The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

http://materias.fi.uba.ar/7538/material/Otros/EymardEtAl%20-%20Finite%20Volume%20Methods.pdf

Finite Volume Methods

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation ∂w/∂t = -(-Δ) α/2 w + w 1+β in the critical dimension d = α/β. By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as t → ∞. In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of α-Laplacians with possibly different parameters a.

/pdf/blow-up-of-semilinear-pde-s-at-the-critical-dimension-a-1wtluvafof.pdf

Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach

Statistics For Spatial Data

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.

/pdf/the-porous-medium-equation-mathematical-theory-42juib4cgk.pdf

The Porous Medium Equation: Mathematical Theory

Soit G un sous ensemble ouvert borne de R N et 1 0 dans G, u=0 sur ∂G ou a∈L ∞ (G) et p*=Np/(N−p)

Quasilinear elliptic equations involving critical Sobolev exponents

Local and global properties of solutions of quasilinear elliptic equations

On determine la structure de l'ensemble des solutions u de −(|u x | p−2 u x ) x +f(u)=λ|u| p−2 u sur (0,1) telles que u(0)=u(1)=0, ou p>1 et λ∈R. On demontre que les solutions avec k zeros sont uniques quand 1 2

/pdf/bifurcation-phenomena-associated-to-the-p-laplace-operator-4r9s0u92x5.pdf

Mohammed Guedda

Papers

Quasilinear elliptic equations involving critical Sobolev exponents

Local and global properties of solutions of quasilinear elliptic equations

Bifurcation phenomena associated to the p-Laplace operator

The Cauchy problem for ut=Δu+|∇u|q

Criticality for Some Evolution Equations