M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno expec he exisence of classical soluions ha is u C R n, ) ) saisfying he equaion everywhere) ;. The Hopf-Lax formula saisfies he equaion almos everywhere. { x y u x, ) = min L y ) } + g y) We see ha exisence is guaraneed as soon as we only require he equaion o be saisfied almos everywhere. Thus i is naural o propose he following as he definiion for weak soluions : u Lipschiz is a weak soluion if u saisfies he equaion a. e. and akes g as is iniial value. However such definiion leads o non-uniqueness, as shown in he following example. Example. Consider he D H-J equaion u + u x = R, ) ; u = R { = }. 3) I is clear ha u is a classical soluion. However, we can ry o find oher soluions using separaion of variables. Se u = T ) + X x). Then he equaion gives ) which gives T ) + X x) ) = 4) T ) = λ, X x) ) = λ. 5) Obviously if we require he above o hold everywhere, hen necessarily λ = Recall Exercise 5 in Homework ). However, if we only require he equaion o be saisfied almos everywhere, hen here are infiniely many soluions. More specifically, we can pick any X such ha X = ± λ piecewise, and hen solve T = λ. We can even ake differen λ for differen x,. One example of almos everywhere soluions obained his way is x u x, ) = x x 6) x x { x x Noe ha u x, ) is in fac assembled using wo soluions : and. I is easy o see ha x x one can assemble any soluions in anyway o ge a funcion saisfying he equaion almos everywhere, he only hing o be careful is o make he resul be consisen wih he iniial value. For example: + gives a almos everywhere soluion noe ha T so he wedge is moving downward, hus saisfying he iniial value). In conras,
is no a good consrucion, as i does no saisfy he iniial value. I is clear ha he soluion is no unique. How do we regain uniqueness? Or more specifically, how o do we ge rid of all non-zero soluions as u is obviously a reasonable soluion for his problem). One way is o add viscosiy. Consider u ε + u ε x ε = ε u x x 7) and hen le ε. In many cases u ε will hen converge o one of he many almos everywhere soluions of he inviscid problem. In oher words, u ε is a smoohed version of he correc soluion. Smoohing he wedge soluion, we see ha x = is a minimizer of u ε. A his poin, we have u <, u x =, ε u x x > 8) a conradicion! Therefore he limi of u ε canno be he wedge soluion! From he above example we see ha wedges should no appear in he correc weak soluion. In oher words, a correc soluion should no have u = + anywhere. Moivaed by his, we inroduce he following noion. Definiion. Semi-concaviy) f is called semi- concave if here is C > such ha for any x, z. f x + z) f x) + f x z) C z 9) I is easy o see ha he condiion is equivalen o f x) C x ) is concave. Hence he name semi-concave. Now we can ry he following definiion of weak soluions for he H-J equaion: Definiion 3. A Lipschiz funcion u is said o be a weak soluion if. u akes g as iniial value;. u saisfies he equaion a. e. ; 3. u is semi- concave: u x + z, ) u x, ) + u x z, ) C + ) z. ) Now we ry o esablish well-posedness. More specifically, we will show ha under cerain assumpions, he soluion given by he Hopf-Lax formula is he unique weak soluion. Theorem 4. Exisence) The funcion u x, ) given by he Hopf-Lax formula is a weak soluion if eiher one of he following is rue. a ) g is semi-concave; b ) H is uniformly convex, ha is for all p, ξ. ξ T D H ξ θ ξ. Of course, for his problem, one can direcly show ha u ε. However he argumen used here is more appropriae for moivaing he general heory. )
