Hyperbolic Partial Differential Equations

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Jason Ramsey
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1 Hyperbolic Partial Differential Equation Evolution equation aociated with irreverible phyical procee like diffuion heat conduction lead to parabolic partial differential equation. When the equation i a model for a reverible phyical proce like propagation of acoutic or electromagnetic wave, then the evolution equation i generally hyperbolic. he mathematical model uually begin with a conervation tatement that i ome verion of Newton econd law d mv F which, when applied to an arbitrary ball B inide the domain U R n where the proce i occuring, often take the form t B t u dx B F x,t n ds B f dx momentum urface traction body force In the uual way, thi lead to B t t u divf f dx 0 B U, ince B i arbitrary, t t ux,t divf x,t fx,t forx,t U0, If u here repreent, ay, the deflection from the equilibrium tate for an elatic membrane, then we will have a a contituitive relation, then F x,t Aux,t tt ux,t Aux,t fx,t forx,t U0,. o complete the pecification for a well poed problem, we would add two initial condition ux,0 u 0 x t ux,0 u x for x U, boundary condition of ome ort, ay ux,t 0 for x U, t 0,. We will repreent thi problem, in general, in the form tt ux,t Lux,t fx,t in U ux,0 u 0 x in U t ux,0 u x in U u 0 on U0, where L denote a econd order operator which i uniformly elliptic on U with coefficient which we will uppoe to be in C U.In thi cae, the initial boundary value problem i aid to be "hyperbolic". Now we are going to try to devie a framework in which thi problem can be viewed a if it behave in more or le the ame way a an ordinary differential equation

2 for a real, vector valued function of t. Becaue of what we already know about the elliptic operator L, if our olution u were to belong to L 0, : V, where V H 0 U,( H H 0 U, V H U ) then Lux,t L 0, : V if f L 0, : H L U L 0, : V, then thi impoe the condition tt u L 0, : V.hen thi create the following ituation with regard to the pace where u it time derivative are found, which ugget u L 0, : V tt u L 0, : V t u L 0, : H; i.e., tt u,u V V ttu,u H t u, t u H t u H. hen, ince L 0, : V L 0, : H L 0, : V, we have u, t u L 0, : H hence u C0, : H t u, tt u L 0, : V hence t u C 0, : V Here we made ue of the fact that if a function it firt derivative are both quare integrable on 0, then the function mut, in fact, be Holder continuou on 0, (of order ).hi permit the following interpretation of the initial condition, u,t u 0 H a t 0 t u,t u V a t 0. hen a reaonable weak formulation of () might be the following: given f L 0, : H, u 0 H, u V find u L 0, : V, with t u L 0, : H tt u L 0, : V uch that tt u,v V V Bu,v,t f,v V V v V a.e. in0, u0 u 0 t u0 u In fact, we are going to aume the initial data i uch that u 0 V, u H then it remain to be een if we can how that thi weak problem ha a unique olution for all admiible data. heorem For all data f L 0, : H u 0 V, u L 0, : V, with t u L 0, : H tt u L u H, there exit a unique 0, : V uch that i) tt u,v V V Bu,v,t f,v H v V a.e. in0, ii) u0 u 0 t u0 u Moreover, the olution mapping i continuou. L 0, : HHV : f,u 0,u u, t u L 0, : VL 0, : H Proof- (uniquene) We will firt prove that the olution i unique if it exit. Let u denote a

3 olution of the weak problem () correponding to data, u 0 u f 0. We would like to proceed by writing 0 u,u V V Bu,u,t 0. However, ince u belong to L 0, : H, not to L 0, : V, neither of the term u,u V V, Bu,u,t i defined. Intead, we mut define for a fixed 0,, vt t u d if 0 t 0 if t hen v V for all t, 0, tt 0 u,v V V Bu,v,t 0 Now u 0 0, v 0 o, 0 u,v V Bu,v,t 0 V Next, we oberve that v t ut for 0 t, hence hi implie that where hi lead to 0 u,u V V Bv,v,t 0 0 d ut H Bv,v,t 0 Cu,v,tDv,v,t Cu,v,t U ub v uv b dx Du,v,t U v tauv t b uuv t cdx. u H Bv0,v0,t 0 Cu,v,tDv,v,t u H v0 V C 0 ut H vt V Now write v0 0 t u d : wt expre thi lat etimate in term of w, u H w V C 0 ut H wt w V But hence wt w V wt V w V 3

4 u H Cw V C 0 ut H wt V If we chooe 0 ufficiently mall that C,then we will have u H w V C 0 ut H wt V for 0 If we let U 0 ut H wt V, thi etimate aert i.e., U C U; U U0 e C. But U0 0 o U 0 for 0 thi implie u 0 for 0. We can repeat thi argument on,,,3,,n,n for n in order to eventually conclude that ut 0 for 0 t. hi prove the uniquene of the weak olution. Oberve that weakening the notion of the olution to the IBVP enlarge the cla of admiible olution o that proving exitence become eaier in general. At the ame time, however, proving uniquene in a larger cla uually become more difficult, a thi proof illutrate. he exitence proof, like the proof for the exitence of a olution to the parabolic problem will proceed in a erie of tep. ) Exitence of Approximate Solution- Let w k denote an orthonormal bai for H that i, imultaneouly, an orthogonal bai for V.hen for each poitive integer N, define N u N t C j,n t w j j where the C j,n are required to atify, for each k, k N, i) u N "t,w k H Bu N,w k,t f,w k H 3 ii) C k,n 0 u 0,w k H C k,n 0 u,w k H Here we are uing the fact that u,v V V u,v H.Now (3) i equivalent to, N t j C k,n Bw j,w k,tc k,n t f k t C k,n 0 u 0,w k H C k,n 0 u,w k H which i a ytem of econd order linear ODE of the form, C N t B jk tc Nt f t, C N0 U 0, C N 0 U, where the coefficient matrix, B jk t i uniformly poitive definite on 0,.It i well known that uch a ytem ha a unique global olution, C Nt, for each N, hence there exit for each N, a unique approximate olution u N t for the weak boundary problem. 4

