Generic Singularities of Solutions to some Nonlinear Wave Equations
|
|
- Stephen Harrison
- 5 years ago
- Views:
Transcription
1 Generic Singularities of Solutions to some Nonlinear Wave Equations Alberto Bressan Deartment of Mathematics, Penn State University (Oberwolfach, June 2016) Alberto Bressan (Penn State) generic singularities 1 / 36
2 Singularity formation For several nonlinear wave equations, solutions with smooth initial data develo singularities in finite time: u u(t, ) C 1 (R) or u(t, ) H s (R) Alberto Bressan (Penn State) generic singularities 2 / 36
3 Generic singularities Prove that, for generic smooth initial data, singularities are localized along finitely many oints, or curves Give a local asymtotic descrition of (structurally stable) singularities generic valid on a countable intersection of oen dense sets in C k Alberto Bressan (Penn State) generic singularities 3 / 36
4 Three basic settings hyerbolic systems of conservation laws: u t + f (u) = 0 Burgers-Hilbert equation: u t + (u 2 /2) = H[u] variational wave equations: u tt c(u)(c(u)u ) = 0 Alberto Bressan (Penn State) generic singularities 4 / 36
5 Generic regularity for scalar conservation laws u t + f (u) = 0 R, t [0, T ] u(0, ) = ū() Theorem (D. Schaeffer, 1973) Assume f smooth, f > 0. For a generic initial data ū C 3 (R), the solution remains smooth outside finitely many shock curves. D. Schaeffer, A regularity theorem for conservation laws. Adv. Math. 11 (1973), C. Dafermos and X. Geng, Generalized characteristics uniqueness and regularity of solutions in a hyerbolic system of conservation laws. Ann. Inst. H. Poincaré 8 (1991), Alberto Bressan (Penn State) generic singularities 5 / 36
6 u t + f (u)u = 0 u(0, ) = ū() equations of characteristics: ẋ = f (u) u = 0 u = f (u)u 2 Along the characteristic starting at y: u (t, (t)) as t T blowu (y) = New shocks can only form at ositive local minima of the ma y T blowu (y) 1 f (ū(y)) ū (y) Alberto Bressan (Penn State) generic singularities 6 / 36
7 Eamle: Burgers equation ( ) u 2 u t + 2 = 0, u(0, ) = ū() New shocks are formed along characteristics originating from negative local minima of ū ū has N local minima = at most N shock curves can aear _ u() t Alberto Bressan (Penn State) generic singularities 7 / 36
8 Piecewise regularity for hyerbolic systems of conservation laws? Question. For generic initial data ū C 3, is the solution smooth outside finitely many shock curves? t 3 3 t 2 2 ossibly true for 2 2 systems false for n n systems, with n 3 L. Caravenna and L. Sinolo, Schaeffer s regularity theorem for scalar conservation laws does not etend to systems, Indiana U. Math. J., to aear Alberto Bressan (Penn State) generic singularities 8 / 36
9 Generic regularity for 2 2 conservation laws? Detailed descrition of singularity formation: De-Xing Kong, Formation and roagation of singularities for 2 2 quasilinear hyerbolic systems. Trans. Amer. Math. Soc. 354 (2002), Generic regularity? t scalar conservation law t 2 2 system Alberto Bressan (Penn State) generic singularities 9 / 36
10 The Burgers-Hilbert equation ( ) u 2 u t + 2 = H[u], u(0, ) = ū (BH) For u L 2 (R), the Hilbert transform is H[u](). = 1 u( y) π P.V. dy = y 1 u( y) lim dy ε 0+ π y >ε y Alberto Bressan (Penn State) generic singularities 10 / 36
11 References J. Biello and J. K. Hunter, Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities. Comm. Pure Al. Math. 63 (2009), Derivation of the model, for nonlinear waves with constant frequency. J. K. Hunter and M. Ifrim, Enhanced life san of smooth solutions of a Burgers-Hilbert equation. SIAM J. Math. Anal. 44 (2012), Local eistence and uniqueness of smooth solutions, estimates on the blow-u time Alberto Bressan (Penn State) generic singularities 11 / 36
12 Entroy-weak solutions in L 2 (R) (A.B., K.Nguyen, SIAM J. Math. Anal., 2014) Theorem (global eistence in L 2 ) Given any initial data ū L 2 (R), the Cauchy roblem (BH) has an entroy weak solution u = u(t, ) defined for all (t, ) [0, [ R. For this solution, the ma t u(t, ) L 2 is non-increasing, while u(t, ) L C(1 + t 1/3 ) for every t > 0. Theorem (uniqueness for satially eriodic, BV solutions) Let u, v be satially eriodic entroy weak solutions with the same initial data. Assume that the total variation of u(t, ) and v(t, ) over [0, 2π] remains uniformly bounded for t [0, T ]. Then u and v coincide for all t [0, T ]. Alberto Bressan (Penn State) generic singularities 12 / 36
