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On the existence of shock fronts for a Burgers equation with a singular source term
Author(s) -
Dias JoãoPaulo,
Figueira Mário,
Sanchez Luis
Publication year - 1998
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199808)21:12<1107::aid-mma987>3.0.co;2-2
Subject(s) - shock front , mathematics , cauchy problem , initial value problem , burgers' equation , shock (circulatory) , mathematical physics , front (military) , combinatorics , mathematical analysis , shock wave , physics , partial differential equation , thermodynamics , medicine , meteorology
In this paper we consider the Cauchy problem for the equation ∂ u /∂ t + u ∂ u /∂ x + u / x = 0 for x > 0, t ⩾ 0, with u ( x , 0) = u 0 − ( x ) for x < x 0 , u ( x , 0) = u 0 + ( x ) for x > x 0 , u 0 − ( x 0 ) > u 0 + ( x 0 ). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u 0 − and u 0 + , a global shock front weak solution u ( x , t ) = u − ( x , t ) for x < ϕ( t ), u ( x , t ) = u + ( x , t ) for x > ϕ( t ), where u − and u + are the strong solutions corresponding (respectively) to u 0 − and u 0 + and the curve t → ϕ( t ) is defined by dϕ/d t ( t ) = 1/2[ u − (ϕ( t ), t ) + u + (ϕ( t ), t )], t ⩾ 0 and ϕ(0) = x 0 . © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.