On the Cauchy Problem for Von Neumann-Landau Wave Equation

Transcription

1 Journal of Applie Mathematics an Physics Publishe Online December 4 in SciRes On the Cauchy Problem for Von Neumann-anau Wave Equation Chuangye iu Minmin iu aboratory of Nonlinear Analysis Department of Mathematics Central China Normal University Wuhan China School of Science Wuhan Institute of Technology Wuhan China chuangyeliu3@6com ocbmml@6com Receive October 4; revise 6 November 4; accepte December 4 Copyright 4 by authors an Scientific Research Publishing Inc This wor is license uner the Creative Commons Attribution International icense (CC BY Abstract In present paper we prove the local well-poseness for Von Neumann-anau wave equation by the T Kato s metho Keywors Von Neumann-anau Wave Equation Strichartz Estimate Cauchy Problem Introuction For the stationary Von Neumann-anau wave equation Chen investigate the Dirichlet problems [] where the generalize solution is stuie by Function-analytic metho The present paper is relate to the Cauchy problem: the Von Neumann-anau wave equation i tu = ( x + y u+ f ( u ( u( xy = u ( xy where = x n i= x i for ( n x = x x n ( utxy is an unnown complex value function on an f is a nonlinear complex value function If the plus + is replace by the minus on right han in Equation ( then the resulte equation is the Schröinger equation For the Schröinger equation the well-poseness problem is investigate for various nonlinear terms f In terms of the nonlinear terms f the problem ( can be ivie into the subcritical case an the critical case for H solutions We are concerne with the subcritical case an obtain a local well- + n How to cite this paper: iu CY an iu MM (4 On the Cauchy Problem for Von Neumann-anau Wave Equation Journal of Applie Mathematics an Physics 4-3

2 C Y iu M M iu poseness result by the T Kato s metho The paper is organize as follows Section contains the list of assumptions on the interaction term f an the main result is presente Section 3 is concerne with the Strichartz estimates Finally in Section 4 the main result is prove Statement of the Main Result In this section we list the assumptions on the interaction term f an state the main result Firstly we recall that the efinition of amissible pair [] Definition Fix = n n We say that a pair ( qr of exponents is amissible if = q r ( an r ( r < if = Remar The pairs ( is always amissible so is the calle the enpoint cases f C satisfy an Seconly let for all uv such that u v M with 4 K( t C ( + t < < where C is a constant inepenent of t Set (3 if > The two pairs are f ( = (4 f ( u f ( v K( M u v (5 ( f u x = f u x (7 + n for all measurable function u an ae x Finally let us mae the notion of solution more precise Definition et I be an interval such that I We say that u is a strong u C I H satisfies the integral equation if t it ( s ( (6 H -solution of ( on I it u t = e u i e f u s s (8 for all t I where = : x + y The main result is the following theorem: Theorem Suppose f C satisfy (4-(6 If f (consiere as a function n et is of class C then the Cauchy problem ( is locally well pose in H ( More specially the following properties hol: (i For any R > there exists a time T = T( R > an constant c = c( such that for each u { ϕ ϕ } B : = H : R there exists a unique strong in the ball R H ( ( in C[ TT ] H ( such that H -solution u to the Equation 5

3 C Y iu M M iu where r = + an ( (ii The map u (iii For every u H u u cu + TT H (9 TT W H ( ( q ( r ( ( qr is an amissible pair u is continuous from R B to C[ TT] H ( ; the unique solution u is efine on a maximal interval ( Tmin Tmax Tmax = Tmax ( u ( ] an Tmin = Tmin ( u ( ]; (iv There is the blowup alternative: If T max < then u( t + as t T H ( max T min < then u( t + as t T H ( min Remar It follows from Strichartz estimates that for any amissible pair ( γ ρ γ loc ( ρ min max u T T W with (respectively if Remar 3 For the Schröinger equations the similar results hol [] It implies a fact that the ellipticity of the operator x y is not the ey point in the local well-poseness problem 3 Strichartz Estimates In this subsection we recall that the Strichartz estimates et ( ξη enote a general Fourier variable in = ( = ( et : ξ ξ ξ n have η η η n x y n = + then by Fourier transform(enoting by or ^ we u (( ξ η uˆ = ( n n for any u H ( It is easy to verify that the is a self-ajoint unboune operator on n it n the omain H ( Then by Stone theorem we see that e is an unitary group on e it can be expresse explicitly by Fourier transform n for any e it ( ξ η ϕ By the irect compute we conclue ( ϕ with Moreover ϕ e it = ˆ ϕ ( ix x iy y it 4t 4t n n ( 4πit e xy = e e ϕ x y x y ( it The following result is the funamental estimate for e emma If p [ ] an t then e it p p maps ( continuously to an p it e ϕ 4π t ϕ p p ( (3 where p is the ual exponent of p efine by the formula + = p p Proof For the proof please see [3] or [4] The following estimates nown as Strichartz estimates are ey points in the metho introuce by T Kato [5] qr be any amissible exponents Then we have the homogeneous Strichartz emma et ( qr an 6

