# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
$\delta ^{\prime }$-shock waves as a new type of solutions to systems of conservation laws. (English) Zbl 1108.35116
Summary: A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called $\delta^{(n)}$-shock wave, where $\delta^{(n)}$ is $n$th derivative of the Dirac delta function $(n=1,2,\dots)$. In this paper the case $n=1$ is studied in details. We introduce a definition of $\delta'$-shock wave type solution for the system $$u_t+\bigl(f(u)\bigr)_x=0,\quad v_t+\bigl(f' (u)v\bigr)_x=0, \quad w_t+\bigl(f''(u)v^2+f'(u)w\bigr)_x=0.$$ Within the framework of this definition, the Rankine-Hugoniot conditions for $\delta'$-shock are derived and analyzed from geometrical point of view. We prove $\delta'$-shock balance relations connected with area transportation. Finally, a solitary $\delta'$-shock wave type solution to the Cauchy problem of the system of conservation laws $u_t+(u^2)_x= 0$, $v_t+2(uv)_x =0$, $w_t+2(v^2+uw)_x=0$ with piecewise continuous initial data is constructed. These results first show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of $\delta$-shocks) but their derivatives as well.

##### MSC:
 35L67 Shocks and singularities 35L65 Conservation laws 76L05 Shock waves; blast waves (fluid mechanics)
Full Text:
##### References:
 [1] Bouchut, F.: On zero pressure gas dynamics. Ser. adv. Math. appl. Sci. 22, 171-190 (1994) · Zbl 0863.76068 [2] Danilov, V. G.; Maslov, V. P.; Shelkovich, V. M.: Algebra of singularities of singular solutions to first-order quasilinear strictly hyperbolic systems. Theoret. math. Phys. 114, No. 1, 1-42 (1998) · Zbl 0946.35049 [3] Danilov, V. G.; Omel’yanov, G. A.; Shelkovich, V. M.: Weak asymptotics method and interaction of nonlinear waves. Amer. math. Soc. transl. Ser. 2 208, 33-165 (2003) [4] Danilov, V. G.; Shelkovich, V. M.: Propagation and interaction of nonlinear waves to quasilinear equations. Internat. ser. Numer. math. 140, 267-276 (2001) · Zbl 1008.35041 [5] Danilov, V. G.; Shelkovich, V. M.: Propagation and interaction of shock waves of quasilinear equation. Nonlinear stud. 8, No. 1, 135-169 (2001) · Zbl 1008.35041 [6] Danilov, V. G.; Shelkovich, V. M.: Propagation and interaction of delta-shock waves of a hyperbolic system of conservation laws. Hyperbolic problems: theory, numerics, applications, 483-492 (2003) · Zbl 1098.35106 [7] Danilov, V. G.; Shelkovich, V. M.: Propagation and interaction of $\delta$-shock waves to hyperbolic systems of conservation laws. Dokl. ross. Akad. nauk 394, No. 1, 10-14 (2004) [8] Danilov, V. G.; Shelkovich, V. M.: Delta-shock wave type solution of hyperbolic systems of conservation laws. Quart. appl. Math. 63, No. 3, 401-427 (2005) [9] Danilov, V. G.; Shelkovich, V. M.: Dynamics of propagation and interaction of delta-shock waves in conservation law systems. J. differential equations 211, 333-381 (2005) · Zbl 1072.35121 [10] Weinan, E.; Rykov, Yu.; Sinai, Ya.G.: Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particles dynamics. Comm. math. Phys. 177, 349-380 (1996) · Zbl 0852.35097 [11] Ercole, G.: Delta-shock waves as self-similar viscosity limits. Quart. appl. Math. 58, No. 1, 177-199 (2000) · Zbl 1157.35430 [12] Evans, L. C.: Partial differential equations. (1998) · Zbl 0902.35002 [13] Joseph, K. T.: Explicit generalized solutions to a system of conservation laws. Proc. indian acad. Sci. math. Sci. 109, No. 4, 401-409 (1999) · Zbl 0941.35050 [14] Keyfitz, B. Lee; Kranzer, H. C.: Spaces of weighted measures for conservation laws with singular shock solutions. J. differential equations 118, 420-451 (1995) · Zbl 0821.35096 [15] Le Floch, P.: An existence and uniqueness result for two nonstrictly hyperbolic systems. IMA vol. Math. appl. 27, 126-138 (1990) · Zbl 0727.35083 [16] Le Floch, P.: Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form. Comm. partial differential equations 13, No. 6, 669-727 (1988) · Zbl 0683.35049 [17] Le Floch, P.; Liu, Tai-Ping: Existence theory for nonlinear hyperbolic systems in nonconservative form. Forum math. 5, No. 3, 261-280 (1993) · Zbl 0804.35086 [18] Li, J.; Zhang, Tong: On the initial-value problem for zero-pressure gas dynamics. Hyperbolic problems: theory, numerics, applications, 629-640 (1999) · Zbl 0926.35120 [19] Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. (1984) · Zbl 0537.76001 [20] Dal Maso, G.; Le Floch, P. G.; Murat, F.: Definition and weak stability of nonconservative products. J. math. Pures appl. 74, 483-548 (1995) · Zbl 0853.35068 [21] Shelkovich, V. M.: Associative and commutative distribution algebra with multipliers, and generalized solutions of nonlinear equations. Mat. zametki 57, No. 5, 765-783 (1995) · Zbl 0868.46028 [22] Shelkovich, V. M.: Delta-shock waves of a class of hyperbolic systems of conservation laws. Patterns and waves, 155-168 (2003) [23] V.M. Shelkovich, A specific hyperbolic system of conservation laws admitting delta-shock wave type solutions, preprint 2003-059 at the url: http://www.math.ntnu.no/conservation/2003/059.html [24] V.M. Shelkovich, Delta-shocks, the Rankine -- Hugoniot conditions, and singular superposition of distributions, Proc. of Internat. Seminar Days on Difraction’2004, 29 June -- 2 July 2004, Fakulty of Physics, St. Petersburg, 2004, pp. 175 -- 196 (see also preprint 2004-059 at the url: http://www.math.ntnu.no/conservation/2004/059.html) [25] V.M. Shelkovich, Delta-shocks, the Rankine -- Hugoniot conditions and the balance laws for \delta -shocks, Fund. Appl. Math., submitted for publication · Zbl 1151.35400 [26] Tan, Dechun; Zhang, Tong; Zheng, Yuxi: Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J. differential equations 112, 1-32 (1994) · Zbl 0804.35077 [27] Volpert, A. I.: The space BV and quasilinear equations. Math. USSR sb. 2, 225-267 (1967) [28] Yang, Hanchun: Riemann problems for class of coupled hyperbolic systems of conservation laws. J. differential equations 159, 447-484 (1999) · Zbl 0948.35079