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# American Institute of Mathematical Sciences

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 & Control Theory, 2014, 3 (4) : 595-626. doi: 10.3934/eect.2014.3.595
##### References:
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##### References:
 [1] O. V. Abramov, High-Intensity Ultrasonics,, Gordon and Breach Science Publishers, (1998).   Google Scholar [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.  doi: 10.1137/S0036142994261518.  Google Scholar [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei, (1976).   Google Scholar [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.  doi: 10.1016/S0377-0427(02)00694-5.  Google Scholar [5] A. C. Biazutti, On a nonlinear evolution equation and its applications,, Nonlinear Analysis, 24 (1995), 1221.  doi: 10.1016/0362-546X(94)00193-L.  Google Scholar [6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer, (1991).  doi: 10.1007/978-1-4612-3172-1.  Google Scholar [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.  doi: 10.4153/CMB-1975-036-1.  Google Scholar [8] L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998).   Google Scholar [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.  doi: 10.1002/nme.1669.  Google Scholar [10] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics,, Academic Press, (1997).   Google Scholar [11] B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations without interior damping,, Applied Mathematics and Optimization, 62 (2010), 381.  doi: 10.1007/s00245-010-9108-7.  Google Scholar [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.  doi: 10.3934/dcdss.2009.2.503.  Google Scholar [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, 80 (2011), 357.  doi: 10.1007/978-3-0348-0075-4_19.  Google Scholar [14] G. Leoni, A First Course in Sobolev Spaces,, American Mathematical Society, (2009).   Google Scholar [15] B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions,, Discrete Contin. Dyn. Syst., II (2011), 763.   Google Scholar [16] M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuators,, Springer, (2004).  doi: 10.1007/978-3-662-05358-4.  Google Scholar [17] A. Raviart and J. M. Thomas, Primal hybrid finite element method for second order elliptic equations,, Mathematics of Computation, 31 (1977), 391.   Google Scholar [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.   Google Scholar [19] G. Teschl, Ordinary Differential Equations and Dynamical Systems,, American Mathematical Society, (2012).   Google Scholar [20] P. J. Westervelt, Parametric acoustic array,, The Journal of the Acoustic Society of America, 35 (1963), 535.  doi: 10.1121/1.1918525.  Google Scholar
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# Is there a direct combinatorial interpretation of this (seemingly coincidental) result?

The original problem is as follows:

Find the probability $p$ that in a bridge game the players North, East, South, West have $a, b, c, d$ spades, respectively (which implies $a+b+c+d=13$).

And after simple counting, you'll get the answer, which can be rewritten as: $$p=\frac{\binom{13}{a}\,\binom{13}{b}\,\binom{13}{c}\,\binom{13}{d}}{\binom{52}{13}}$$

And it seems like this result can be directly obtained by interpreting the numerator and denominator respectively, but I'm stuck with how to do this. Any ideas?

• Don't think your answer is correct. Have a look at math.stackexchange.com/q/1416902/321264. – StubbornAtom Aug 19 '16 at 11:08
• @StubbornAtom Actually they are the same, and you can rewrite it as above. – Daniel Aug 19 '16 at 11:13

Suppose that there are 52 positions in a row where you can put 13 spades into different positions there. So you'll have $\binom{52}{13}$ choices in total. And now you want there to be $a$ spades in the first 13 positions, $b$ spades in the second 13 positions, $c$ spades in the third 13 positions, and $d$ spades in the fourth 13 positions, where $a+b+c+d=13$. So in this case you'll have $\binom{13}{a}\,\binom{13}{b}\,\binom{13}{c}\,\binom{13}{d}$ choices in total. Thus the result follows.

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