13 Generic singularities for the Burgers-Hilbert equation Describe the local behavior of a solution near a shock Describe how a shock is formed Describe the interaction of two shocks Is a generic solution iecewise smooth? t Alberto Bressan (Penn State) generic singularities 13 / 36
14 Piecewise regular solutions Burgers Burgers Hilbert u( τ,) u(t,) 0 0 For Burgers equation, at the time τ when a new shock is formed: u(τ, ) = a b( 0 ) 1/3 + for 0 Alberto Bressan (Penn State) generic singularities 14 / 36
15 Burgers Burgers Hilbert u( τ,) u(t,) 0 0 For Burgers-Hilbert, near a shock located at = 0: u(t, ) = u + u ln π + b + O(1) 3/2 if < 0 2 ln π + b + + O(1) 3/2 if > 0 A.B., Tianyou Zhang, Piecewise smooth solutions to the Burgers-Hilbert equation. Comm. Math. Sci., to aear. (local eistence and uniqueness) Alberto Bressan (Penn State) generic singularities 15 / 36
16 The variational wave equation u tt c(u) ( c(u)u ) = 0 { u(0, ) = u0 () u t (0, ) = u 1 () (u 0, u 1 ) H 1 (R) L 2 (R) c : R R + is a smooth, uniformly ositive function ±c(u) = wave seeds Ping Zhang and Yui Zheng, Proc. Royal Soc. Edinburgh (2002), Ping Zhang and Yui Zheng, Arch. Rat. Mech. Anal. (2003), Ping Zhang and Yui Zheng, Ann. Inst. H. Poincaré, (2004). Alberto Bressan (Penn State) generic singularities 16 / 36
17 Auiliary variables {. R = ut + c(u)u,. S = u t c(u)u, u t = R + S 2, u = R S 2c Evolution equation for R, S: R t cr = c 4c (R2 S 2 ) S t + cs = c 4c (S 2 R 2 ) Possible blow-u: R, S in finite time c 0 = D Alembert solution of wave equation Alberto Bressan (Penn State) generic singularities 17 / 36
18 Conserved quantities (for smooth solutions) Balance laws for R 2, S 2 : (R 2 ) t (cr 2 ) = c 2c (R2 S RS 2 ) (S 2 ) t + (cs 2 ) = c 2c (R2 S RS 2 ) R 2 and S 2 reresent the energy of backward and forward moving waves. Energy is transferred from forward to backward waves, and vice-versa Total energy: E(t) = 1 2 (u 2 t + c 2 u 2 ) d = constant Natural domain: (u, u t) H 1 (R) L 2 (R) = solutions remain Hölder continuous Alberto Bressan (Penn State) generic singularities 18 / 36
19 Recent results (A.B., Geng Chen, Tao Huang, Fang Yu) For an oen, dense set of initial data (u 0, u 1 ) D U =. ( ) C 3 (R) H 1 (R) ( ) C 2 (R) L 2 (R) the conservative solution u = u(t, ) is C 2 outside a finite set of singular oints and C 2 singular curves. A detailed asymtotic descrition of u can be given near each oint of singularity. t 3 q 2 2 q 1 Alberto Bressan (Penn State) generic singularities 19 / 36 1
20 Basic tools from differential geometry: Sard s theorem, Thom s transversality theorem aly to C k mas. For solutions to nonlinear wave equations, such regularity is not available. Key idea: By a change of deendent and indeendent coordinates, one obtains an equivalent system whose solutions remain globally smooth Alberto Bressan (Penn State) generic singularities 20 / 36
21 Coordinate change: indeendent variables Equations for characteristics ẋ + = c(u), ẋ = c(u) s + (s, t, ) (s, t, ) As coordinates (X, Y ) of a oint (t, ) we use the quantities X =. (0, t, ), Y =. + (0, t, ) t X = const. (,t) Y = const. s + (s,,t) (s,,t) Alberto Bressan (Penn State) generic singularities 21 / 36
22 Coordinate change: deendent variables w. = 2 arctan R, z. = 2 arctan S w, z R/(2π Z) R, S ± w, z π. = 1 + R2 X, q. = 1 + S 2 Y Alberto Bressan (Penn State) generic singularities 22 / 36
23 A semilinar system in characteristic variables A.B., Yui Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys. 266 (2006), w Y = c (u) (cos z cos w) q 8c 2 (u) Y = c (u) (sin z sin w) q 8c 2 (u) z X = c (u) (cos w cos z) 8c 2 (u) q X = c (u) (sin w sin z) q 8c 2 (u) u X = sin w 4c(u) u Y = sin z 4c(u) q X = Y = (1+cos w) 4 (1+cos z) q 4 t X = t Y = (1+cos w) 4c(u) (1+cos z) q 4c(u) Λ : (X, Y ) (, t) Alberto Bressan (Penn State) generic singularities 23 / 36
24 Boundary data - comatible solutions Y 0 X γ 0. Along the curve γ 0 = {X + Y = 0} corresonding to {t = 0}, the boundary data ( w, z,, q, ū) L are defined by { w = 2 arctan R(, 0) { = 1 + R 2 (, 0) z = 2 arctan S(, 0) q = 1 + S 2 (, 0) = X = Y, ū = u 0 () Alberto Bressan (Penn State) generic singularities 24 / 36
