A complete global solution to the pressure gradient equation

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1 J. Diffeentil Equtions ) A complete glol solution to the pessue gdient eqution Zhen Lei,,c, Yuxi Zheng d, School of Mthemticl Sciences, Fudn Univesity, Shnghi 20033, PR Chin School of Mthemtics nd Sttistics, Nothest Noml Univesity, Chngchun 3002, PR Chin c Key Lotoy of Mthemtics fo Nonline Sciences Fudn Univesity), Ministy of Eduction, PR Chin d Deptment of Mthemtics, Pennsylvni Stte Univesity, Stte College, PA 6802, USA Received 5 Septeme 2006; evised 29 Jnuy 2007 Aville online 3 Feuy 2007 Astct We study the domin of existence of solution to Riemnn polem fo the pessue gdient eqution in two spce dimensions. The Riemnn polem is the expnsion of qudnt of gs of constnt stte into the othe thee vcuum qudnts. The glol existence of smooth solution ws estlished in Di nd Zhng Z. Di, T. Zhng, Existence of glol smooth solution fo degenete Goust polem of gs dynmics, Ach. Rtion. Mech. Anl ) ] up to the fee oundy of vcuum. We pove tht the vcuum oundy is the coodinte xes Elsevie Inc. All ights eseved. MSC: pimy 35L65, 35J70, 35R35; secondy 35J65 Keywods: Regulity; Vcuum oundy; Two-dimensionl Riemnn polem; Chcteistic decomposition; Gs dynmics; Shock wves; Eule equtions; Goust polem. Intoduction The pessue gdient system u t + p x 0, v t + p y 0, E t + pu) x + pv) y 0,.) * Coesponding utho. E-mil ddesses: leizhn@yhoo.com Z. Lei), yzheng@mth.psu.edu Y. Zheng) /$ see font mtte 2007 Elsevie Inc. All ights eseved. doi:0.06/j.jde

2 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) Fig.. Solution with pesumed vcuum ule. whee E u 2 + v 2 )/2 + p, ppeed fist in the flux-splitting method of Li nd Co 9] nd Agwl nd Hlt ] in numeicl computtion of the Eule system of compessile gs. Lte, n symptotic deivtion ws given in Zheng 3,6] fom the two-dimensionl full Eule system fo n idel fluid ρ t + ρu) 0, ρu) t + ρu u + pi) 0, ρe) t + ρeu + pu) 0, whee u u, v), E u 2 + v 2 )/2 + p/γ )ρ), nd γ> is gs constnt. We efe the ede to the ooks of Zheng ] nd Li et l. 6] fo moe ckgound infomtion, nd the ppes 2,3,7,0,2,3,5,6] fo ecent studies. Afte eing decoupled fom system.), the pessue stisfies the following second ode qusi-line hypeolic eqution: pt p ) t x,y) p 0..2) Di nd Zhng 2] studied Riemnn polem fo system.), see lso Yng nd Zhng ] y the hodogph method. In the self-simil viles ξ x/t,η y/t, the vlue of the pessue vile of the Riemnn dt is pξ,η) ξ 2 fo 0 <ξ, η p, ξ p ) 2 + η 2 p, pξ,η) η 2 fo 0 <ξ, η p, ξ 2 + η p ) 2 p..3) Hee p is ny positive nume. They showed tht the Goust polem fo Eq..2) in the selfsimil plne dmits glol solution, which is smooth with possile vcuum ne the oigin see Fig., whee p ). We e inteested in the size of the vcuum oundy ξ, η) pξ,η) 0} whee the pessue gdient eqution.2) is degenete. Somewht supisingly, ou esult shows tht the vcuum ule is tivil nd the entie vcuum oundy is the tivil

