American Institute of Mathematical Sciences

Advanced
June  2004, 3(2): 197-216. doi: 10.3934/cpaa.2004.3.197

Existence of solutions to equations for the flow of an incompressible fluid with capillary effects

D. L. Denny 1,

1. 

Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071, United States

Received  March 2003 Revised  November 2003 Published  March 2004

We study the initial-value problem for a system of equations that models the low-speed flow of an inviscid, incompressible fluid with capillary stress effects. The system includes hyperbolic equations for the density and velocity, and an algebraic equation (the equation of state). We prove the local existence of a unique, classical solution to an initial-value problem with suitable initial data. We also derive a new, a priori estimate for the density, and then use this estimate to show that, if the regularity of the initial data for the velocity alone is increased, then the regularity of the solution for the density and the velocity may be increased, by a bootstrapping argument.
Keywords: Existence, capillary, incompressible, fluid..
Mathematics Subject Classification: 35A07.
Citation: D. L. Denny. Existence of solutions to equations for the flow of an incompressible fluid with capillary effects. Communications on Pure & Applied Analysis, 2004, 3 (2) : 197-216. doi: 10.3934/cpaa.2004.3.197
[1]

Horst Heck, Gunther Uhlmann, Jenn-Nan Wang. Reconstruction of obstacles immersed in an incompressible fluid. Inverse Problems & Imaging, 2007, 1 (1) : 63-76. doi: 10.3934/ipi.2007.1.63
[2]

Youcef Amirat, Kamel Hamdache. On a heated incompressible magnetic fluid model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 675-696. doi: 10.3934/cpaa.2012.11.675
[3]

Martin Kružík, Ulisse Stefanelli, Jan Zeman. Existence results for incompressible magnetoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2615-2623. doi: 10.3934/dcds.2015.35.2615
[4]

Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control & Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315
[5]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations & Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007
[6]

Scott W. Hansen, Andrei A. Lyashenko. Exact controllability of a beam in an incompressible inviscid fluid. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 59-78. doi: 10.3934/dcds.1997.3.59
[7]

I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759
[8]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[9]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553
[10]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337
[11]

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068
[12]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[13]

Bernard Ducomet, Šárka Nečasová. On the motion of rigid bodies in an incompressible or compressible viscous fluid under the action of gravitational forces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1193-1213. doi: 10.3934/dcdss.2013.6.1193
[14]

Alexander Khapalov, Giang Trinh. Geometric aspects of transformations of forces acting upon a swimmer in a 3-D incompressible fluid. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1513-1544. doi: 10.3934/dcds.2013.33.1513
[15]

Tian Ma, Shouhong Wang. Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 459-472. doi: 10.3934/dcds.2004.10.459
[16]

Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib, Said Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems & Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044
[17]

Quan Wang. Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 543-563. doi: 10.3934/dcdsb.2014.19.543
[18]

Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765
[19]

Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541
[20]

Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences