Your browser do not support script. Here are news slide. Research Staff

# Liu, Tai-Ping

 Dr. Liu's research interests center around Nonlinear partial differential equations, shock wave theory, and kinetic theory.   Email : tpliu AT math.sinica.edu.tw Phone:+886 2 2368-5999 ext. 704 Fax: +886 2 2368-9771

### Publications :

 1 The Riemann problem for general 2 X 2 conservation Laws, Transactions of A.M.S. , 199(1974), 89-112 2 The Riemann Problem for general systems of conservation laws, J. Diff. Equations , 18(1975), 218-234 3 Existence and uniqueness theorems for Riemann problems, Transactions of A.M.S. , 212(1975), 375-382 4 Uniqueness theorem of the Cauchy problem for general 2 X 2 conservation laws, J. Diff. Eqs. , 20(1976), 369-388 5 Shock waves in nonisentropic gas flow, J. Diff. Equations , 22(1976), 442-452 6 The entropy condition and the admissibility of shocks, J. of Math. Anal. and Applications , 53(1976), 75-88 7 Solutions in the large for equations of nonisentropic gas dynamics, Indiana Univ. J. , 26(1977), 144-177 8 Initial-boundary value problems for gas dynamics, Arc. Rat. Mech. and Anal. , 64(1977), 137-168 9 Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math , 30(1977), 586-611 10 Large-time behavior of solutions of initial and initial-boundary value problem of general system of hyperbolic conservation laws, Comm. Math. Phys. , 55(1977), 163-177 11 Linear and nonlinear large-time behaviors of solutions of hyperbolic conservation laws, Comm. Pure Appl. Math. , 30(1977), 767-796 12 The deterministic version of the Glimm scheme, Comm. Math. Phys. , 57(1977), 135-148 13 Asymptotic behavior of solutions of general systems of nonlinear hyperbolic conservation laws, Indiana Univ. J. , 27(1978), 211-253 14 Invariants and asymptotic behavior of solution of a conservation law, Proceedings of A.M.S. , 71(1978), 227-231 15 The free piston problem for gas dynamics, J. of Diff. Equations , 39(1978), 175-191 16 (with S. Antman) Traveling waves in hyperelastic rods, Quarterly of Appl. Math. , 36(1979), 377-399 17 Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations, J. Diff. Equations, , 33(1979), 92-111 18 Quasilinear hyperbolic systems68, Comm. Math. Phys. , 68(1979), 142-172 19 A system of quasilinear hyperbolic partial differential equations. Trends in applications of pure mathematics to mechanics, Vol. III (Edinburgh, 1979), 179-189., (1979) 20 probleme du vide pour les equations de la dynamique des gaz isentropiques. (French) [The problem of the vacuum for equations in isentropic gas dynamics] Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), 319--326, 395., (1979/1980) 21 (with J. Smoller) On the vacuum state for isentropic gas dynamics equations, Advances in Math. , 1(1980), 345-359 22 (with Conlon, Joseph G.) Admissibility criteria for hyperbolic conservation laws, Indiana Univ. Math. J. , 30(1981), no. 5, 641-652 23 Admissible solutions of hyperbolic conservation laws, Memoirs, Amer. Math. Soc. , 240(1981) 24 Transonic gas flow in a duct of varying area, Arch. Rat. Mech. and Anal. , 80(1982), 1-18 25 Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. , 83(1982), 243-260 26 (with A. Chorin and Z. H. Teng) Riemann problems for reacting gas with applications to transitions, SIAM J. Appl. Math. , 42(1982), 964-981 27 Resonance for quasilinear hyperbolic equation, Bulletin Amer. Math. Soc. , 6(1982), 463-465 28 (with C. Li) Asymptotic states for hyperbolic conservation laws with a moving source, Advances in Appl. Math. , 4(1983), 353-379 29 (with M. Pierre) Source-solutions and asymptotic behavior in conservation law, J. Diff. Equ. , 51(1984), 419-441 30 (with H. Galz) The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow, Advances Appl. , 5(1984), 111-146 31 Stability of nonlinear waves. Metastability and incompletely posed problems, Minneapolis, Minn. (1985), 259-267 32 (with C. H. Wang) On a system of nonstrictly hyperbolic system, J.D.E. , 57(1985), 1-14 33 Nonlinear stability of shock waves for viscous conservation laws, Amer. Math. Soc. (N.S.) , 12(1985), no. 2, 233-236 34 Nonlinear stability of shock waves for viscous conservation laws, Memoirs, Amer. Math. Soc. , 328(1985) 35 Shock waves and diffusion waves. Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), 283--291, Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, 1986., (1986) 36 (with L. P. Huang) Piecewise steady elements and Godunov-type difference scheme for duct flow, Comp. & Maths. with Appls. , 12A(1986), 377-388 37 Shock waves for compressible Navier-Stokes equations are stable, Comm. Pure Appl. Math. , 39(1986), 565-594 38 (with R. Caflisch) Stability of shock waves for the Broadwell equations, Comm. Math. Phys. , 39(1986), 565-594 39 (with I-L. Chern) Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. , 110(1987), 503-517 40 Hyperbolic conservation laws and related topics. Partial differential equations of hyperbolic type and applications, 113--139, World Sci. Publishing, Singapore, 1987., (1987) 41 Pointwise convergence to $N$-waves for hyperbolic conservation laws, Bulletin, Inst. Math., Academia Sinica , 15(1987), 1-17 42 Hyperbolic conservation laws with relaxation, Comm. Math. Phys. , 108(1987), 153-175 43 Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. , 28(1987), 2593-2602 44 Nonlinear waves for viscoelasticity with fading memory, J. Diff. Equ. , 76(1988), 26-46 45 (with Z. Xin) Nonlinear stability of rarefaction waves for compressible Navier--Stokes equations, Comm. Math. Phys. (1988), 451-465 46 Quasilinear hyperbolic-parabolic partial differential equations, Algebra, analysis and geometry (Taipei) (1988), 185-197 47 (with Hsu, Sze-Bi) Local uniqueness of interior layer solution and its related stability property., Nonlinear analysis (Taipei) (1989), 51-60 48 (with D. Hoff) The inviscid limit for the Navier-Stokes equations of compressible, instropic flow with shock data, Indiana Univ. J. , 38(1989), 861-915 49 Geometric theory of shock waves. Multidimensional hyperbolic problems and computations (Minneapolis, MN, 1989), 198-202., (1989) 50 Nonlinear analysis. Papers from the conference held in Taipei, June 19--24, 1989, (1989) 51 Nonlinear hyperbolic waves. Nonlinear hyperbolic equations in applied sciences. Rend. Sem. Mat. Univ. Politec. Torino 1988, Special Issue, 269--277 (1989)., (1989) 52 (with Z. Xin) Over compressive shock waves nonlinear evolution equations that change types, ed. B. Keyfitz and M. Shearer, IMA volumes in Math Appls. , 27(1990), 139-145 53 (with S. Hsu) Nonlinear singular Sturm-Liouville problem and application to transonic flow through a nozzle, Comm. Pure Appl. Math. , 43(1990), 31-61 54 On the viscosity criterion for hyperbolic conservation laws. Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990), 105-114., (1990) 55 (with T. Zheng) A scalar combustion model, Arch. Rational Mech. Anal. , 114(1991), 297-312 56 Viscosity criterion for hyperbolic conservation laws, Proc. SIAM. Ed. M. Shearer (1991) 57 Nonlinear hyperbolic-parabolic p.d.e., Nonlinear Analysis, Academia Sinica, R.O.C., World Scientific (1991), 161-170 58 Interaction of nonlinear hyperbolic waves, Nonlinear Analysis, Academia Sinica, R.O.C., World Scientific (1991), 171-184 59 (with Z. Xin) Stability of viscous shock waves associated with a nonstrictly hyperbolic conservantion law, Comm. Pure Appl. Math., XLV (1992), 361-388., (1992) 60 (with S. Cox) Zero-memory limit for a viscoelasticity model with Riemann data (preprint)., 61 (with L. Hsiao) (with ) Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. , 143(1992), 599-605 62 (with G.