25 Global conservative solutions Theorem (A.B. - Yui Zheng, 2006) Given smooth initial data (u, u t ) = (u 0, u 1 ), the semilinear system has a t=0 unique smooth solution (, t, u, w, z,, q)(x, Y ) defined for all (X, Y ) R 2. The function u = u(, t) whose grah is grah(u) = {((X, Y ), t(x, Y ), u(x, Y )) ; (X, Y ) R 2} is the unique conservative solution to the wave equation u tt c(u)(c(u)u ) = 0 Singularities can only arise because the ma Λ : (X, Y ) (, t) is not smoothly invertible ( ) (1+cos w) (1+cos z) q X DΛ = Y 4 4 = t X t Y (1+cos w) 4c(u) (1+cos z) q 4c(u) Alberto Bressan (Penn State) generic singularities 25 / 36
26 Structure of the singular set The set of oints (, t) where u is not smooth is contained in the image of the level sets S w. = {(X, Y ) ; w(x, Y ) = π}, S z. = {(X, Y ) ; z(x, Y ) = π} w = π Y w > π z = π P 3 Q 2 P P 2 z > π γ 0 w < π Q 1 0 P 1 X Alberto Bressan (Penn State) generic singularities 26 / 36
27 Generic regularity u tt c(u)(c(u)u ) = 0 ( ) (A) The function c is smooth and uniformly ositive. Moreover, c (u) = 0 = c (u) 0 Theorem (A.B., Geng Chen, Ann. Inst. H.Poincaré, 2016) Let (A) hold. Then there eists an oen dense set of initial data ( ) ( ) D C 3 (R) H 1 (R) C 2 (R) L 2 (R) such that the solution u = u(t, ) is iecewise smooth in the -t lane. Alberto Bressan (Penn State) generic singularities 27 / 36
28 Classification of generic singularities w = π Y w > π z = π P 3 Q 2 P P 2 z > π γ 0 w < π Q 1 0 P 1 X t 3 q 2 2 q 1 Alberto Bressan (Penn State) generic singularities 28 / 36 1
29 Three tyes of singular oints (X, Y ) Tye 1: w = π, w X 0 (oints along a singular curve) Tye 2: w = π, w X = 0 = w Y 0, w XX 0 (oints were two singular curves of the same family originate or terminate) Tye 3: w = π, z = π = w X 0, z Y 0 (oints where two curves of oosite families cross) Note: the imlication = is true for a generic solution Alberto Bressan (Penn State) generic singularities 29 / 36
30 Thom s transversality theorem = Fi a bounded domain Ω in the X -Y lane. Then there is an oen dense set of comatible solutions (u,, t, w, z,, q) to the semilinear system such that the following values are NEVER attained on Ω: { (w, wx, w XX ) = (π, 0, 0), (z, z Y, z YY ) = (π, 0, 0), { (w, z, wx ) = (π, π, 0), (w, z, z Y ) = (π, π, 0), { (w, wx, c (u)) = (π, 0, 0), (z, z Y, c (u)) = (π, 0, 0). (1) (2) (3) Alberto Bressan (Penn State) generic singularities 30 / 36
31 f y X f(x) For a fied ȳ = (ȳ 1, ȳ 2, ȳ 3 ), a generic smooth ma f : R 2 R 3 does NOT attain the value ȳ. BUT: a generic solution of a system containing the equation w Y = c (u) 8c 2 (cos z cos w) q (u) can still attain the value (w, z, w Y ) = (0, 0, 0). Results on a generic solution to a system of PDEs require more detailed analysis. J. Damon, Generic roerties of solutions to artial differential equations. Arch. Rational Mech. Anal. 140 (1997) Alberto Bressan (Penn State) generic singularities 31 / 36
32 Asymtotic descrition of singularities w = π Y w > π z = π P 3 Q 2 P P 2 z > π γ 0 w < π Q 1 0 P 1 X t 3 q 2 2 q 1 1 Alberto Bressan (Penn State) generic singularities 32 / 36
33 Theorem (A.B., T.Huang, F. Yu, Bull. Inst. Math. Acad. Sinica, 2015) Let (A) hold. Then a generic solution to the wave equation has only three tyes of singular oints ( 0, t 0). At oints of Tye 1 (along a singular curve γ) one has [ ] 2/3 ( ) u(, t) = u 0 a c(u 0)(t t 0) + ( 0) + O(1) t t At oints of Tye 2 (where two new singular curves γ, γ + originate) one has [ ] 3/5+ ( ) 4/5 u(, t) = u 0 + a c(u 0)(t t 0) + ( 0) O(1) t t At oints of Tye 3 (where two singular curves γ, γ cross), one has [ ] 2/3 u(, t) = u 0 + a 1 c(u 0)(t t 0) + ( 0) [ ] 2/3 ( ) +a 2 c(u 0)(t t 0) ( 0) + O(1) t t Alberto Bressan (Penn State) generic singularities 33 / 36
34 At a time t 0 when a new singularity forms: u(, t 0 ) u 0 a ( 0 ) 3/5 After the singularity has formed: u(, t 0 ) u 0 + a ( 0 ) 2/3 u(,t ) 0 u(,t) u 0 u t 3 q 2 2 q 1 1 Alberto Bressan (Penn State) generic singularities 34 / 36
35 Singular curves and characteristics Y t t 0 P w= π X 0 γ Characteristics curves satisfy ẋ(t) = ± c(u(t, (t)) Singular curves are enveloes of characteristics The distance between a singular curve γ( ) and the characteristic ( ) assing through the same oint ( 0, t 0 ) is (t) γ(t) κ (t t 0 ) 3 Alberto Bressan (Penn State) generic singularities 35 / 36
