Initial-boundary value problem for Euler equations with incompatible data
Author:
Dening Li
Journal:
Quart. Appl. Math. 76 (2018), 47-64
MSC (2010):
Primary 35L50, 35Q31; Secondary 35L67, 76N10
DOI:
https://doi.org/10.1090/qam/1490
Published electronically:
October 16, 2017
MathSciNet review:
3733094
Full-text PDF
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Additional Information
Abstract: We study the initial-boundary value problem for the general 3-D Euler equations with data which are incompatible in the classical sense, but are “shock-compatible”. We show that such data are also shock-compatible of infinite order and the initial-boundary value problem has a piece-wise smooth solution containing a shock.
References
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- An Ton Bui and De Ning Li, Double shock fronts for hyperbolic systems of conservation laws in multidimensional space, Trans. Amer. Math. Soc. 316 (1989), no. 1, 233–250. MR 935939, DOI https://doi.org/10.1090/S0002-9947-1989-0935939-1
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- Shuxing Chen and Dening Li, Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system, J. Differential Equations 257 (2014), no. 6, 1939–1988. MR 3227287, DOI https://doi.org/10.1016/j.jde.2014.05.027
- Jean-François Coulombel and Paolo Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 85–139 (English, with English and French summaries). MR 2423311, DOI https://doi.org/10.24033/asens.2064
- R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1948 original; Applied Mathematical Sciences, Vol. 21. MR 0421279
- K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 100718, DOI https://doi.org/10.1002/cpa.3160110306
- Mitsuru Ikawa, Mixed problem for a hyperbolic system of the first order, Publ. Res. Inst. Math. Sci. 7 (1971/72), 427–454. MR 0330785, DOI https://doi.org/10.2977/prims/1195193549
- Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 437941, DOI https://doi.org/10.1002/cpa.3160230304
- De Ning Li, Rarefaction and shock waves for multidimensional hyperbolic conservation laws, Comm. Partial Differential Equations 16 (1991), no. 2-3, 425–450. MR 1104106, DOI https://doi.org/10.1080/03605309108820764
- Dening Li, Compatibility of jump Cauchy data for non-isentropic Euler equations, J. Math. Anal. Appl. 425 (2015), no. 1, 565–587. MR 3299680, DOI https://doi.org/10.1016/j.jmaa.2014.12.053
- Ta Tsien Li and Wen Ci Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series, V, Duke University, Mathematics Department, Durham, NC, 1985. MR 823237
- Andrew Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR 683422, DOI https://doi.org/10.1090/memo/0275
- Jeffrey B. Rauch and Frank J. Massey III, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303–318. MR 340832, DOI https://doi.org/10.1090/S0002-9947-1974-0340832-0
- J. Rauch, Boundary value problems with nonuniformly characteristic boundary, J. Math. Pures Appl. (9) 73 (1994), no. 4, 347–353. MR 1290491
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- David S. Tartakoff, Regularity of solutions to boundary value problems for first order systems, Indiana Univ. Math. J. 21 (1971/72), 1113–1129. MR 440182, DOI https://doi.org/10.1512/iumj.1972.21.21089
References
- S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14 (1989), no. 2, 173–230 (French, with English summary). MR 976971, DOI https://doi.org/10.1080/03605308908820595
- An Ton Bui and De Ning Li, Double shock fronts for hyperbolic systems of conservation laws in multidimensional space, Trans. Amer. Math. Soc. 316 (1989), no. 1, 233–250. MR 935939, DOI https://doi.org/10.2307/2001282
- Shuxing Chen, Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary, Front. Math. China 2 (2007), no. 1, 87–102. Translated from Chinese Ann. Math. 3 (1982), no. 2, 222–232 [MR0663102]. MR 2289911, DOI https://doi.org/10.1007/s11464-007-0006-5
- Shuxing Chen and Dening Li, Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system, J. Differential Equations 257 (2014), no. 6, 1939–1988. MR 3227287, DOI https://doi.org/10.1016/j.jde.2014.05.027
- Jean-François Coulombel and Paolo Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 85–139 (English, with English and French summaries). MR 2423311, DOI https://doi.org/10.24033/asens.2064
- R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1948 original; Applied Mathematical Sciences, Vol. 21. MR 0421279
- K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 0100718, DOI https://doi.org/10.1002/cpa.3160110306
- Mitsuru Ikawa, Mixed problem for a hyperbolic system of the first order, Publ. Res. Inst. Math. Sci. 7 (1971/72), 427–454. MR 0330785, DOI https://doi.org/10.2977/prims/1195193549
- Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 0437941, DOI https://doi.org/10.1002/cpa.3160230304
- De Ning Li, Rarefaction and shock waves for multidimensional hyperbolic conservation laws, Comm. Partial Differential Equations 16 (1991), no. 2-3, 425–450. MR 1104106, DOI https://doi.org/10.1080/03605309108820764
- Dening Li, Compatibility of jump Cauchy data for non-isentropic Euler equations, J. Math. Anal. Appl. 425 (2015), no. 1, 565–587. MR 3299680, DOI https://doi.org/10.1016/j.jmaa.2014.12.053
- Ta Tsien Li and Wen Ci Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series, V, Duke University, Mathematics Department, Durham, NC, 1985. MR 823237
- Andrew Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR 683422, DOI https://doi.org/10.1090/memo/0275
- Jeffrey B. Rauch and Frank J. Massey III, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303–318. MR 0340832, DOI https://doi.org/10.2307/1996861
- J. Rauch, Boundary value problems with nonuniformly characteristic boundary, J. Math. Pures Appl. (9) 73 (1994), no. 4, 347–353. MR 1290491
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- David S. Tartakoff, Regularity of solutions to boundary value problems for first order systems, Indiana Univ. Math. J. 21 (1971/72), 1113–1129. MR 0440182, DOI https://doi.org/10.1512/iumj.1972.21.21089
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Additional Information
Dening Li
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-0001
MR Author ID:
194475
Email:
li@math.wvu.edu / dnli@hotmail.com
Keywords:
Initial boundary value problem,
Euler system,
shock.
Received by editor(s):
March 25, 2017
Published electronically:
October 16, 2017
Article copyright:
© Copyright 2017
Brown University
