## On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient.(English)Zbl 1015.35049

This paper concerns the following Cauchy problem: $$\partial_tu+\partial_x(\gamma(x)f(u))=\partial_x^2A(u)$$ in $$\Pi_T=\mathbb{R}\times(0,T)$$, $$u(x,0)=u_0(x)$$. The coefficient $$\gamma(x)$$ appearing in the transport part satisfies suitable conditions and may be discontinuous. The function $$f(s)$$ belongs to $$C^2([0,1])$$, satisfies $$f(0)=f(1)=0$$ and there is no subinterval of $$[0,1]$$ on which it is linear. The function $$A(s)$$ belongs to $$C^2([0,1])$$, satisfies $$A'(s)\geq 0$$ and $$A(0)=0$$. This weak parabolicity is general enough to include the hyperbolic conservation law $$\partial_tu+\partial_x(\gamma(x)f(u))=0$$. The initial data $$u_0(x)\in [0,1]$$ belongs to $$L^\infty(\mathbb{R})\cap L^1(\mathbb{R})$$. If $$A'(s)=0$$ on an interval, solutions may be discontinuous and they are not uniquely determined by the initial data: an entropy condition is imposed in this case. To prove existence of a weak solution the authors introduce a family of approximation problems where $$\gamma(x)$$ is replaced by a smooth coefficient $$\gamma_\varepsilon(x)$$ and $$A(u)$$ is replaced by a strictly increasing function $$A_\varepsilon(u)$$. Then they prove that a sequence of solutions $$u_\varepsilon(x)$$ converges in the $$L^1$$ norm to a solution of the previous problem. Furthermore, a subsequence of $$A_\varepsilon(u_\varepsilon)$$ converges uniformly on compact sets to a Hölder continuous function which coincides with $$A(u)$$ almost everywhere. If $$\gamma(x)$$ is discontinuous, the total variation of $$u_\varepsilon$$ cannot be bounded uniformly with respect to $$\varepsilon$$, and the standard BV compactness argument cannot be applied. To get around this difficulty, they establish a strong compactness of the diffusion function $$A_\varepsilon(u_\varepsilon)$$ as well as the total flux $$\gamma_\varepsilon(x)f(u_\varepsilon)-\partial_xA_\varepsilon(u_\varepsilon)$$ and apply the compensated compactness method. Also the purely hyperbolic case $$A'=0$$ is discussed by derivation of strong convergence via some a priori energy estimates that may have independ interest.

### MSC:

 35K65 Degenerate parabolic equations 35R05 PDEs with low regular coefficients and/or low regular data 35L80 Degenerate hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000)
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