36 New singular curves Y w= π Y 0 P 1 P 2 P 0 t = τ > t t = t 0 0 t 1 γ 2 _ γ + t 0 0 X 0 X 0 At the oint ( 0, t 0 ) where two new singular curves γ, γ + are formed, their distance is γ + (t) γ (t) = κ (t t 0 ) 5/2 + O(1) (t t 0 ) 3 Alberto Bressan (Penn State) generic singularities 36 / 36
Structurally Stable Singularities for a Nonlinear Wave Equation
Structurally Stable Singularities for a Nonlinear Wave Equation Alberto Bressan, Tao Huang, and Fang Yu Department of Mathematics, Penn State University University Park, Pa. 1682, U.S.A. e-mails: bressan@math.psu.edu,
More informationPiecewise Smooth Solutions to the Burgers-Hilbert Equation
Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang
More informationOn Asymptotic Variational Wave Equations
On Asymptotic Variational Wave Equations Alberto Bressan 1, Ping Zhang 2, and Yuxi Zheng 1 1 Department of Mathematics, Penn State University, PA 1682. E-mail: bressan@math.psu.edu; yzheng@math.psu.edu
More informationLipschitz Metrics for a Class of Nonlinear Wave Equations
Lipschitz Metrics for a Class of Nonlinear Wave Equations Alberto Bressan and Geng Chen * Department of Mathematics, Penn State University, University Park, Pa 1680, USA ** School of Mathematics Georgia
More informationConservative Solutions to a Nonlinear Variational Wave Equation
Conservative Solutions to a Nonlinear Variational Wave Equation Alberto Bressan and Yuxi Zheng Department of Mathematics, The Pennsylvania State University E-mail: bressan@math.psu.edu; yzheng@math.psu.edu
More informationSingularity formation for compressible Euler equations
Singularity formation for compressible Euler equations Geng Chen Ronghua Pan Shengguo Zhu Abstract In this paper, for the p-system and full compressible Euler equations in one space dimension, we provide
More informationENERGY CONSERVATIVE SOLUTIONS TO A NONLINEAR WAVE SYSTEM OF NEMATIC LIQUID CRYSTALS
ENERGY CONSERVATIVE SOLUTIONS TO A NONLINEAR WAVE SYSTEM OF NEMATIC LIQUID CRYSTALS GENG CHEN, PING ZHANG, AND YUXI ZHENG Abstract. We establish the global existence of solutions to the Cauchy problem
More informationOn the piecewise smoothness of entropy solutions to scalar conservation laws for a large class of initial data
On the piecewise smoothness of entropy solutions to scalar conservation laws for a large class of initial data Tao Tang Jinghua Wang Yinchuan Zhao Abstract We prove that if the initial data do not belong
More informationContractive Metrics for Nonsmooth Evolutions
Contractive Metrics for Nonsmooth Evolutions Alberto Bressan Department of Mathematics, Penn State University, University Park, Pa 1682, USA bressan@mathpsuedu July 22, 212 Abstract Given an evolution
More informationarxiv: v1 [math.ap] 19 Mar 2011
Life-San of Solutions to Critical Semilinear Wave Equations Yi Zhou Wei Han. Abstract arxiv:113.3758v1 [math.ap] 19 Mar 11 The final oen art of the famous Strauss conjecture on semilinear wave equations
More informationApplications of the compensated compactness method on hyperbolic conservation systems
Applications of the compensated compactness method on hyperbolic conservation systems Yunguang Lu Department of Mathematics National University of Colombia e-mail:ylu@unal.edu.co ALAMMI 2009 In this talk,
More informationA generalized Fucik type eigenvalue problem for p-laplacian
Electronic Journal of Qualitative Theory of Differential Equations 009, No. 18, 1-9; htt://www.math.u-szeged.hu/ejqtde/ A generalized Fucik tye eigenvalue roblem for -Lalacian Yuanji Cheng School of Technology
More informationGLOBAL SOLUTIONS TO A ONE-DIMENSIONAL NONLINEAR WAVE EQUATION DERIVABLE FROM A VARIATIONAL PRINCIPLE
Electronic Journal of Differential Equations, Vol. 207 207, No. 294, pp. 20. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL SOLUTIONS TO A ONE-DIMENSIONAL NONLINEAR
More informationREGULARITY THROUGH APPROXIMATION FOR SCALAR CONSERVATION LAWS
SIAM J. MATH. ANAL. c 1988 Society for Industrial and Applied Mathematics Vol. 19, No. 4, pp. 1 XX, July 1988 003 REGULARITY THROUGH APPROXIMATION FOR SCALAR CONSERVATION LAWS BRADLEY J. LUCIER Abstract.