3 282 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) Fig. 2. The glol solution. coodinte xes in the self-simil plne see Fig. 2, whee p ). The esult is stted in ou min theoem t the end of Section 3. Futhe motivtion fo the study of the cuent polem is tht the study of oundies such s sonic cuve is impotnt in estlishing the glol existence of solution to genel two-dimensionl Riemnn polem of the pessue gdient system. In ddition, the solution of the cuent polem coves wve intection polems in which only some fctions of the plne wves e involved. Wve intections of these kinds e common in two-dimensionl Riemnn polems. Finlly, the study of the pessue gdient system hs motivted wok on two-dimensionl full Eule systems, see Li 5], Zheng 7,8], nd Li nd Zheng 8]. In pticul, the ltest wok 8] hs esolved completely the loction of the vcuum oundy fo the gs expnsion polem fo the ditic Eule system. 2. Integtion long chcteistics In the self-simil viles ξ x/t, η y/t, the pessue gdient eqution.2) tkes the fom ξ ξ + η η ) 2 p p In the pol coodintes ξ,η) p + ξ ξ + η η )p p ξ ξ + η η )p) 2 p ) ξ 2 + η 2, θ ctn η ξ, Eq. 2.) cn e decomposed long the chcteistics into the following fom see 2,,7]): + p mp p, + p mp + p, 2.2)

4 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) povided tht p< 2, whee m λ 2p 2, nd ± θ ± λ, λ p 2 2 p). 2.3) Note tht the eqution is invint unde the following scling tnsfomtion: ξ η ξ,η,p),, p ) p p p p > 0). Thus, without loss of genelity, the coesponding oundy condition.3) in the pol coodintes cn e set in the fom p ξ 2 2 cos 2 θ cos θ on 2 cos θ, π/ θ π/2; p η 2 2 sin 2 θ sin θ on 2sinθ, 0 θ π/. 2.) The solution exists in the intection zone up to possile vcuum ule. See Fig. 3. The chcteistic fom 2.2) of the pessue gdient eqution enjoys nume of useful popeties. Fo exmple, the quntities ± p keep thei positivities/negtivities long chcteistics of plus/minus fmily, nd the sign-peseveing quntities yield monotonicity of the pimy vile p see 2,7], lso ] whee the uthos popose to cll them Riemnn sign-peseveing viles), nd the fct tht stte djcent to constnt stte fo the pessue gdient system must e simple wve in which p is constnt long the chcteistics of plus/minus fmily 7]. The chcteistic decomposition 2.2) hs plyed n impotnt ole nd ws poweful tool fo uilding the existence of smooth solutions in the wok of Di nd Zhng 2]. We e inteested in the size of the vcuum oundy, θ) p,θ) 0} whee the pessue gdient eqution is degenete. Fo this pupose, we ewite 2.2) s + p q + p p q p) 2, + p q + p p q + p) 2, 2.5) whee q 2 p 2 p). 2.6) Define the chcteistic cuves θ) nd + θ) y d θ) dθ λ θ),θ), ) 2sin, nd d + θ) dθ λ + θ),θ), + ) 2 cos. 2.7)

5 28 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) We point out hee tht fo convenience, we lso use the nottion, θ) +, θ), espectively) which epesents the chcteistic cuve pssing though the point, θ) nd intesecting the lowe uppe, espectively) oundy t point, espectively). See Fig. 3. Now let us ewite system 2.5) in the fom + p exp q + p+ φ), φ) dφ) q exp q + p+ φ), φ) dφ, + p exp q p φ), φ) dφ) q exp q p φ), φ) dφ. Integte the ove equtions long the positive nd negtive chcteistics + θ) nd θ) fom nd to θ, espectively, with espect to θ, one otin the itetive expessions of + p nd p: p exp q + p+ φ), φ) dφ p 2 cos, ) + q+ ψ), ψ) exp ψ q + p+ φ), φ) dφ dψ, + p exp q p φ), φ) dφ + p 2sin, ) + q ψ), ψ) exp ψ q p φ), φ) dφ dψ. On the othe hnd, we use oundy condition 2.) to find exp exp q + p + φ), φ ) dφ p + ) 2 + p + p φ), φ) dφ p + θ), θ) exp 2 cos θ p + θ),θ) p2cos, ) 2.8) 2 + ) p dp }. 2.9) Similly, one hs exp q p φ), φ ) dφ p θ), θ ) exp 2sinθ p θ),θ) p2sin, ) 2 ) p dp }. 2.0) Then, we hve

