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**Multiplication of distributions and travelling wave solutions for the Keyfitz-Kranzer system.**
*(English)*
Zbl 1412.46049

Since the impossibility result on the multiplication of distributions, it is well known that the space of distributions is not suitable for studying nonlinear operations, in particular, nonlinear partial differential equations. Nevertheless, certain authors have introduced an intrinsic multiplication of distributions which works for some classes of distributions. The author of this paper first recalls the definition of an intrinsic multiplication called \(\alpha\)-product of distributions depending on a regularizing test function \(\alpha\) satisfying \(\int \alpha =1,\) and gives some properties of this product. Rephrasing the following system of nonlinear partial differential equations
\[
\begin{aligned} u_{t}+(u^{2}-v)_{x}&=0, \\ v_{t}+(u^{3}/3-u)_{x}&=0,\end{aligned}
\]
on the basis of his definition, he introduces the concept of \(\alpha\)-solutions for the system. A comparison with classical solutions is given. In the last section, travelling wave solutions with distributional profiles for the cited system are studied as well as necessary and sufficient conditions for the propagation of distributional wave profiles.

Reviewer: Chikh Bouzar (Oran)

### MSC:

46F10 | Operations with distributions and generalized functions |

35D99 | Generalized solutions to partial differential equations |

35L67 | Shocks and singularities for hyperbolic equations |

35F50 | Systems of nonlinear first-order PDEs |

### Keywords:

product of distributions; conservation laws; travelling shock waves; travelling delta waves; travelling waves which are not measures; Keyfitz-Kranzer system
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\textit{C. O. R. Sarrico}, Taiwanese J. Math. 22, No. 3, 677--693 (2018; Zbl 1412.46049)

### References:

[1] | A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Bull. Un. Mat. Ital. B (7) 2 (1988), no. 3, 641–656. · Zbl 0653.49002 |

[2] | J.-J. Cauret, J.-F. Colombeau, A. Y. LeRoux, Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations, J. Math. Anal. Appl. 139 (1989), no. 2, 552–573. · Zbl 0691.35057 |

[3] | J.-F. Colombeau and A. LeRoux, Multiplications of distributions in elasticity and hydrodynamics, J. Math. Phys. 29 (1988), no. 2, 315–319. · Zbl 0646.76007 |

[4] | G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. · Zbl 0853.35068 |

[5] | V. G. Danilov, V. P. Maslov and V. M. Shelkovich, Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems, Theoret. Mat. Phys. 114 (1998), no. 1, 1–42. · Zbl 0946.35049 |

[6] | V. G. Danilov and D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differerential Equations 245 (2008), no. 12, 3704–3734. · Zbl 1192.35120 |

[7] | Yu. V. Egorov, On the theory of generalized functions, Uspekhi Mat. Nauk 45 (1990), no. 5, 3–40. · Zbl 0754.46034 |

[8] | B. T. Hayes and P. G. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), no. 6, 1547–1563. · Zbl 0908.35075 |

[9] | H. Kalisch and D. Mitrović, Singular solutions of a fully nonlinear \(2 × 2\) system of conservation laws, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 3, 711–729. · Zbl 1286.35162 |

[10] | B. L. Keyfitz, Conservation laws, delta-shocks and singular shocks, in: Nonlinear Theory of Generalized Functions (Vienna, 1997), 99–111, Chapman & Hall/CRC Res. Notes Math 401, Chapman & Hall/CRC, Boca Raton, FL, 1999. · Zbl 0933.35134 |

[11] | ——–, Singular shocks: retrospective and prospective, Confluentes Math. 3 (2011), no. 3, 445–470. · Zbl 1237.35113 |

[12] | B. L. Keyfitz and H. C. Kranzer, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. Differential Equations 47 (1983), no. 1, 35–65. · Zbl 0521.35035 |

[13] | ——–, Spaces of weighed measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451. · Zbl 0821.35096 |

[14] | V. P. Maslov, Non-standard characteristics in asymptotic problems, Russian Math. Surveys 38 (1983), no. 6, 1–42. · Zbl 0562.35007 |

[15] | V. P. Maslov and G. A. Omel’yanov, Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys 36 (1981), no. 3, 73–149. · Zbl 0494.35080 |

[16] | D. Mitrovic, V. Bojkovic and V. G. Danilov, Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process, Math. Methods Appl. Sci. 33 (2010), no. 7, 904–921. · Zbl 1189.35178 |

[17] | C. O. R. Sarrico, About a family of distributional products important in the applications, Portugal. Math. 45 (1988), no. 3, 295–316. · Zbl 0664.46042 |

[18] | ——–, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003), no. 2, 641–656. · Zbl 1026.35078 |

[19] | ——–, New solutions for the one-dimensional nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 317 (2006), no. 2, 496–509. · Zbl 1099.35121 |

[20] | ——–, Collision of delta-waves in a turbulent model studied via a distribution product, Nonlinear Anal. 73 (2010), no. 9, 2868–2875. · Zbl 1198.35051 |

[21] | ——–, Products of distributions and singular travelling waves as solutions of advection-reaction equations, Russ. J. Math. Phys. 19 (2012), no. 2, 244–255. · Zbl 1284.35112 |

[22] | ——–, Products of distributions, conservation laws and the propagation of \(δ'\)-shock waves, Chin. Ann. Math. Ser. B 33 (2012), no. 3, 367–384. · Zbl 1288.46030 |

[23] | ——–, The multiplication of distributions and the Tsodyks model of synapses dynamics, Int. J. Math. Anal. (Ruse) 6 (2012), no. 21-24, 999–1014. |

[24] | ——–, A distributional product approach to \(δ\)-shock wave solutions for a generalized pressureless gas dynamics system, Int. J. Math. 25 (2014), no. 1, 1450007, 12 pp. · Zbl 1298.46038 |

[25] | ——–, The Brio system with initial conditions involving Dirac masses: a result afforded by a distributional product, Chin. Ann. Math. Ser. B 35 (2014), no. 6, 941–954. · Zbl 1315.46044 |

[26] | ——–, The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product, Russ. J. Math. Phys. 22 (2015), no. 4, 518–527. · Zbl 1344.46034 |

[27] | C. O. R. Sarrico and A. Paiva, Products of distributions and collision of a \(δ\)-wave with a \(δ'\)-wave in a turbulent model, J. Nonlinear Math. Phys. 22 (2015), no. 3, 381–394. |

[28] | D. G. Schaeffer, S. Schecter and M. Shearer, Non-strictly hyperbolic conservation laws with a parabolic line, J. Differential Equations 103 (1993), no. 1, 94–126. · Zbl 0809.35056 |

[29] | S. Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), no. 1, 185–210. · Zbl 1077.35095 |

[30] | L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. |

[31] | M. Willem, Analyse harmonique réele, Hermann, Paris, 1995. · Zbl 0839.43001 |

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