More informationGlobal conservative solutions of the Camassa-Holm equation
Global conservative solutions of the Camassa-Holm equation Alberto Bressan Deptartment of Mathematics, Pennsylvania State University, University Park 168, U.S.A. e-mail: bressan@math.psu.edu and Adrian
More informationPresenter: Noriyoshi Fukaya
Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), 347-373. Presenter: Noriyoshi
More informationHyperbolic Systems of Conservation Laws. in One Space Dimension. II - Solutions to the Cauchy problem. Alberto Bressan
Hyperbolic Systems of Conservation Laws in One Space Dimension II - Solutions to the Cauchy problem Alberto Bressan Department of Mathematics, Penn State University http://www.math.psu.edu/bressan/ 1 Global
More informationHyperbolic Systems of Conservation Laws
Hyperbolic Systems of Conservation Laws III - Uniqueness and continuous dependence and viscous approximations Alberto Bressan Mathematics Department, Penn State University http://www.math.psu.edu/bressan/
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationMATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )
More informationBlowup for Hyperbolic Equations. Helge Kristian Jenssen and Carlo Sinestrari
Blowup for Hyperbolic Equations Helge Kristian Jenssen and Carlo Sinestrari Abstract. We consider dierent situations of blowup in sup-norm for hyperbolic equations. For scalar conservation laws with a
More informationLIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP
More informationA Remark on IVP and TVP Non-Smooth Viscosity Solutions to Hamilton-Jacobi Equations
2005 American Control Conference June 8-10, 2005. Portland, OR, USA WeB10.3 A Remark on IVP and TVP Non-Smooth Viscosity Solutions to Hamilton-Jacobi Equations Arik Melikyan, Andrei Akhmetzhanov and Naira
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationSemicontinuous filter limits of nets of lattice groupvalued
Semicontinuous ilter limits o nets o lattice grouvalued unctions THEMATIC UNIT: MATHEMATICS AND APPLICATIONS A Boccuto, Diartimento di Matematica e Inormatica, via Vanvitelli, I- 623 Perugia, Italy, E-mail:
More informationDynamic Blocking Problems for Models of Fire Propagation
Dynamic Blocking Problems for Models of Fire Propagation Alberto Bressan Department of Mathematics, Penn State University bressan@math.psu.edu Alberto Bressan (Penn State) Dynamic Blocking Problems 1 /
More informationFinite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation. CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO
Finite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation JOSÉ ALFREDO LÓPEZ-MIMBELA CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO jalfredo@cimat.mx Introduction and backgrownd
More informationSOLUTIONS OF SEMILINEAR WAVE EQUATION VIA STOCHASTIC CASCADES
Communications on Stochastic Analysis Vol. 4, No. 3 010) 45-431 Serials Publications www.serialspublications.com SOLUTIONS OF SEMILINEAR WAVE EQUATION VIA STOCHASTIC CASCADES YURI BAKHTIN* AND CARL MUELLER
More informationGLOBAL EXISTENCE FOR THE ONE-DIMENSIONAL SEMILINEAR TRICOMI-TYPE EQUATIONS
GLOBAL EXISTENCE FOR THE ONE-DIMENSIONAL SEMILINEAR TRICOMI-TYPE EQUATIONS ANAHIT GALSTYAN The Tricomi equation u tt tu xx = 0 is a linear partial differential operator of mixed type. (For t > 0, the Tricomi
More informationNecessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrödinger Equation
Necessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrödinger Equation Pascal Bégout aboratoire Jacques-ouis ions Université Pierre et Marie Curie Boîte Courrier 187,
More informationGROUND STATES OF LINEARLY COUPLED SCHRÖDINGER SYSTEMS
Electronic Journal of Differential Equations, Vol. 2017 (2017), o. 05,. 1 10. ISS: 1072-6691. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu GROUD STATES OF LIEARLY COUPLED SCHRÖDIGER SYSTEMS
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationSurfaces of Revolution with Constant Mean Curvature H = c in Hyperbolic 3-Space H 3 ( c 2 )
Surfaces of Revolution with Constant Mean Curvature H = c in Hyerbolic 3-Sace H 3 ( c 2 Kinsey-Ann Zarske Deartment of Mathematics, University of Southern Mississii, Hattiesburg, MS 39406, USA E-mail:
More informationSYMPLECTIC STRUCTURES: AT THE INTERFACE OF ANALYSIS, GEOMETRY, AND TOPOLOGY
SYMPLECTIC STRUCTURES: AT THE INTERFACE OF ANALYSIS, GEOMETRY, AND TOPOLOGY FEDERICA PASQUOTTO 1. Descrition of the roosed research 1.1. Introduction. Symlectic structures made their first aearance in
More informationPropagation of discontinuities in solutions of First Order Partial Differential Equations
Propagation of discontinuities in solutions of First Order Partial Differential Equations Phoolan Prasad Department of Mathematics Indian Institute of Science, Bangalore 560 012 E-mail: prasad@math.iisc.ernet.in
