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling. Evolution Equations and Control Theory, 2014, 3 (4) : 595-626. doi: 10.3934/eect.2014.3.595
References:
[1]

O. V. Abramov, High-Intensity Ultrasonics, Gordon and Breach Science Publishers, Amsterdam, 1998.

[2]

A. Bamberger, R. Glowinski and Q. H. Tran, A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change, SIAM Journal on Numerical Analysis, 34 (1997), 603-639, doi: 10.1137/S0036142994261518.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti Romania and Noordhoff International Publishing, Leyden Netherlands, 1976.

[4]

A. Bermudez, R. Rodriguez and D. Santamarina, Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations, Journal of Computational and Applied Mathematics, 152 (2003), 17-34. doi: 10.1016/S0377-0427(02)00694-5.

[5]

A. C. Biazutti, On a nonlinear evolution equation and its applications, Nonlinear Analysis, 24 (1995), 1221-1234. doi: 10.1016/0362-546X(94)00193-L.

[6]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, Berlin, 1991. doi: 10.1007/978-1-4612-3172-1.

[7]

J. C. Clements, On the existence and uniqueness of solutions of the equation $u_{t t}-\partial \sigma _i(u_{x_i}) / {\partial x_i} - D_Nu_t=f$, Canadian Mathematical Bulletin, 18 (1975), 181-187. doi: 10.4153/CMB-1975-036-1.

[8]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.

[9]

B. Flemisch, M. Kaltenbacher and B. I. Wohlmuth, Elasto-acoustic and acoustic-acoustic coupling on nonmatching grids, International Journal of Numerical Methods in Engineering, 67 (2006), 1791-1810. doi: 10.1002/nme.1669.

[10]

M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, New York, 1997.

[11]

B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations without interior damping, Applied Mathematics and Optimization, 62 (2010), 381-410. doi: 10.1007/s00245-010-9108-7.

[12]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 503-523. doi: 10.3934/dcdss.2009.2.503.

[13]

B. Kaltenbacher, I. Lasiecka and S. Veljović, Well-posedness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundary data, J. Escher et. al. (Eds): Progress in Nonlinear Differential Equations and Their Applications, Springer Basel AG, 80 (2011), 357-387. doi: 10.1007/978-3-0348-0075-4_19.

[14]

G. Leoni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, 2009.

[15]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., II (2011), 763-773.

[16]

M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuators, Springer, Berlin, 2004. doi: 10.1007/978-3-662-05358-4.

[17]

A. Raviart and J. M. Thomas, Primal hybrid finite element method for second order elliptic equations, Mathematics of Computation, 31 (1977), 391-413.

[18]

M. A. Rammaha and Z. Wilstein, Hadamard well-posedness for wave equations with $p$-Laplacian damping and supercritical sources, Advances in Differential Equations, 17 (2012), 105-150.

[19]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, Providence, 2012.

[20]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537. doi: 10.1121/1.1918525.

show all references

References:
[1]

O. V. Abramov, High-Intensity Ultrasonics, Gordon and Breach Science Publishers, Amsterdam, 1998.

[2]

A. Bamberger, R. Glowinski and Q. H. Tran, A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change, SIAM Journal on Numerical Analysis, 34 (1997), 603-639, doi: 10.1137/S0036142994261518.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti Romania and Noordhoff International Publishing, Leyden Netherlands, 1976.

[4]

A. Bermudez, R. Rodriguez and D. Santamarina, Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations, Journal of Computational and Applied Mathematics, 152 (2003), 17-34. doi: 10.1016/S0377-0427(02)00694-5.

[5]

A. C. Biazutti, On a nonlinear evolution equation and its applications, Nonlinear Analysis, 24 (1995), 1221-1234. doi: 10.1016/0362-546X(94)00193-L.

[6]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, Berlin, 1991. doi: 10.1007/978-1-4612-3172-1.

[7]

J. C. Clements, On the existence and uniqueness of solutions of the equation $u_{t t}-\partial \sigma _i(u_{x_i}) / {\partial x_i} - D_Nu_t=f$, Canadian Mathematical Bulletin, 18 (1975), 181-187. doi: 10.4153/CMB-1975-036-1.

[8]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.

[9]

B. Flemisch, M. Kaltenbacher and B. I. Wohlmuth, Elasto-acoustic and acoustic-acoustic coupling on nonmatching grids, International Journal of Numerical Methods in Engineering, 67 (2006), 1791-1810. doi: 10.1002/nme.1669.

[10]

M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, New York, 1997.

[11]

B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations without interior damping, Applied Mathematics and Optimization, 62 (2010), 381-410. doi: 10.1007/s00245-010-9108-7.

[12]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 503-523. doi: 10.3934/dcdss.2009.2.503.

[13]

B. Kaltenbacher, I. Lasiecka and S. Veljović, Well-posedness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundary data, J. Escher et. al. (Eds): Progress in Nonlinear Differential Equations and Their Applications, Springer Basel AG, 80 (2011), 357-387. doi: 10.1007/978-3-0348-0075-4_19.

[14]

G. Leoni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, 2009.

[15]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., II (2011), 763-773.

[16]

M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuators, Springer, Berlin, 2004. doi: 10.1007/978-3-662-05358-4.

[17]

A. Raviart and J. M. Thomas, Primal hybrid finite element method for second order elliptic equations, Mathematics of Computation, 31 (1977), 391-413.

[18]

M. A. Rammaha and Z. Wilstein, Hadamard well-posedness for wave equations with $p$-Laplacian damping and supercritical sources, Advances in Differential Equations, 17 (2012), 105-150.

[19]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, Providence, 2012.

[20]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537. doi: 10.1121/1.1918525.

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