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Existence and uniqueness of entropy solution to initial boundary value problem for the inviscid Burgers equation. (English) Zbl 1168.35393

Summary: This paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid Burgers equation \[ \begin{cases} u_t + (\frac{u^2}{2}) =0 \qquad x > 0 \quad t > 0 \\ u(x,0) = u_0 (x) \qquad x \leqslant 0 \\ u(0,t)=0 \qquad t \geqslant \end{cases} \] To apply the method of vanishing viscosity to study the existence of the entropy solution, we first introduce the initial boundary value problem for the viscous Burgers equation, and as in [L. Evans, “Partial differential equations” (Graduate Studies in Mathematics 19, American Mathematical Society, Providence) (1998; Zbl 0902.35002)] and E. Hopf [Commun. Pure Appl. Math. 3, 201–230 (1950; Zbl 0039.10403)], give the formula of the corresponding viscosity solutions by Hopf-Cole transformation. Secondly, we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution. Finally, we give an example to show how our main result can be applied to solve the initial boundary value problem for the Burgers equation.

MSC:

35L65 Hyperbolic conservation laws
35D05 Existence of generalized solutions of PDE (MSC2000)
35K05 Heat equation
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