-Q. Chen) Zero relaxation and dissipation limits for hyperbolic conservation laws, Comm. Pure Appl. Math. , 46(1993), 755-781 63 (with H. Freistühler) Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscious conservation laws, Comm. Math. Physics , 153(1993), 147-158 64 (with P. Le Floch) Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. , 5(1993), 261-280 65 (with L. Hsiao) Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chin.Ann. Math. , 14B(1993), 465-480 66 Nonlinear evolutionary partial differential equations., Proceedings of the International Conference held in Beijing (1993-06), 21-25 67 Nonlinear waves for viscous conservation laws, Nonlinear evolutionary partial differential equations (1993), 65-74 68 (with Hsiao, Ling) Nonlinear diffusive phenomena of nonlinear hyperbolic systems. A Chinese summary appears in Chinese Ann. Math. Ser. A 14 (1993), no. 6, 740. Chinese Ann. Math. Ser. B 14 (1993), no. 4, 465--480., (1993) 69 (with Bolliat, G.) Dafermos, C. M.; Lax, P. D. Recent mathematical methods in nonlinear wave propagation. Lectures given at the 1st C.I.M.E. Session held in Montecatini Terme, May 23--31, 1994., (1994-05) 70 (with G.-Q. Chen and D. Levermore) Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Applied Math. , 47(1994), 787-830 71 (with L. Ying) Nonliner stability strong detonations for a combustion model, SIAM J. of Math. Anal. , 26(1995), 519-528 72 (with K. Zumbrun) Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Physics , 168(1995), 163-186 73 (with K. Zumbrum) On nonlinear stability of general undercompressive viscous shock waves, Comm. Math. Phys. , 175(1995), 319-345 74 Stability and viscosity criterion of shocks. Proceedings of the VIII International Conference on Waves and Stability in Continuous Media, Part I (Palermo, 1995). Rend. Circ. Mat. Palermo (2) Suppl. No. 45, part I (1996), 357--366., (1996) 75 Nonlinear hyperbolic-dissipative partial differential equations. Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994), 103--136, Lecture Notes in Math., 1640, Springer, Berlin, 1996., Recent mathematical methods in nonlinear wave propagation (1994/1996) 76 Gas flows with damping and vacuum, Japan J. Indus. and Appl. Math. , 13(1996), 25-32 77 (with D. Li and D. Tan) Stability of strong detonation travelling waves to combustion model, J. Math. Anal. Appl. , 201(1996), 516-531 78 (with Y. Zeng) Large time behavior of solutions of general quasilinear hyperbolic-parabolic systems of conservation laws, Memoirs, Amer. Math. Soc. , 599(1997) 79 (with S. Yu) Propagation of Burgers stationary shocks under boundary effect, Archive Rational Mech. Anal. , 139(1997), 57-82 80 Pointwise convergence to shock waves for viscous conservaton laws, Comm. Pure Appl. Math. L (1997), Comm. Pure Appl. Math. , L(1997), 1113-1182 81 (with T. Yang) Compressible Euler equations with vacuum, J.D.E. , 140(1997), 223-237 82 (with Z. Xin) Pointwise decay to contact continuities for viscous conservation laws, Asian J. Math. , 1(1997), 34-84 83 (with K. Nishihara) Asymptotic behaviour for scalar viscous conservation law with boundary effect, J.D.E. , 133(1997), 296-320 84 (with T. Yang) L1 stability of constant states for hyperbolic conservation laws, Methods and Appl. Analysis , 4(1997), 339-355 85 Regularization of solution operators for quasilinear hyperbolic-parabolic partial differential equations, J. Partial Differential Equations , 10(1997), no. 4, 347-354 86 (with Y. Zeng) Compressible Navier-Stokes equations with zero heat conductivity, J. D. E. (1998) 87 (with S.-H. Yu) Continuum shock profiles for discrete conservation laws, I. Construction, Comm. Pure Appl. Math. (1998) 88 (with W. Wang) The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd dimensions, Comm. Math. Phys. , 196(1998), 145-173 89 (with W.