More informationThe Scalar Conservation Law
The Scalar Conservation Law t + f() = 0 = conserved qantity, f() =fl d dt Z b a (t, ) d = Z b a t (t, ) d = Z b a f (t, ) d = f (t, a) f (t, b) = [inflow at a] [otflow at b] f((a)) f((b)) a b Alberto Bressan
More informationA SINGULAR PERTURBATION PROBLEM FOR THE p-laplace OPERATOR
A SINGULAR PERTURBATION PROBLEM FOR THE -LAPLACE OPERATOR D. DANIELLI, A. PETROSYAN, AND H. SHAHGHOLIAN Abstract. In this aer we initiate the study of the nonlinear one hase singular erturbation roblem
More informationInstability of Finite Difference Schemes for Hyperbolic Conservation Laws
Instability of Finite Difference Schemes for Hyperbolic Conservation Laws Alberto Bressan ( ), Paolo Baiti ( ) and Helge Kristian Jenssen ( ) ( ) Department of Mathematics, Penn State University, University
More informationCOMPONENT REDUCTION FOR REGULARITY CRITERIA OF THE THREE-DIMENSIONAL MAGNETOHYDRODYNAMICS SYSTEMS
Electronic Journal of Differential Equations, Vol. 4 4, No. 98,. 8. ISSN: 7-669. UR: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu COMPONENT REDUCTION FOR REGUARITY CRITERIA
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationLecture Notes on Hyperbolic Conservation Laws
Lecture Notes on Hyperbolic Conservation Laws Alberto Bressan Department of Mathematics, Penn State University, University Park, Pa. 16802, USA. bressan@math.psu.edu May 21, 2009 Abstract These notes provide
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationz x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.
Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These
More informationSBV REGULARITY RESULTS FOR SOLUTIONS TO 1D CONSERVATION LAWS
SBV REGULARITY RESULTS FOR SOLUTIONS TO 1D CONSERVATION LAWS LAURA CARAVENNA Abstract. A well-posedness theory has been established for entropy solutions to strictly hyperbolic systems of conservation
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationA PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL
A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL LAPLACE EQUATIONS Abstract. We establish ointwise a riori estimates for solutions in D 1, of equations of tye u = f x, u, where
More informationAn extension to the theory of trigonometric functions as exact periodic solutions to quadratic Liénard type equations
An extension to the theory of trigonometric functions as exact eriodic solutions to quadratic Liénard tye equations D. K. K. Adjaï a, L. H. Koudahoun a, J. Akande a, Y. J. F. Komahou b and M. D. Monsia
More informationWEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS-II
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 94, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu WEAK ASYMPTOTIC
More informationPointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang
Pointwise convergence rate for nonlinear conservation laws Eitan Tadmor and Tao Tang Abstract. We introduce a new method to obtain pointwise error estimates for vanishing viscosity and nite dierence approximations
More informationAuthor(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)
Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL
More informationIntersection Models and Nash Equilibria for Traffic Flow on Networks
Intersection Models and Nash Equilibria for Traffic Flow on Networks Alberto Bressan Department of Mathematics, Penn State University bressan@math.psu.edu (Los Angeles, November 2015) Alberto Bressan (Penn
More informationHyperbolic Systems of Conservation Laws. I - Basic Concepts
Hyperbolic Systems of Conservation Laws I - Basic Concepts Alberto Bressan Mathematics Department, Penn State University Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 1 / 27 The
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationHyperbolic Conservation Laws Past and Future
Hyperbolic Conservation Laws Past and Future Barbara Lee Keyfitz Fields Institute and University of Houston bkeyfitz@fields.utoronto.ca Research supported by the US Department of Energy, National Science
More informationShock formation in the compressible Euler equations and related systems
Shock formation in the compressible Euler equations and related systems Geng Chen Robin Young Qingtian Zhang Abstract We prove shock formation results for the compressible Euler equations and related systems
More informationPropagation of Singularities
Title: Name: Affil./Addr.: Propagation of Singularities Ya-Guang Wang Department of Mathematics, Shanghai Jiao Tong University Shanghai, 200240, P. R. China; e-mail: ygwang@sjtu.edu.cn Propagation of Singularities
More informationThe first order quasi-linear PDEs
Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.
