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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.

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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