4 C Y iu M M iu estimate the ual homogeneous Strichartz estimate an the inhomogeneous Strichartz estimate t it ( s it e (4 ϕ ( ϕ q r qr ( φ ( t t φ qr q r (5 ( e it φ( s s qrqr φ q r ( ( J J ( t e q r (6 for any interval J an real number t Proof For the proof please see [3] or [4] in the non-enpoint case On the other han the proof in the enpoint case follows from the theorem in [6] an the lemma in the present paper 4 The Proof of Theorem Proof et χ C be such that ( z χ = for = χ = ( χ f z z f z f z z f z one easily verifies that for any zw f z f w z w Set fl ( u( x fl ( u( x z an ( z f z f w z + w z w χ = for z Setting = for l = Using (7 we euce from Höler s inequality that f u f v u v ( ( f( u f( v r u v + u v An it follows from Remar 3 (vii in [] that f u u r r r ( ( ( ( r ( ( f u u u r r ( ( ( We now procee in four steps Step Proof of (i Fix AT> to be chosen later an let r missible pair an set I ( TT equippe with the istance = Consier the set = + q be such that ( q r { ( ( : ( ( } q r I H IW (7 (8 (9 qr is an a- E= u IH IW u A u A ( + q r ( I ( I u v = u v u v ( We claim that ( E is a complete metric space Inee let { u } { u } q r is also a Cauchy sequence in I ( an E be a Cauchy sequence Clearly I In particular there exists a 7

5 C Y iu M M iu function ( q ( r ( u I I such that u Applying theorem 5 in [] twice we conclue that an that thus u u in E as Taing up any uv E Since ( ( q r ( u in I u IH IW u liminf u A ( ( ( IH IH u liminf u A; ( ( ( ( q r q r IW IW I as an q r ( f is continuous ( ( it follows that f ( u : I ( is measurable an we euce easily that f ( u ( I ( Similarly since r ( r ( q r we see that f ( u I (iii in [] We euce the following: an f is continuous Using inequalities (8 an (9 an Remar q r ( ( ( f u I H f u I w f u u ( IH ( ( IH ( q r f u u u f u f v u v r q r ( Iw ( ( I ( ( IW ( ( I ( ( I ( f ( u f ( v u v u v q r + Using the embeing H ( r ( estimates that an an Given u H r r q r ( I ( ( I ( ( I ( ( I ( an Höler s inequality in time we euce from the above qq f( u ( ( + f u q r T T A A I H IW + + ( qq f( u f( v + f ( ( u f ( v q r ( ( ( I I T + T + A u v (3 For any u E let ( u be efine by it t it ( s = It follows from ( an Strichartz estimates (lemma that ( u t e u i e f u s s (4 q r ( ( ( [ ] u C TT H TT W (5 ( u ( ( u ( C ( u qq q r T T ( A A I H IW H (6 8

6 C Y iu M M iu Also we euce from (3 that qq ( u ( v ( ( u ( v ( ( ( ( + I q I r C T + T + A u v (7 Finally note that q > q We now procee as follows For any an we let T T( R = be the unique positive number so that It then follows from (6 an (8 that for any u BR Thus ( u qq C ( T + T ( + A = R u we set = H ( ( A C R ( u + ( ( u q r C ( ( u + A C ( R+ A= A I H IW H (9 E an by (7 we obtain (8 ( ( u ( v ( uv (3 In particular is a strict contraction on E By Banach s fixe-point theorem has a unique fixe point u E; that is u satisfies (8 By (5 u= ( u C[ TT ] H ( con- clue that u is a strong then the estimate (9 hols for c C ( H -solution of ( on [ TT] = by letting For uniqueness assume that uv are two strong u Then we have For simplicity we set for l = an w u v obtain t it ( s By the efinition we Note that T( R is ecreasing on R R = u in (9 H H -solution of ( on [ TT ] ( ( with the same initial value ut vt = i e f us f vs s (3 ( ( t it ( s l = l l w t i e f u s f v s s = For any interval J ( TT ( ( by (8 an Strichartz estimates (6 then we w w f u f v w + (3 ( J ( ( q r J J J Similarly for w we have w w f u f v + ( ( q ( ( ( ( r ( ( r ( q r J J q J u + v w IH IH J Note that w = w w + Then it follows from that where the constant w q r ( ( ( q r + w C + B w + w J J ( J ( J (34 IH IH B = u + v an the constant C is inepenent of J by above ( ( inequalities Note that q < q we conclue that w = by the lemma 4 in [] So u = v ( Step Proof of (ii Suppose that u u in B R as By the part (i we enote (33 u an u by 9