More informationFirst order Partial Differential equations
First order Partial Differential equations 0.1 Introduction Definition 0.1.1 A Partial Deferential equation is called linear if the dependent variable and all its derivatives have degree one and not multiple
More informationSIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM
Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SIMULTANEOUS AND NON-SIMULTANEOUS
More informationThe inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method
The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method Alexis Vasseur, and Yi Wang Department of Mathematics, University of Texas
More informationIMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES
IMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES OHAD GILADI AND ASSAF NAOR Abstract. It is shown that if (, ) is a Banach sace with Rademacher tye 1 then for every n N there exists
More informationRadon measure solutions for scalar. conservation laws. A. Terracina. A.Terracina La Sapienza, Università di Roma 06/09/2017
Radon measure A.Terracina La Sapienza, Università di Roma 06/09/2017 Collaboration Michiel Bertsch Flavia Smarrazzo Alberto Tesei Introduction Consider the following Cauchy problem { ut + ϕ(u) x = 0 in
More informationAlberto Bressan. Department of Mathematics, Penn State University
Non-cooperative Differential Games A Homotopy Approach Alberto Bressan Department of Mathematics, Penn State University 1 Differential Games d dt x(t) = G(x(t), u 1(t), u 2 (t)), x(0) = y, u i (t) U i
More informationNon-radial solutions to a bi-harmonic equation with negative exponent
Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei
More informationSimultaneous vs. non simultaneous blow-up
Simultaneous vs. non simultaneous blow-up Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F.C.E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility
More informationOn uniqueness of weak solutions to transport equation with non-smooth velocity field
On uniqueness of weak solutions to transport equation with non-smooth velocity field Paolo Bonicatto Abstract Given a bounded, autonomous vector field b: R d R d, we study the uniqueness of bounded solutions
More informationI Results in Mathematics
Result.Math. 45 (2004) 293-298 1422-6383/04/040293-6 DOl 10.1007/s00025-004-0115-3 Birkhauser Verlag, Basel, 2004 I Results in Mathematics Further remarks on the non-degeneracy condition Philip Korman
More informationSimultaneous vs. non simultaneous blow-up
Simultaneous vs. non simultaneous blow-up Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F..E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility
More informationConical Shock Waves for Isentropic Euler System
Conical Shock Waves for Isentropic Euler System Shuxing Chen Institute of Mathematical Research, Fudan University, Shanghai, China E-mail: sxchen@public8.sta.net.cn Dening Li Department of Mathematics,
More informationGlobal existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases
Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases De-Xing Kong a Yu-Zhu Wang b a Center of Mathematical Sciences, Zhejiang University Hangzhou
More informationFORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY
Jrl Syst Sci & Complexity (2007) 20: 284 292 FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY Muhammad USMAN Bingyu ZHANG Received: 14 January 2007 Abstract It
More informationOn a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws
On a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws Stefano Bianchini and Alberto Bressan S.I.S.S.A., Via Beirut 4, Trieste 34014 Italy. E-mail addresses: bianchin@mis.mpg.de,
More informationDeng Songhai (Dept. of Math of Xiangya Med. Inst. in Mid-east Univ., Changsha , China)
J. Partial Diff. Eqs. 5(2002), 7 2 c International Academic Publishers Vol.5 No. ON THE W,q ESTIMATE FOR WEAK SOLUTIONS TO A CLASS OF DIVERGENCE ELLIPTIC EUATIONS Zhou Shuqing (Wuhan Inst. of Physics and
More informationOn Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations
On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations G. Seregin, V. Šverák Dedicated to Vsevolod Alexeevich Solonnikov Abstract We prove two sufficient conditions for local regularity
More informationExtremal Solutions of Differential Inclusions via Baire Category: a Dual Approach
Extremal Solutions of Differential Inclusions via Baire Category: a Dual Approach Alberto Bressan Department of Mathematics, Penn State University University Park, Pa 1682, USA e-mail: bressan@mathpsuedu
More informationMath 311, Partial Differential Equations, Winter 2015, Midterm
Score: Name: Math 3, Partial Differential Equations, Winter 205, Midterm Instructions. Write all solutions in the space provided, and use the back pages if you have to. 2. The test is out of 60. There
More informationIntegrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows
Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows Alexander Chesnokov Lavrentyev Institute of Hydrodynamics Novosibirsk, Russia chesnokov@hydro.nsc.ru July 14,
More informationSTRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2
STRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2 ANGELES ALFONSECA Abstract In this aer we rove an almost-orthogonality rincile for
More informationin Bounded Domains Ariane Trescases CMLA, ENS Cachan
CMLA, ENS Cachan Joint work with Yan GUO, Chanwoo KIM and Daniela TONON International Conference on Nonlinear Analysis: Boundary Phenomena for Evolutionnary PDE Academia Sinica December 21, 214 Outline