6 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) q + ψ), ψ ) exp ψ q + p + φ), φ ) } dφ dψ 2 cos exp 2 p 3 2 p + ψ), ψ ) p + ψ),ψ) p2cos, ) } 2 + ) p dp dψ, 2.) nd q ψ), ψ ) exp ψ q p φ), φ ) dφdψ 2sin exp 2 p 3 2 p ψ), ψ ) p ψ),ψ) p2sin, ) } 2 ) p dp dψ. 2.2) Finlly, y sustituting 2.0) nd 2.2) into the second equlity of 2.8), we otin new itetive expession fo + p: + p,θ) exp q p φ), φ) dφ + p 2sin, ) + q ψ), ψ) exp ψ q p φ), φ) dφ dψ p, θ) exp p θ),θ) p2sin, ) 2 ) p dp } p 3/ 2sin 2 cos 2 p ψ), ψ) exp p ψ),ψ) p2sin, ) 2 ) p dp } dψ. 2.3) Similly, y sustituting 2.9) nd 2.) into the fist equlity of 2.8), one otins the new itetive fomul fo p : p,θ) exp q + p+ φ), φ) dφ p 2 cos, ) + q+ ψ), ψ) exp ψ q + p+ φ), φ) dφ dψ p, θ) exp p + θ),θ) } p2cos, ) 2 + ) p dp 2 2 p 3/ 2cos 2 sin 2 p + ψ), ψ) exp p + ψ),ψ) p2cos, ) }. 2 + ) p dp dψ 2.)

7 286 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) Fig. 3. Pesumed vcuum in the pol coodintes. 3. The oundy of the vcuum ule In this section, we use the itetive fomuls 2.3) nd 2.) to pove tht the vcuum ule, θ) p,θ) 0,θ 0,π/2)} is in fct the tivil oigin 0, 0)} in the self-simil plne. We use the method of contdiction. Assume to the conty tht thee is ule with oundy 0 θ) 0 fo ll θ 0, π 2 ), nd 0θ) > 0fosomeθ 0, π 2 ). Tht mens p 0θ), θ) 0fo θ 0, π 2 ), nd the solution is smooth in the domin ounded y the ule oundy 0θ) nd the uppe nd lowe chcteistic oundies. We intend to deduce contdictions. Befoe we stt the ove pocedue, we point out two osevtions which oughly imply the nonexistence of the vcuum ule. Fo pesenting the osevtions, we ssume futhe tht 0 θ) > 0 fo ll θ 0, π 2 ). Fist, we cn compute esily the chcteistic slope dθ/d λ 2sinθ 2 long the uppe chcteistic oundy nd in the limit s θ π/2. Simil esult holds on the lowe oundy. Now let us ssume tht thee is nontivil ule with oundy 0 θ) > 0fo ll θ 0, π 2 ). We see fom the definition of λ in 2.3) tht p dθ/d λ 2 2 p) 0 s p 0 t the vcuum oundy) except fo 0, 0) nd 0, π 2 ). The ove computtion evels tht thee is some kind of inconsistency fo the slopes of the chcteistics t 0, 0) nd 0, π 2 ) in the pol coodinte plne i.e. the slope jumps fom 0 to /2). Next, let us clculte the decy te of p long the middle line θ π/ see Fig. 3). By using the symmety of system 2.), we clim tht ± p ±M 0 p /2 symptoticlly fo some M 0 0. To show the detils, we popose ± p ±M 0 p +δ

