# Well-posedness and longtime behavior for the Westervelt equation with absorbing boundary conditions of order zero

@article{Simonett2016WellposednessAL, title={Well-posedness and longtime behavior for the Westervelt equation with absorbing boundary conditions of order zero}, author={Gieri Simonett and Mathias Wilke}, journal={Journal of Evolution Equations}, year={2016}, volume={17}, pages={551-571} }

We investigate the Westervelt equation from nonlinear acoustics, subject to nonlinear absorbing boundary conditions of order zero, which were recently proposed in Kaltenbacher and Shevchenko (Discrete Contin Dyn Syst 1000–1008, 2015), Shevchenko and Kaltenbacher (J Comput Phys 302:200–221, 2015). We apply the concept of maximal regularity of type Lp to prove global well-posedness for small initial data. Moreover, we show that the solutions regularize instantaneously, which means that they are C…

## 6 Citations

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## References

SHOWING 1-10 OF 23 REFERENCES

### Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions

- Mathematics
- 2011

In this paper we show wellposedness of two equations of nonlinear
acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous…

### Optimal Regularity and Long-Time Behavior of Solutions for the Westervelt Equation

- Mathematics
- 2011

We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently…

### Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation

- MathematicsJ. Comput. Phys.
- 2015

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

- Mathematics
- 2012

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…

### An analysis of nonhomogeneous Kuznetsov's equation: Local and global well‐posedness; exponential decay

- Mathematics
- 2012

This paper deals with well‐posedness of the Kuznetsov equation, which is an enhanced model for nonlinear acoustic wave propagation, e.g., in the context of high intensity ultrasound therapy. This is…

### Global existence and exponential decay rates for the Westervelt equation

- Mathematics
- 2009

We consider the Westervelt equation which models propagation of sound in a fluid medium.
This is an accepted in nonlinear acoustics model which finds a multitude of applications in medical imaging…

### Absorbing boundary conditions for the Westervelt equation

- Mathematics
- 2014

The focus of this work is on the construction of a family of nonlinear absorbing boundary
conditions for the Westervelt equation in one and two space dimensions. The principal ingredient
used in…

### Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping

- Mathematics
- 2010

In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear…

### Nonlinear analytic semiflows

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 1990

Synopsis In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems (x′(t) = f(x(t))), and quasilinear initial value problems (x′(t)=A(x(t))x(t) +…