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Large-time behavior for a simplified 1D model of fluid-solid interaction. (English) Zbl 1071.74017

Summary: We consider a simple model in one space dimension for the interaction between a fluid and a solid represented by a point mass. The fluid is governed by the viscous Burgers equation and the solid mass, which shares the velocity of the fluid, is accelerated by the difference of pressure at both sides of it. We describe the asymptotic behavior of solutions for integrable data using energy estimates and scaling techniques. We prove that the asymptotic profile of the fluid is a self-similar solution of the Burgers equation with an appropriate total mass, and we describe the parabolic trajectory of the point mass. We also prove that, asymptotically, the difference of pressure to both sides of the point mass vanishes.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
76D99 Incompressible viscous fluids
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References:

[1] Burgers J. M., The Nonlinear Diffusion Equation (1974) · Zbl 0302.60048
[2] Conca C., C. R. Acad. Sci. Paris 328 pp 473– (1999) · Zbl 0937.76012
[3] Conca C., Commun. Partial Diff. Eqns. 25 pp 1019– (2000) · Zbl 0954.35135
[4] Desjardins B., Archive Rat. Mech. Anal. 146 pp 59– (1999) · Zbl 0943.35063
[5] Desjardins B., Comm. Partial Diff. Eqns. 25 pp 1399– (2000) · Zbl 0953.35118
[6] Errate D., C. R. Acad. Sci. Paris 318 pp 275– (1994)
[7] Escobedo M., Archive Rat. Mech. Anal. 124 (1) pp 43– (1993) · Zbl 0807.35059
[8] Gunzburger M., J. Math. Fluid. Mech. 2 (3) pp 219– (2000) · Zbl 0970.35096
[9] San Martin J., Archive Rat. Mech. Anal. 161 (2) pp 113– (2002) · Zbl 1018.76012
[10] Serre D., Japan J. Appl. Math. 4 pp 99– (1987) · Zbl 0655.76022
[11] Simon J., Ann. Mat. Pura Appl. 146 (4) pp 65– (1987)
[12] Véron L., Ann. Fac. Sci. Toulouse 1 pp 171– (1979) · Zbl 0426.35052
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