More informationGENERALIZED FRONTS FOR ONE-DIMENSIONAL REACTION-DIFFUSION EQUATIONS
GENERALIZED FRONTS FOR ONE-DIMENSIONAL REACTION-DIFFUSION EQUATIONS ANTOINE MELLET, JEAN-MICHEL ROQUEJOFFRE, AND YANNICK SIRE Abstract. For a class of one-dimensional reaction-diffusion equations, we establish
More informationWave breaking in the short-pulse equation
Dynamics of PDE Vol.6 No.4 29-3 29 Wave breaking in the short-pulse equation Yue Liu Dmitry Pelinovsky and Anton Sakovich Communicated by Y. Charles Li received May 27 29. Abstract. Sufficient conditions
More informationHyperbolic Gradient Flow: Evolution of Graphs in R n+1
Hyperbolic Gradient Flow: Evolution of Graphs in R n+1 De-Xing Kong and Kefeng Liu Dedicated to Professor Yi-Bing Shen on the occasion of his 70th birthday Abstract In this paper we introduce a new geometric
More informationA global solution curve for a class of free boundary value problems arising in plasma physics
A global solution curve for a class of free boundary value problems arising in plasma physics Philip Korman epartment of Mathematical Sciences University of Cincinnati Cincinnati Ohio 4522-0025 Abstract
More informationINFINITELY MANY SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 2016 2016, No. 188,. 1 9. ISSN: 1072-6691. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu INFINITELY MANY SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS
More informationGlobal unbounded solutions of the Fujita equation in the intermediate range
Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,
More informationThe Nemytskii operator on bounded p-variation in the mean spaces
Vol. XIX, N o 1, Junio (211) Matemáticas: 31 41 Matemáticas: Enseñanza Universitaria c Escuela Regional de Matemáticas Universidad del Valle - Colombia The Nemytskii oerator on bounded -variation in the
More informationSUBORBITAL GRAPHS FOR A SPECIAL SUBGROUP OF THE NORMALIZER OF. 2p, p is a prime and p 1 mod4
Iranian Journal of Science & Technology, Transaction A, Vol. 34, No. A4 Printed in the Islamic Reublic of Iran, Shiraz University SUBORBITAL GRAPHS FOR A SPECIAL SUBGROUP * m OF THE NORMALIZER OF S. KADER,
More informationMATH 220: PROBLEM SET 1, SOLUTIONS DUE FRIDAY, OCTOBER 2, 2015
MATH 220: PROBLEM SET 1, SOLUTIONS DUE FRIDAY, OCTOBER 2, 2015 Problem 1 Classify the following PDEs by degree of non-linearity (linear, semilinear, quasilinear, fully nonlinear: (1 (cos x u x + u y =
More informationFeedback-error control
Chater 4 Feedback-error control 4.1 Introduction This chater exlains the feedback-error (FBE) control scheme originally described by Kawato [, 87, 8]. FBE is a widely used neural network based controller
More informationDependence on Initial Conditions of Attainable Sets of Control Systems with p-integrable Controls
Nonlinear Analysis: Modelling and Control, 2007, Vol. 12, No. 3, 293 306 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls E. Akyar Anadolu University, Deartment
More informationAn analytical approximation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory
Proceedings of the 27 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July -3, 27 ThA8.4 An analytical aroimation method for the stabilizing solution of the Hamilton-Jacobi
More information1 Introduction. Controllability and observability
Matemática Contemporânea, Vol 31, 00-00 c 2006, Sociedade Brasileira de Matemática REMARKS ON THE CONTROLLABILITY OF SOME PARABOLIC EQUATIONS AND SYSTEMS E. Fernández-Cara Abstract This paper is devoted
More informationRelaxation methods and finite element schemes for the equations of visco-elastodynamics. Chiara Simeoni
Relaxation methods and finite element schemes for the equations of visco-elastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy)
More informationGlobal regularity of a modified Navier-Stokes equation
Global regularity of a modified Navier-Stokes equation Tobias Grafke, Rainer Grauer and Thomas C. Sideris Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Department of Mathematics,
More informationRecent Trends in Differential Inclusions
Recent Trends in Alberto Bressan Department of Mathematics, Penn State University (Aveiro, June 2016) (Aveiro, June 2016) 1 / Two main topics ẋ F (x) differential inclusions with upper semicontinuous,
More informationAN ESTIMATE FOR THE BLOW-UP TIME IN TERMS OF THE INITIAL DATA
AN ESTIMATE FOR THE BLOW-UP TIME IN TERMS OF THE INITIAL DATA JULIO D. ROSSI Abstract. We find an estimate for the blow-up time in terms of the initial data for solutions of the equation u t = (u m ) +
More informationVarious behaviors of solutions for a semilinear heat equation after blowup
Journal of Functional Analysis (5 4 7 www.elsevier.com/locate/jfa Various behaviors of solutions for a semilinear heat equation after blowup Noriko Mizoguchi Department of Mathematics, Tokyo Gakugei University,
More informationRemovable singularities for some degenerate non-linear elliptic equations
Mathematica Aeterna, Vol. 5, 2015, no. 1, 21-27 Removable singularities for some degenerate non-linear ellitic equations Tahir S. Gadjiev Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9,
More informationOn semilinear elliptic equations with nonlocal nonlinearity
On semilinear elliptic equations with nonlocal nonlinearity Shinji Kawano Department of Mathematics Hokkaido University Sapporo 060-0810, Japan Abstract We consider the problem 8 < A A + A p ka A 2 dx
More information