7 C Y iu M M iu the unique solution of ( corresponing to the initial value ( u an u respectively We will show that u u in C[ TT ] H ( as Note that it ( ( ( u t u t = e u u + u u (35 an the estimate (9 which implies that (7 hols for v = u Note that the choosing of the time T in (8 it follows from (7 with (3 that Hence we have ( ( u u u u + ( u u (36 u u u u u u (37 ( + TT TT Next we nee to estimate ( u u (( TT ( ( ( q ( r ( ( Note that commutes with e it an so t it ( s ( it u t = e u i e f u s s (38 u We use the assumption A similar ientity hols for f C which implies that f u f u u f u is a real matrix Therefore we may write = where it ( t it ( s ( Note that f an f are also t it ( s ( u u t = e u u i e f u u u s i e f u f u u s C so that f f f = + an from (7 we euce that 3 (39 f z C an f z z for any z an some constant C 3 Therefore arguing as in Step we obtain the estimate ( (( ( u u + u u ( (( ( q r TT TT By choosing T T( R ( u u + T u u TT qq + T u u u ( ( ( TT q r ( ( TT ( ( ( TT ( f u f u u + + ( f ( u f ( u u q r (( TT = as (8 an noting that u B from (4 we obtain that R (4 ( (( ( u u + u u ( (( ( q r TT TT ( ( ( ( ( ( TT ( u u + f u f u u (4 + ( f ( u f ( u u q r (( TT ( There if we prove that 3

8 C Y iu M M iu as then we have ( ( (( ( ( ( (( ( f u f u u + f u f u u (4 q r TT TT ( ( ( ( ( ( u u + u u (43 q r TT TT as which combine with (37 yiels the esire convergence we prove (4 by contraiction an we assume that there exists ε > an a subsequence which we still enote by { u } ( ( (( ( ( ( (( ( such that f u f u u + f u f u u ε q r TT TT By using (37 an possibly extracting a subsequence we may assume that such that u an that there exists v q ( TT r ( ( f ( u f ( u u an ( f ( u f ( u u an converge to ae on ( TT ( f ( u f ( u u C3 u ( T T ( u (44 u ae on ( TT v ae on ( TT In particular both Since q r ( f ( u f ( u u ( u + u u ( v + u u ( T T ( we obtain from the ominate convergence a contraiction with (44 u H an let Step 3 Proof of (iii Consier = { > [ ]} Tmax u sup T : there exists a solution of on T = { > [ ]} Tmin u sup T : there exists a solution of on T It follows from part (i there exists a solution ( min max u C T T H of ( Step 4 Proof of (iv Suppose now that T max < an assume that there exist M < an a sequence tj Tmax such that u( tj M H ( initial ata u t one can exten u up to t T( M et be such that t T( M T ( u ( + > By part (i from the max + which contraicts maximality Therefore u t as t T H max One shows by the same argument that if T min < then This completes the proof Acnowleegments u t as t T ( H min x We are grateful to the anonymous referee for many helpful comments an suggestions which have been incorporate into this version of the paper C iu was supporte in part by the NSFC uner Grants No an the Funamental Research Funs for the Central Universities An M iu was 3

9 C Y iu M M iu supporte by science research founation of Wuhan Institute of Technology uner grants No 4 References [] Chen Z (9 Dirichlet Problems for Stationary von Neumann-anau Wave Equations Acta Mathematica Scientia [] Cazenave T (3 Semilinear Schröinger Equations Courant ecture Notes in Mathematics New Yor University Courant Institute of Mathematical Sciences AMS [3] Tao T (6 Nonlinear Dispersive Equations: ocal an Global Analysis CBMS Regional Conference Series in Mathematics Vol 8 American Mathematical Society Provience [4] inares F an Ponce G (9 Introuction to Nonlinear Dispersive Equations [5] Kato T (987 On nonlinear Schröinger Equations Annales e l IHP Physique Théorique [6] Keel M an Tao T (998 Enpoint Strichartz Estimates American Journal of Mathematics

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