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The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics
A simple proof of wellposedness for the freesurface incompressible Euler equations
1.  Maxwell Institute for Mathematical Sciences and department of Mathematics, HeriotWatt University, Edinburgh, EH14 4AS, United Kingdom 
2.  Department of Mathematics, University of California, Davis, CA 95616 
[1] 
Chengchun Hao. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities. Discrete and Continuous Dynamical Systems  B, 2015, 20 (9) : 28852931. doi: 10.3934/dcdsb.2015.20.2885 
[2] 
Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations and Control Theory, 2012, 1 (2) : 297314. doi: 10.3934/eect.2012.1.297 
[3] 
Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems  B, 2018, 23 (1) : 193202. doi: 10.3934/dcdsb.2018013 
[4] 
Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3D water waves. Discrete and Continuous Dynamical Systems  B, 2002, 2 (1) : 134. doi: 10.3934/dcdsb.2002.2.1 
[5] 
R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbationparameter techniques. Communications on Pure and Applied Analysis, 2012, 11 (4) : 14971522. doi: 10.3934/cpaa.2012.11.1497 
[6] 
Kunquan Li, Yaobin Ou. Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry. Discrete and Continuous Dynamical Systems  B, 2022, 27 (1) : 487522. doi: 10.3934/dcdsb.2021052 
[7] 
Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems  B, 2016, 21 (5) : 14551468. doi: 10.3934/dcdsb.2016006 
[8] 
Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete and Continuous Dynamical Systems  S, 2018, 11 (3) : 465476. doi: 10.3934/dcdss.2018025 
[9] 
Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386397. doi: 10.3934/proc.2001.2001.386 
[10] 
Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete and Continuous Dynamical Systems  B, 2014, 19 (3) : 789799. doi: 10.3934/dcdsb.2014.19.789 
[11] 
Elena Kartashova. Nonlinear resonances of water waves. Discrete and Continuous Dynamical Systems  B, 2009, 12 (3) : 607621. doi: 10.3934/dcdsb.2009.12.607 
[12] 
Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 31033131. doi: 10.3934/dcds.2015.35.3103 
[13] 
Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 33573385. doi: 10.3934/dcds.2018144 
[14] 
Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible freeboundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503542. doi: 10.3934/eect.2019025 
[15] 
Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 68256842. doi: 10.3934/dcds.2019233 
[16] 
Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations and Control Theory, 2017, 6 (3) : 319344. doi: 10.3934/eect.2017017 
[17] 
Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete and Continuous Dynamical Systems  S, 2014, 7 (4) : 673693. doi: 10.3934/dcdss.2014.7.673 
[18] 
Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic twophase free boundary problems. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 69616978. doi: 10.3934/dcds.2019239 
[19] 
Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365374. doi: 10.3934/proc.2013.2013.365 
[20] 
Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 415421. doi: 10.3934/dcdsb.2018179 
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