文档介绍：Introduction to the theory of partialdi?erential equationsEduard FeireislChapter Partial di?erential equationsIn general, apartial di?erential equationis an equation involving anunknown function of two or more independent variables and its partialderivatives with respect to these =u(y1, ..., yN) and a multiindexα={α1, ..., αN}, where ponent is a non-negative integer, we denote?αu=?αy1,...,yNu≡?|α|u?yα11...?yαNN,andDku≡{?αu| |α|=k},withk= 0,1, ...De?nition relation of the formF(Dku(y), Dk?1u(y), ..., Du(y), y) = 0, y∈Q?RN,is called a partial di?erential equation of ,F:RNk×...×RN×R×Q→Ris a given, in general non-linear, function, anduis termed the un- 1. INTRODUCTIONA partial di?erential equation is calledlinearif the functionFislinear with respect tou,Du, ... ,Dku. IfU:Q→RMis a vector ?eld andF:RNk×M×...×RM×Q→RKa (non-linear) mapping, then a relationF?DkU(y), ..., DU,U, y?= 0 fory∈Qis called asystemofK?equations andM?unknowns of best known examples prise:Linear transport equation??tu+NXi=1?xju= ’s equation??u≡NXi=1?2xi,xiu= equation??tu??u= equation??2t,tu??u= follows examples ofnon-linearequations:. DATA, BOUNDARY CONDITIONS3Burger’s equation??tu+u?xu= -deVries equation??tu+u?xu+?3x,x,xu= -Jacobi equation??tu+H(Dxu, x) = Data, boundary conditionsBydatawe mean prescribed quantities appearing as parameters of agiven problem. The functionFitself may be considered as a “datum”.Typical examples of data are related to theboundary conditions-restrictions of the formG(Dku, ..., Du, u, y) = 0 prescribed on a part of the boundary? following example shows that the boundary data must be some-how related to the equation governing the behaviour of solutions a linear transport equation?tu(t, x)??xu(t, x) = 0fort >0, x >0supplemented with boundary conditions?????u(0, x) =u0(x)forx >0,u(t,0) =ub(t)fort >0,?????with giv