5 ) Energy Etimate a) max 0t u Nt V u N t H C u 0 V u H f L U 4 b) u N L 0,:V C u 0 V u H f L U It follow from (3) that u N "t,u N H Bu N,u N,t f,u N H u N "t,u N H d u N t,u N H d u N t H Bu N,u N,t u U N Au N dx u U N bu N c u N u N dx : B u N,u N,tB u N,u N,t We can aume WLOG that the matrix A i ymmetric, which lead to B u N,u N,t u U N Au N dx d u N Au N dx U u N A u N dx U B u N,u N,t d U u N Au N dx Cu N V In addition, B u N,u N,t C u N V u N H, combining thee lead to d u N t H U u N Au N dx C u N H u N V f H Note here that in order to etimate f,u N H in term of u N H, we need f in L 0, : H not in L 0, : V where we might have thought it hould be. Next we oberve that by the ellipticity aumption, we have U u N Au N dx a 0 u N V which implie that If we let, d u N t H U u N Au N dx C u N H U u N Au N dx f H Ut : u N t H U u N Au N dx. then d Ut C Ut d ect Ute Ct C f H. it follow that Ut atifie, 5

6 Ut U0 C 0 t fh d e Ct But, U0 u N 0 H U u N 0 Au N 0dx thi lead to C u H u 0 V u N t H u N U t Au N tdx C u H u 0 V f L U. Finally, we can make ue of the ellipticity aumption together with the Poincare inequality to arrive at the concluion u N t H u N t V C u H u 0 V f L U 5 ince thi lat etimate hold for all t 0,, max u N t 0t H u N t V C u H u 0 V f L U hi prove (4a). Alternatively, we could integrate (5) to obtain u N L 0,,H u N L 0,,V C u H u 0 V f L U.. 6 Next, fix v V with v V, write v v v where v panw,,w N M N v M N. hen v V v V u N,v V V u N,v H f,v H Bu N,v. It follow then, uing the Cauchy-Schwartz inequality the boundedne of B, that hence u N,v V V u N t V up v V Cft H u N t V u N,v V V Cft H u N t V un t 0 V C fth u N t 0 V thi i equivalent to (4b). 3. Exitence of Weak Solution C f L U he energy etimate (4) imply that u N t i bounded in L 0, : V u N t i bounded in L 0, : H u N t i bounded in L 0, : V u H u 0 V 6

7 it follow that there exit a ubequence u n t u N t uch that u n t converge weakly to u in L 0, : V u n t converge weakly to v in L 0, : H u n t converge weakly to w in L 0, : V In the uual way, making ue of the fact that L 0, : V L 0, : H L 0, : V D 0, : V we get that u v, u v w.it remain now to how that thi weak limit point i a weak olution, i.e., that it atifie (). Let For m n, m V m vt d j t w j : d j t C 0, j m. j 0 un "t,v H Bu n,v,t 0 f,vh for all v V m For m n, let n tend to infinity ue the weak convergence reult to get 0 u"t,vh Bu,v,t 0 f,vh for all v V m m0 Since w k i a bai for V,it follow that V m L 0, : V therefore, m0 0 u"t,vh Bu,v,t 0 f,vh for all v L 0, : V. In addition, u C0, : H, u C0, : V hence ut u0 in H u t u 0 in V a t 0. Finally, for any v C 0, : V uch that v v 0, we have 0 u,v H Bu,v,tf,v H u 0,v0 u0,v 0 hen 0 un,v H Bu n,v,tf,v H u n 0,v0 u n 0,v 0 0 uun,v H Buu n,v,t u 0 u n 0,v0 u0 u n 0,v 0 the weak convergence of the ubequence u n implie that the left ide of thi equation tend to zero a n tend to infinity. On the right ide, recalling (3ii), we have u n 0 converge in V to u 0 u n 0 converge in H to u,which implie u0 u 0 u 0 u. hen it follow that u i a weak olution of the IBVP. But then every ubequence of the equence of approximate olution, u N,mut converge to a weak olution. Since the weak olution ha been hown to be unique, it follow that all ubequence have the ame weak 7

8 limit. But in thi cae, the equence u N, mut itelf converge, weakly, to the weak olution. Note that the etimate (5) applie to u lim u N which how that the mapping L 0, : HHV : f,u 0,u u, t u L 0, : VL 0, : H i continuou. Finally notice that if the initial condition in (), ux,0 u 0 x t ux,0 u x, were replaced by final condition, ux, u 0 x t ux, u x, then the new problem till admit a unique weak olution which depend continuouly on the data. hi i in contrat to the parabolic problem where the final value problem i not well poed. 8

c n b n 0. c k 0 x b n < 1 b k b n = 0. } of integers between 0 and b 1 such that x = b k. b k c k c k

c n b n 0. c k 0 x b n < 1 b k b n = 0. } of integers between 0 and b 1 such that x = b k. b k c k c k 1. Exitence Let x (0, 1). Define c k inductively. Suppoe c 1,..., c k 1 are already defined. We let c k be the leat integer uch that x k An eay proof by induction give that and for all k. Therefore c n