Proof. a) Since g is semi-concave, here is C > such ha g x + z) g x) + g x z) C z. 3) Now for any, le y be such ha ) x y u x, ) = L + g y). 4) Then we have 3 ) x y u x ± z, ) L + g y ± z) and he conclusion follows. 6) b) Again we choose y such ha ) x y u x, ) = L + g y). 7) Now ) x ± z y u x ± z, ) L + g y). 8) Thus [ ) ) ) ] x + z y x y x z y u x + z, ) u x, ) + u x z, ) L L + L. 9) The conclusion follows from he fac ha ) p + p H uniformly convex H ) q + q L Which is Problem 3. 5. 9. 4 Now we esablish uniqueness. H p ) H p ) θ 8 p p L q ) L q ) 8 θ q q. ) Theorem 5. Uniqueness) Assume H is C, convex, wih super-linear growh a infiniy and g Lipschiz. Then here exiss a mos one weak soluion of he H-J equaion. Proof. Suppose u, ũ are wo weak soluions. hen leing w = u ũ, we have w is differeniable a. e. and Leing w x, ) = u x, ) ũ x, ) = H D ũ x, ) ) H D u x, ) ) d = H r D u + r) D ũ) dr dr = D H r D u + r) Dũ) D w x, ) dr [ ] = D H r D u + r) Dũ) dr D w x, ). ) b x, ) D H r D u + r) Dũ) dr ). Equivalenly, H θ p is sill convex. 3. By he Hopf-Lax formula Take y = y ± z. { x ± z y u x ± z, ) = min L y 4. Hin: For any q, q, ake p, p such ha H p i) + L q i) = p i q i. ) } + g y ) 5)
we conclude ha w saisfies w + b D w = a. e. ; w = =. 3) If w is a classical soluion o he above equaion, hen obviously w and we are done. Bu his is no he case. Neverheless, if b is Lipschiz, we sill can show w as follows. Le x ) be such ha x = b. Then w saisfies d w x ), ) = a. e. 4) d As w is Lipschiz in x, and x ) Lipschiz in, W ) w x ), ) is also Lipschiz and hus absoluely coninuous. As a consequence we have W ) W ) = W s) ds = 5) as W s) = almos everywhere. Now since W ) = we conclude ha W. Then i s clear ha w = almos everywhere. However he problem is ha b x, ) D H r D u + r) Dũ) dr 6) is definiely no Lipschiz. The idea now is o approximae b by a Lipschiz funcion b ε, ge some esimae and hen le ε. Then we can wrie w + b ε D w = b ε b) D w a. e. 7) If we ry o do energy-ype esimae of he above, we quickly realize ha a uniform upper bound of b ε is needed. This means ha he naïve way b ε = η ε b 8) will no work as his way b ε /ε. The correc smoohing here is we use subscrip o emphasize ha his is no he usual mollificaion) b ε D H r D u ε + r) D ũ ε ) dr 9) where u ε, ũ ε are usual mollificaion f ε = η ε f. This way we have [ b ε = D H: r D u ε + r) D ũ ] ε 3) where A: B ) i, j A i j B i j. Now using he fac ha boh u, ũ are semi-concave, we have D u ε, D ũ ε C + I. Noe ha as he mollificaion is aken over space-ime, such bound only holds for > C ε. Using his we have b ε C + ). 3 ) Now le v φ w) o be fixed laer. Then v saisfies he same equaion as w. Se R max { D H p) p max { Lip u), Lip ũ) } }. 3) We ry o show ha v in he cone 5 Se C { x, ), x x R ) }. 33) e ) B x, R ) ) v x, ) dx. 34) 5. Recall our proof of he uniqueness for he wave equaion! Here he slope R is an upper bound of he speed of characerisics.
We compue e ) = v R v ds B x, R ) ) = b ε v) + b ε ) v + b ε b) D v dx R v ds B B x, R ) ) = v b ε n + R) ds + b ε ) v + b ε b) D v. 35) B B ) The firs erm is due o he definiions of b ε and R. The second erm is bounded by C + e ), he hird erm vanishes as ε due o dominaed convergence. Thus we have e ) C + ) e ) 36) for a. e. < <. Finally, aking φ such ha φ z) = for z ε [ Lip u) + Lip ũ) ] and posiive oherwise, we have v = φ w) = 37) for ε. Now for > ε, + + and we have ε e ) C + ) e ) a. e., > ε; e ) = ε. 38) ε which gives e ) =. This means u ũ = w ε [ Lip u) + Lip ũ) ] 39) in he cone C. By he arbirariness of ε we conclude ha u ũ = almos everywhere and ends he proof.