On uniqueness techniques for degenerate convection-diffusion problems. (English) Zbl 1263.35007
Let \(\Omega \subset \mathbb{R}^d\) be a bounded domain, \(Q = (0,T)\times \Omega\). The following equation is considered:
\[
j(v)_t - \text{ div } a(w,\nabla w)=f, \quad w=\varphi(v) \, \text{ in Q }, \quad j(v)|_{t=0} = j_0 \,\text{ on } \Omega
\]
with Dirichlet boundary conditions \(w|_{(0,T)\times \partial\Omega} = g\) or with the Neumann boundary condition \(a(w,\nabla w)\cdot n|_{(0,T)\times \partial\Omega} = s,\) where \(a(r,\xi) = S(r)\,a_0(\xi) + F(r)\) \( 1/C \leq S(r) \;\leq C \) or \(a(r,\xi) = a_0(\xi) + F(r)\), \( 0 \leq S(r) \;\leq C \), \(a_0: \mathbb{R}^d \to \mathbb{R}^d\), \(F: \mathbb{R} \to \mathbb{R}^d\), \(a_0(\xi) \cdot \xi \geq 1/C |\xi|^p\), \((a_0(\xi) - a_0(\eta)) \cdot (\xi - \eta) \geq 0\), \(|a_0(\xi)|^{p'} \leq C(1 + |\xi|^p)\), \(|F(r)|^{p'} \;\leq C(1 + |r|^p)\) \(p \in (1, +\infty)\), \(1/p + 1/p' = 1\), \(C = \text{const} > 0\), \(j, \, \varphi \) are continuous non-decreasing functions such that \(j(0) = \varphi(0) = 0, \) \(n\) is a unit normal to \(\partial\Omega\).
The authors prove the uniqueness of the weak solutions to the two problems with the help of the Kato inequality.
The authors prove the uniqueness of the weak solutions to the two problems with the help of the Kato inequality.
Reviewer: Galina Bizhanova (Almaty)
MSC:
35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
35K65 | Degenerate parabolic equations |
35K55 | Nonlinear parabolic equations |
35D30 | Weak solutions to PDEs |
35R35 | Free boundary problems for PDEs |