双重退化抛物-椭圆营养趋肤系统的弱全局可解性
Weak global solvability of a doubly degenerate parabolic-elliptic nutrient taxis system
AI总结：
本文通过正则化方法研究了双重退化抛物-椭圆营养趋肤系统的弱全局可解性，利用Harnack不等式和正则性性质证明了在特定条件下问题的解存在性。
摘要：
本文研究了以下双重退化抛物-椭圆营养趋肤系统$$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$在有界区间$Ω⊂\mathbb{R}$内，受无净流边界条件和非负初始值$u(x,0) = u_0(x) ≥ 0$的约束，其中$f(x,t) ≥ 0$是已知的营养物外部供给。证明了对于任何非负的$u_0 ∈ W^{1,∞}(Ω)$和$f ∈ C^1(\barΩ × [0,∞))$，且$f ≠ 0$，通过正则化方法可以构造该问题的全局弱解。分析的核心在于一个Harnack型不等式，它允许我们克服缺乏统一 coercivity 的问题。结合时间正则性性质，通过结合Arzelà-Ascoli定理和Aubin-Lions引理，我们获得了相对紧致性。
英文摘要：
This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $Ω\subset \mathbb{R}$, under no-flux boundary conditions and nonnegative initial value $u(x,0) = u_0(x) \geq 0$, where $f(x,t) \geq 0$ is known external supply of the nutrient. It is shown that for any nonnegative $u_0 \in W^{1,\infty}(Ω)$ and $f \in C^1\big(\barΩ \times [0,\infty) \big)$, $f \not \equiv 0$, a global weak solution of the problem can be constructed by means of a regularization approach. The core of the analysis lies on a Harnack-type inequality for the second that allows us to overcome the lack of uniform coercivity. Together with time regularity properties, we obtain relative compactness through a combination of the Arzelà-Ascoli theorem and the Aubin-Lions lemma.
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