-C. Lien) Nonlinear stability of a self-similar 3-dimensional gas flow, Comm. Math. Phys. , 204(1999), no. 3, 525-549 90 Zero dissipation and stability of shock waves, Methods and Appl. Analysis , 5(1998), 81-94 91 (with Z. Xin and T. Yang) Vacuum states for compressible flow, Disc. Cont. Dyn. Ana. , 4(1998), 1-32 92 (with T. Yang) (with ) L1 stability of 2x2 systems of hyperbolic conservation laws, J. Amer. Math. Soc. (1998), 729-772 93 (with T. Yang) L1 stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. J. (1998) 94 (with SIAM J. Math. Anal.) Behaviours of solutions for the Burgers equation with boundary corresponding to rarefaction waves, A. Matsumura and K. Nishihar , 29(1998), 293-308 95 (with A. Bressan and T. Yang) L1 stability estimates for nxn conservatiob laws, Archive Rational Mech. Anal. (1998) 96 (with T. Yang) A generalized entropy for scalar conservation laws, Comm. Pure Appl. Math. , LII(1999), 1427-1442 97 (with S.-H. Yu) Continuum shock profiles for discrete conservation laws, II. Stability. Comm. Pure Appl. Math. LII (1999), 1047-1073., (1999) 98 (with T. Yang) Well-posedness theory for hyperbolic conservation l aws, Comm. Pure Appl. Math. LII, Comm. Pure Appl. Math. , II(1999), 1553-1586 99 (with S.-H. Yu) Nonlinear stability of weak detonation waves for a combustion model, Comm. Math. Phys. , 204(1999), 551-586 100 (with W.-C. Lien) Nonlinear stability of a self-similar 3-dimensional gas flow, Comm. Math. Phys. , 204(1999), no. 3, 525-549 101 (with Yang, Tong) Well-posedness theory for system of hyperbolic conservation laws. Hyperbolic problems: theory, numerics, applications, Vol.II, (Z�rich,1998), 681-691, Internat. Ser. Numer. Math., 130, Birkh酳ser, Basel, 1999., , II(1998/1999) 102 L1 stability for systems of hyperbolic conservation laws.'' Nonlinear partial differential equations (Evanston, IL, 1998), 183-192, Contemp. Math., 238, Amer. Math. Soc., Providence, RI, 1999., Yang, Tong (1999) 103 (with S.H. Yu) 145Hyperbolic conservation laws and dynamic systems, Discrete Contin. Dynam. Systems , 6(2000), no. 1, 143-145 104 Hyperbolic and Viscous Conservation Laws. CBMS-NSF Regional Conference Series in Applied Mathematics, 72. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000., (2000) 105 Shock waves. Proceedings of the International Congress of Mathematicians, Higher Ed. Press (Beijing) , III(2002), 185-188 106 (with T. Yang and Y. Zheng) Some nonlinear functionals for scalar conservation laws, Methods Appl. Anal. , 8(2001), no. 4, 609-622 107 (with T. Yang) Weak Solutions to General Systems of Hyperbolic Conservation Laws, Comm. Math. Phys. , 230(2002), no. 2, 289-327 108 (with T. Ruggeri) Entropy production and admissibility of shocks, Acta Math. Appl. Sin. Engl. Ser. , 19(2003), 1-12 109 (with Y. Deng, T. Yang, and T. Yao) Solutions with vacuum for Euler-Poisson equations, Arch. Rational Mech. Anal. , 164(2002), no. 3, 261-285 110 (with Yu, Shih-Hsien) Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys. , 246(2004), no. 1, 133-179 111 (with Yang, Tong; Yu, Shih-Hsien) Energy method for Boltzmann equation, Phys. D , 188(2004), no. 3-4, 178-192 112 (with Yu, Shih-Hsien) (with ) The Green's function and large-time behavior of solutions for one-dimensional Boltzmann equation, Comm. Pure. Appl. Math. , 57(2004), no. 12, 1543-1608 113 (with C.-C. Chen and T. Yang) Existence of boundary layer solutions to the Boltzmann equation, Anal. Appl. (Singap.) , 2(2004), no. 4, 337-363 114 (with C.-Y., Lan;H.-E., Lin;S.-H. Yu) Propagation of viscous shock waves away from the boundary., SIAM J. Math. Anal. , 36(2004), no. 2, 580-617 115 (with V. Elling) The ellipticity principle for self-similar potential flows., J. Hyperbolic Differ. Equ. , 2(2005), no. 4, 909-917 116 (with T. Yang, S.-H. Yu, H.-J. Zhao) Nonlinear stability of rarefaction waves for the Boltzmann equation., Arch. Rational Mech. Anal. , 181(2006), no. 2, 333-371 117 (with S.