8 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) Fig.. Domin nd nottion. s, θ) tends to point of the ule oundy. Then, y 2.3), we hve + p,θ) p, θ) exp p θ),θ) p2sin, ) 2 p p) p ψ), ψ ) exp δm 0 p +δ + high ode tems, which foces δ, i.e., δ should e. Thus, y using 2.3), we hve } / 2 ) p dp 2 2sin 2 cos p M 0 2 p p ψ),ψ) p2sin, ) } } 2 ) p dp dp symptoticlly s, θ) tends to point of the ule oundy. Theefoe, p c exp M ) 0 3.) symptoticlly s, θ) tends to point of the ule oundy on the line θ π, which implies tht thee is no inteio vcuum t lest on θ π. We point out incidentlly tht Zheng s pevious numeicl computtion of the ule, efeed to in Di nd Zhng 2], is poly cused y the fst exponentil decy 3.). In wht follows, we concentte on estlishing the ove foml intuition igoously. With the symmety of system 2.5), let us estict ou gument θ to e in 0, π ]. Fix point,θ) on the ule oundy. Let us use D,θ) to epesent the ounded domin suounded y

9 288 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) the positive nd negtive chcteistic cuves stting fom,θ) nd 2, π ). See Fig.. Fo 0 <ɛ<, define cuve ɛ θ), θ 0, π 2 ),y Fom 2.3) nd 2.), it is esy to see tht p ɛ θ), θ ) ɛ. 3.2) + p,θ) > 0, p,θ) < 0, fo 0 <θ π. 3.3) Thus, y 2.3) nd 3.3), we hve p, θ) > 0, 3.) which implies tht the cuve ɛ θ), θ 0, π 2 ), defined in 3.2) is smooth if ɛ 0, ). Heewe point out tht we still denote y θ) nd + θ) the chcteistics pssing though ɛθ), θ) lthough in fct oth ɛ nd e dependent on ɛ nd θ. Futhe, we still denote y, θ) nd +, θ) the chcteistics pssing though, θ), whee, depend on, θ). Let us now fix n ɛ 0 0, ). Thus the cuve ɛ0 θ) exists. Define M mx p 2 + p, p 2 p }, 3.5) ove S whee S :, θ) θ 0,π/2), ɛ0 θ) 2 }. Then we hve + p M p 2, p M p 2, 3.6) fo ll, θ) with ɛ0 θ). Note tht M depends only on ɛ 0 nd does not depend on,θ). Next, fo, θ) D,θ),wefistlet e the intesection of the minus chcteistic cuve though, θ) with the oundy, nd the coesponding countept. We then let Then, let M 2 mx A, θ) B,θ) mx,θ) D,θ) exp p θ),θ) p2sinθ,θ ) ] 2 dp ) p 2, 2sin 2 cos exp p + θ),θ) p2cos, ) ] 2 + dp ) p 2. 2cos 2 sin p 2sin, ), mx A, θ),θ) D,θ) 3.7) p } 2 cos, ), 3.8) B,θ)

10 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) nd M 3 mxm,m 2 + }. 3.9) Just s wht we hd pointed out efoe, the intesections nd in 3.7) nd 3.8) vy with, θ), which e not expessed explicitly fo nottionl evity. We intend to pove tht inequlities 3.6) e still vlid fo ll points, θ) D,θ) with M eing eplced y M 3. Nmely, thee hold + p M 3 p 2,, θ) D,θ). 3.0) p M 3 p 2, Note tht the positive constnt M 3 M ) is independent of, θ) D,θ), ut depends on the fixed,θ). We stt to pove 3.0). Suppose tht 3.0) is coect up to line segment ɛ θ) D,θ), then we impove 3.0) to stict inequlities on the line segment ɛ θ) in the sme domin. In fct, fo, θ) D,θ) with ɛ0 θ), let us compute: / + p,θ) p 2 p 3, θ) A, θ) + 2 p ψ), ψ ) exp p ψ),ψ) p θ),θ) ] } 2 ) p dp dψ / p p 2 p 3, θ) A, θ) + p) 2 p) ψ), ψ ) exp p ψ),ψ) p θ),θ) M 3 p, θ) / M 3 A, θ) exp p ψ),ψ) p θ),θ) ] } 2 ) p dp dψ + p θ),θ) p2sin, ) ] } 2 ) p dp dp M 3 p, θ) M 3 A,θ) + fθ ) p θ),θ) p2sin, ) p 5 dp p 5 2 ψ), ψ ) 2 p M 3 p, θ), 3.) M 3 A,θ) + fθ )p, θ) p 2sin, )]

