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Distributional products and global solutions for nonconservative inviscid Burgers equation. (English) Zbl 1026.35078
The paper deals with the existence of global distributional solutions for the nonconservative Burgers equation (NCB) $$\partial _{t}u+u\partial _{x}u=0,$$ $$t\geq 0$$ and $$x\in \mathbb{R}$$. First the author introduces his definition of the product of distributions used in the paper, called $$\alpha -$$product, and sketches some properties of this product; then he defines the concept of global $$\alpha -$$solution for the (NCB) equation. The following sections of the paper are devoted to the conditions on $$C^{1}$$ functions $$u(x,t)$$ having a jump discontinuity along a $$C^{1}$$ curve $$\gamma$$ of the $$(x,t)-$$plane, to be global $$\alpha -$$solutions for the (NCB) equation. The relationship with the global weak solutions of the conservative Burgers equation $$\partial _{t}u+\partial _{x}(\frac{1}{2} u^{2})=0$$ is also considered.
Reviewer: C.Bouzar (Oran)

##### MSC:
 35L65 Hyperbolic conservation laws 35D05 Existence of generalized solutions of PDE (MSC2000) 46F10 Operations with distributions and generalized functions
##### Keywords:
generalized solutions; product of distributions
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##### References:
 [1] Bressan, A., Hyperbolic systems of conservation laws (the one dimensional Cauchy problem), (2000), Oxford Univ. Press · Zbl 0997.35002 [2] Bressan, A.; Rampazzo, F., On differential systems with vector-valued impulsive controls, Bull. un. mat. ital. (7), 2B, 641-656, (1988) · Zbl 0653.49002 [3] Burgers, J.M., A mathematical model illustrating the theory of turbulence, Adv. appl. mech., 1, 171-179, (1948) [4] Cole, J.D., On a quasilinear parabolic equation occurring in aerodynamics, Quart. appl. math., 9, 225-236, (1951) · Zbl 0043.09902 [5] Colombeau, J.; Le Roux, A., Multiplication of distributions in elasticity and hydrodynamics, J. math. phys., 29, 315-319, (1988) · Zbl 0646.76007 [6] Hopf, E., The partial differential equation ut+uux=μuxx, Comm. pure appl. math., 3, 201-230, (1950) [7] Hörmander, L., Lectures on nonlinear hyperbolic differential equations, (1997), Springer · Zbl 0881.35001 [8] Dal Maso, G.; LeFloch, P.; Murat, F., Definitions and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068 [9] Sarrico, C.O.R., About a family of distributional products important in the applications, Portugal. math., 45, 295-316, (1988) · Zbl 0664.46042 [10] Sarrico, C.O.R., Distributional products with invariance for the action of unimodular groups, Riv. mat. univ. parma, 4, 79-99, (1995) · Zbl 0888.46013 [11] Schwartz, L., Théorie des distributions, (1966), Hermann Paris [12] Smoler, J., Shock waves and reaction – diffusion equations, (1994), Springer
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