A Free Boundary Approach to Two-Dimensional Steady Capillary Gravity Water Waves

We prove that suitable weak solutions of the two-dimensional steady capillary gravity water wave problem (even in the presence of vorticity) are smooth. On a technical level, solutions are closely related to critical points of the Mumford–Shah functional, so that our main task is to exclude cusps pointing into the water phase. We prove that cusps do not exist, and that the free surface is smooth.     

边界元法边界层处理的一种新方法—拟多连域法

提出了处理边界元法边界层问题的新方法。这种方法在求得边界未知量后，在距边界较远的域中构造一工作边界，将实际边界与工作边界看作一多连域问题求得工作边界的未知量。再将工作边界与边界层边界视为一多连域问题求解，可得到满足精度要求的边界附近点的位移与应力。这种方法理论简洁、计算方便、有效、精度高，对于需求边界上多点的未知量问题很具优越性     

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation

Energy Approach to the Formulation of Boundary Problems of Nonlinear Thermomechanics for Elastic Systems

An energy approach is proposed to derive the physical constitutive equations of nonlinear thermomechanics for inertial elastic systems. A potential of local inertial thermodynamic state and a potential of thermoelastic energy dissipation are introduced. The variational formulation of nonlinear boundary problems of thermoelasticity is implemented on the basis of the Hamiltonian energy functional. Sufficient conditions for the convexity of the functional are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 52–59, September 2005.     

A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations

In this paper we study the fully nonlinear free boundary problem $$\left\{\begin{array}{ll}F(D^{2}u) = 1 & {\rm almost \, everywhere \, in}\, B_{1} \cap \Omega\\ |D^{2} u| \leqq K & {\rm almost \, everywhere \, in} \, B_{1} \setminus \Omega,\end{array}\right.$$ where K > 0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W ^2,n solutions are locally C ^1,1 inside B ₁. Under the extra condition that ${\Omega \supset \{D{u} \neq 0 \}}$ and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary ${\partial \Omega \cap B_1}$ .     

Hausdorff Measure Estimates and Lipschitz Regularity in Inhomogeneous Nonlinear Free Boundary Problems

In this paper, we prove a Hausdorff measure estimate for the free boundaries of subsolutions of fully nonlinear and quasilinear equations of the type \({F(D^2u,x)\geqq f(x)}\) and \({{\rm div}\,A(x,\nabla u)\geqq \mu}\) where \({f \in L^{q}, q >N}\) and μ is a signed Radon measure with some appropriate growth condition. Gradient estimates for nonnegative harmonic functions with bounded normal derivatives along the boundary obtained by Caffarelli and Salsa (Geometric Approach to Free Boundary Problems, 2005) are extended to the context of inhomogeneous problems involving fully nonlinear and p-Laplace equations. As an application, Lipschitz regularity is obtained for one phase solutions of inhomogeneous nonlinear free boundary problems.     

Homogenization of the Free Boundary Velocity

In this paper we investigate some free boundary problems with space-dependent free boundary velocities. Based on maximum principle-type arguments, we show the uniform convergence of the solutions in the homogenization limit. The main step is to show the uniqueness of the limiting free boundary velocity, which turns out to be a continuous function of the normal direction of the free boundary.     

Entrainment Boundary Conditions for Free Shear Flows

In this paper the performance of different free stream boundary conditions has been investigated. By performing two dimensional free jet calculations, a test case for which an analytical solution is available, it has been shown that boundary conditions in which the pressure is prescribed perform significantly better than boundary conditions in which the velocity is prescribed. Furthermore, it has been shown that pressure boundary conditions allow for much smaller computational domains than velocity (free-slip) boundary conditions.     

Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      11. A Parabolic Free Boundary Problem Modeling Electrostatic MEMS      Joachim Escher  Philippe Laurençot  Christoph Walker 《Archive for Rational Mechanics and Analysis》2014,211(2):389-417 The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right-hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time, while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.      12. An Approach to Rotating Boundary Layers Based on Vector Radon Measures      Yoshikazu Giga  Jürgen Saal 《Journal of Mathematical Fluid Mechanics》2013,15(1):89-127 In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.      13. Backward Free Convection Boundary Layers in Porous Media      Magyari  E.  Keller  B. 《Transport in Porous Media》2004,55(3):285-300 The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = –1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reported.      14. 局部损伤梁动力问题的近似计算方法   总被引：1，自引：0，他引：1   潘旦光  董聪 《应用力学学报》2005,22(1):119-122 应用模态摄动法求解局部损伤梁动力特性。这一方法是在无损伤梁的模态子空间内实施,将含局部损伤梁的微分方程的求解转化为代数方程求解。通过算例,表明这一方法可有效的简化计算,同时计算结果具有较高的精度。      15. On Hydrostatic Free Surface Problems      Michael Renardy 《Journal of Mathematical Fluid Mechanics》2011,13(1):89-93 We consider the hydrostatic Euler equations in a strip bounded by two free surfaces. We show that the initial value problem is well-posed only if the horizontal velocity is uniform in every cross section of the strip.      16. Invariant Boundary‐Value Problems of an Optimally Controlled Boundary Layer      K. G. Garaev  V. A. Ovchinnikov 《Journal of Applied Mechanics and Technical Physics》2003,44(1):27-31 The group properties of equations of the variational problem on finding a continuous law for velocity distribution of liquid injection into an incompressible laminar boundary layer in a planar case, which ensures the minimum friction force acting on the airfoil, are considered. It is shown that the optimal injection velocity on a wedge with x=0 is finite.      17. Instabilities in a Two-Dimensional Combustion Model with Free Boundary   总被引：2，自引：0，他引：2   Claude-Michel Brauner  Alessandra Lunardi 《Archive for Rational Mechanics and Analysis》2000,154(2):157-182 We prove instability of the planar travelling wave solution in a two-dimensional free boundary problem modelling the propagation of near- equidiffusional premixed flames in the whole plane. We reduce the problem to a fixed boundary fully nonlinear parabolic system. The spectrum of the linearized operator contains an interval [0,ω c ], ω c > 0, so we cannot construct backward solutions. We use an argument about stability of dynamical systems in Banach spaces in order to prove pointwise instability of the moving front. (Accepted: January 31, 2000)?Published online August 21, 2000      18. Application of the Local Boundary Integral Equation Method to Boundary-Value Problems   总被引：5，自引：0，他引：5   Sladek  J.  Sladek  V.  Atluri  S. N. 《International Applied Mechanics》2002,38(9):1025-1047 A review of the meshless formulations based on local boundary integral equation (LBIE) methods is presented. Physical quantities are approximated by the moving least-squares method. A summary of recent developments in the application of the LBIE method to potential problems, elastostatics, elastodynamics, thermoelasticity, and plate bending problems is given. The efficiency and generality of the present formulation in a wide class of engineering problems are confirmed.      19. Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian      Ray Yang 《Archive for Rational Mechanics and Analysis》2013,208(3):693-723 We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli et al. (J Eur Math Soc (JEMS) 12(5):1151–1179, 2010). A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be zero along the boundary.      20. SPH固壁边界处理方法的改进      龚凯  刘桦  王本龙 《力学季刊》2008,29(4) 提出一种适用于光滑质点水动力学（SPH）方法的改进的边界处理方法。在这种方法中,边界粒子的压力可通过其周围的流体粒子的压力插值得到,从而改进了耦合边界法在边界上压力不准的问题。运用这种改进的边界处理方法模拟了二维方形水箱中的非线性晃荡问题以及二维楔形体自由入水问题。模拟结果与实验结果吻合较好,证明了此改进的边界处理方法是有效的。

A Parabolic Free Boundary Problem Modeling Electrostatic MEMS

The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right-hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time, while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.     

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.     

Backward Free Convection Boundary Layers in Porous Media

The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = –1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reported.     

局部损伤梁动力问题的近似计算方法

应用模态摄动法求解局部损伤梁动力特性。这一方法是在无损伤梁的模态子空间内实施,将含局部损伤梁的微分方程的求解转化为代数方程求解。通过算例,表明这一方法可有效的简化计算,同时计算结果具有较高的精度。     

On Hydrostatic Free Surface Problems

We consider the hydrostatic Euler equations in a strip bounded by two free surfaces. We show that the initial value problem is well-posed only if the horizontal velocity is uniform in every cross section of the strip.     

Invariant Boundary‐Value Problems of an Optimally Controlled Boundary Layer

The group properties of equations of the variational problem on finding a continuous law for velocity distribution of liquid injection into an incompressible laminar boundary layer in a planar case, which ensures the minimum friction force acting on the airfoil, are considered. It is shown that the optimal injection velocity on a wedge with x=0 is finite.     

Instabilities in a Two-Dimensional Combustion Model with Free Boundary

We prove instability of the planar travelling wave solution in a two-dimensional free boundary problem modelling the propagation of near- equidiffusional premixed flames in the whole plane. We reduce the problem to a fixed boundary fully nonlinear parabolic system. The spectrum of the linearized operator contains an interval [0,ω ^c ], ω ^c > 0, so we cannot construct backward solutions. We use an argument about stability of dynamical systems in Banach spaces in order to prove pointwise instability of the moving front. (Accepted: January 31, 2000)?Published online August 21, 2000     

Application of the Local Boundary Integral Equation Method to Boundary-Value Problems

A review of the meshless formulations based on local boundary integral equation (LBIE) methods is presented. Physical quantities are approximated by the moving least-squares method. A summary of recent developments in the application of the LBIE method to potential problems, elastostatics, elastodynamics, thermoelasticity, and plate bending problems is given. The efficiency and generality of the present formulation in a wide class of engineering problems are confirmed.     

Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli et al. (J Eur Math Soc (JEMS) 12(5):1151–1179, 2010). A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be zero along the boundary.     

SPH固壁边界处理方法的改进

提出一种适用于光滑质点水动力学（SPH）方法的改进的边界处理方法。在这种方法中,边界粒子的压力可通过其周围的流体粒子的压力插值得到,从而改进了耦合边界法在边界上压力不准的问题。运用这种改进的边界处理方法模拟了二维方形水箱中的非线性晃荡问题以及二维楔形体自由入水问题。模拟结果与实验结果吻合较好,证明了此改进的边界处理方法是有效的。