11 290 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) whee fθ ) 2 2 p θ ), θ ) exp > exp p θ ),θ ) p θ),θ) 2 2 dp ] p θ ),θ ) p θ),θ) ] 2 ) p dp p exp θ ), θ ) p ] θ), θ) > 3.2) 8 with some <θ <nd A, θ) is defined in 3.7). Thus, y 3.9), 3.) nd 3.2), we hve M 3 p, θ) + p,θ) < M 3 A,θ) +p, θ) p 2sin, )] M 3 p 2, θ). 3.3) Similly, y 2.8), 2.9) nd 2.), we hve whee / p,θ) p 2 p 3, θ) B,θ) 2 p + ψ), ψ ) exp p + ψ),ψ) p + θ),θ) ] } 2 + ) p dp dψ M 3 p, θ), 3.) M 3 B,θ) + gθ + )p, θ) p 2 cos, )] g θ +) 2 2 p + θ + ),θ +) exp > exp p + θ + ),θ + ) p + θ),θ) ] dp 2 2 p exp + θ + ), θ + ) p+ 8 p + θ + ),θ + ) p + θ),θ) θ), θ) ] > ] 2 + ) p dp

12 Z. Lei, Y. Zheng / J. Diffeentil Equtions ) with some θ<θ + < nd B,θ) is given in 3.7). Thus, the simil estimte s 3.3) holds: p,θ) < M 3 p 2, θ). 3.5) Since M 3 depends only on,θ), nd in pticully, it is independent of, θ) D,θ),the poof of 3.0) follows y 3.3) nd 3.5). Now with the id of 2.3), we dd up 3.3) nd 3.5) to yield p M p) p 2M 3 2 fo ll, θ) in smll neighohood of,θ) in D,θ). Thus, simple integtion of the ove inequlity with espect to fom to yields p,θ) p,θ)exp 2 M } 3 ) fo <. 3.6) 2 On the othe hnd, y 3.) nd the fct tht p 0 θ), θ) 0, we hve p,θ) > 0 fo ll, θ) with > 0 θ), which togethe with 3.6) esult in tht p,θ)>0. Since,θ) is point on the ule, we ive t contdiction. Summing up, we hve in fct poven the following theoem. Theoem. The Riemnn polem.2), 2.) fo the pessue gdient eqution dmits unique smooth solution. The pessue of the solution is stictly positive in ξ>0,η>0. Acknowledgments This wok ws done when Zhen Lei ws visiting the Deptment of Mthemtics of the Pennsylvni Stte Univesity. He would like to expess his thnks fo its hospitlity. In pticul, he thnks Pofesso Chun Liu fo his encougement. Zhen Lei ws ptilly suppoted y the Ntionl Science Foundtion of Chin unde gnt nd Foundtion fo Excellent Doctol Dissettion of Chin. Yuxi Zheng ws ptilly suppoted y Ntionl Science Foundtion unde gnts NSF-DMS , 0305, nd Refeences ] R.K. Agwl, D.W. Hlt, A modified CUSP scheme in wve/pticle split fom fo unstuctued gid Eule flows, in: D.A. Cughey, M.M. Hfez Eds.), Fonties of Computtionl Fluid Dynmics, 99. 2] Zihun Di, Tong Zhng, Existence of glol smooth solution fo degenete Goust polem of gs dynmics, Ach. Rtion. Mech. Anl ) ] Eun Heui Kim, Kyungwoo Song, Clssicl solutions fo the pessue-gdient equtions in non-smooth nd nonconvex domins, J. Mth. Anl. Appl ) ] Zhen Lei, Yuxi Zheng, A chcteistic decomposition of nonline wve eqution in two spce dimensions, pepint, Feuy ] Jiequn Li, On the two-dimensionl gs expnsion fo compessile Eule equtions, SIAM J. Appl. Mth ) p

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