-H. Yu) Green's function of Boltzmann equation, 3-D, Bull. Inst. Math. Acad. Sin. (N.S.) , 1(2006), no. 1, 1-78 118 (with Chen, Chiun-Chuan; Chen, I-Kun; Sone, Yoshio) Thermal transpiration for the linearized Boltzmann equation., Comm. Pure Appl. Math. , 60(2007), no. 2, 147-163 119 (with Lee, Ming-Yi; Yu, Shih-Hsien) Large-time behavior of solutions for the Boltzmann equation with hard potentials., Comm. Math. Phys. , 269(2007), no. 1, 17-37 120 (with Yu, Shih-Hsien) Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation., Comm. Pure Appl. Math. , 60(2007), no. 3, 295-356 121 (with Elling, Volker) Supersonic flow onto a solid wedge., Comm. Pure Appl. Math. , 61(2008), no. 10, 1347-1448 122 (with Yu, Shih-Hsien) Diffusion under gravitational and boundary effects., Bull. Inst. Math. Acad. Sin. (N.S.) , 3(2008), no. 1, 167-210 123 (with S.-H., Yu) Boltzmann equation, boundary effects., Discrete Contin. Dyn. Syst. , 24(2009), no. 1, 145-157 124 (with Y., Zeng) Time-asymptotic behavior of wave propagation around a viscous shock profile., Comm. Math. Phys. , 290(2009), no. 1, 23-82 125 (with S.-H., Yu) Viscous shock wave tracing, local conservation laws, and pointwise estimates, Bull. Inst. Math. Aca. (N.S.) , 4(2009), no. 3, 167-210 126 (with Zeng, Yanni) On Green's function for hyperbolic-parabolic systems, Acta Math. Sci. Ser. B Engl. Ed. , 29(2009), no. 6, 1556-1572 127 (with Kuo, Hung-Wen; Noh, Se Eun) Mixture lemma, Bull. Inst. Math., Acad. Sin. (N.S.) , 5(2010), no. 1, 1-10 128 (with Yu S.-H.) Viscous rarefaction waves, Bull. Inst. Math. Acad. Sin. (N.S.) , 5(2010), no. 2, 123-179 129 (with S.-H. Yu; Y. Zeng) Viscous Conservation Laws, Part I: Scalar Laws, Bull. Inst. Math., Acad. Sin. (N.S.) , 5(2010), no. 3, 233-310 130 (with Yu, Shih-Hsien) Solving Boltzmann equation, Part I: Green's function, Bull. Inst. Math. Acad. Sin. (N.S.) , 6(2011), no. 1, 115-243 131 (with Yu S.-H.) On boundary relation for some dissipative systems, Bull. Inst. Math. Acad. Sin. (N.S.) , 6(2011), no. 3, 245-267 132 Multi-dimensional gas flow: some historical perspectives, Bull. Inst. Math. Acad. Sin. (N.S.) , 6(2011), no. 3, 269-291 133 (with Yu S.-H.) Dirichlet-Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. (N.S.) , 7(2012), no. 4, 477-543 134 (with Kuo, Hung-Wen and Tsai, Li-Cheng) Free molecular flow with boundary effect, Comm. Math. Phys. , 318(2013), no. 2, 375-409 135 Aspects of dissipation for compressible fluids and kinetic theory, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 371(2013), no. 2005 136 (with Yu, Shih-Hsien) Invariant manifolds for steady Boltzmann equation and applications, Archive Rational Mechanics and Analysis , 209(2013), no. 3, 869-997 137 (with Chen, I-Kun and Takata, Shigeru) Boundary singularity for thermal transpiration problem of the linearized Boltzmann equation, Arch. Ration. Mech. ANal. , 212(2014), no. 2, 575-595 138 Equilibrating effects of boundary and collision in rarefied gases., Comm. Math. Phys. , 318(2014), no. 2, 421-480 139 (with Yu, Shih-Hsien) Fujita equation with boundary effect, Bull. Inst. Math. Acad. Sin. (N.S.) , 9(2014), no. 2, 267-282 140 (with Yu, Shih-Hsien) Boundary wave propagator for compressible Navier-Stokes equations., Found. Comput. Math. , 14(2014), no. 6, 1287-1335 141 (with Yu S.-H.) Multi-dimensional wave propagation over a Burgers shock profile, Archive for Rational Mechanics and Analysis , 229(2018), 231-337 142 (with Zeng, Yanni) Shock waves in conservation laws with physical viscosity., Mem. Amer. Math. Soc. , 234(2015), no. 1105 143 (with Noh, Se Eun) Wave propagation for the compressible Navier-Stokes equations., J. Hyperbolic Differ. Equ. , 12(2015), no. 2, 385-445 144 (with Chen, I-Kun; Funagane, Hitoxhi; and Takata, Shigeru) Singularity of the velocity distribution function in molecular velocity space., Comm. Math. Phys. , 341(2016), no. 1, 105-134 145 Hopf-Cole transformation, Bull. Inst. Math. Acad. Sin. (N.S.) , 12(2017), no. 1, 71-101