H 1 -Solutions for the Hele-Shaw Equation
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H 1 -Solutions for the Hele-Shaw Equation Giuseppe Maria Coclite1
· Lorenzo di Ruvo2
Received: 6 March 2020 / Accepted: 20 July 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract The Hele-Shaw equation arises in modeling the motion of viscous droplets spreading over a solid surface. It models also the diffusion of dopant in semiconductors. In this paper, we prove the existence of the solutions of the Cauchy problem associated with this equation. Keywords Existence · Hele-Shaw equation · Cauchy problem Mathematics Subject Classification (2010) 35G25 · 35K55
1 Introduction In this study, we investigate the existence of solutions of the following Cauchy problem: ∂t u = −∂x (|u|p ∂x3 u), t > 0, x ∈ R, (1.1) u(0, x) = u0 (x), x ∈ R, where we assume that (1.2) u0 ∈ H 1 (R), 0 < p < 8. Equation (1.1) arises in modeling the motion of viscous droplets spreading over a solid surface (see [4, 22, 25]). [4, 22, 25] generalize [15–17, 21], where the authors deduce (1.1) with p = 3, which corresponds to viscous flow on a solid surface without slip driven by surface tension. (1.1) is deduced in [13, 14, 26] with p = 1, to model a thin layer of fluid in the Hele-Shaw cell. Moreover, (1.1) is also deduced in [18–20], to model the diffusion of dopant in semiconductors. From a mathematical point of view, in [1, 2], the initial boundary problem for (1.1) is studied when p ≥ 1, while, in [5], the existence of the periodic solutions is proved, for To our dear friend Enrique Zuazua in occasion of his 60th birthday. Giuseppe Maria Coclite
[email protected] Lorenzo di Ruvo [email protected] 1
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, via E. Orabona 4, 70125 Bari, Italy
2
Dipartimento di Matematica, Universit`a di Bari, via E. Orabona 4, 70125 Bari, Italy
G.M. Coclite, L. di Ruvo
0 < p < 3. Instead, in [3], the authors prove that the solution of (1.1) preserves mass. Observe that, both in [1, 2] and in [5], it is proven that if the initial datum is positive, then the solution u of (1.1) is positive. In [25], some explicit solutions and heuristic asymptotic analysis with respect to p is studied, while, in [6], the existence of the traveling-wave solution for (1.1) is proven. Moreover, in [4, 23], numerical schemes are studied. Following [2], we give the following definition of solution: Definition 1.1 We say that u is a distributional solution of the Cauchy problem (1.1), if for each test function with compact support φ ∈ C ∞ (R2 ), we have ∞ ∞ u∂t φdtdx + |u|p ∂x3 u∂x φdtdx + u0 (x)φ(0, x)dx = 0. (1.3) 0
R
0
R
R
Following [10], the main result of this paper is the following theorem. Theorem 1.1 Fix T > 0. Assume (1.2). There exists a distributional solution u of (1.1), such that (1.4) u ∈ L∞ (0, T ; H 1 (R)). Compared to [1, 2, 5], we do not require that the solution of (1.1) is positive, and the proof of Theorem 1.1 is based on the Aubin-Lions Lemma (see [8, 11, 12, 24]). The paper is organized as fol
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