Abstract
We consider the two-dimensional capillary-gravity water waves problem where the free surface \(\Gamma _t\) intersects the bottom \(\Gamma _b\) at two contact points. In our previous works (Ming and Wang in SIAM J Math Anal 52(5):4861–4899; Commun Pure Appl Math 74(2), 225–285, 2021), the local well-posedness for this problem has been proved with the contact angles less than \(\pi /16\). In this paper, we study the case where the contact angles belong to \((0, \pi /2)\). It involves much worse singularities generated from corresponding elliptic systems, which have this strong influence on the regularities for the free surface and the velocity field. Combining the theory of singularity decompositions for elliptic problems with the structure of the water waves system, we obtain a priori energy estimates. Based on these estimates, we also prove the local well-posedness of the solutions in a geometric formulation.
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1 Introduction
We consider the irrotational incompressible capillary-gravity water waves problem in a two-dimensional domain \(\Omega _t\), where \(\Omega _t\) is a bounded domain with an upper free surface \(\Gamma _t\) and a fixed bottom \(\Gamma _b\). This moving domain contains two moving contact points, \(p_{l}, p_{r}\) (left and right), with the contact angles \(\omega _l, \omega _r\in (0, \pi /2)\), which are the intersection points of \(\Gamma _t, \Gamma _b\):
Moreover, the fixed bottom \(\Gamma _b\) is assumed to be smooth enough, and it becomes straight near the contact points \(p_i\) (\(i=l, r\)) for the sake of simplicity.
The water waves problem has been widely studied in centuries; see, for example, [49, 63]. This problem focuses on the motion of an ideal fluid and describes the evolution of the free surface \(\Gamma _t\) as well as the velocity field v. Mathematically, it is described by Euler’s equation with boundary conditions and initial conditions, and in our case, we also need some boundary conditions at contact points.
We express the water waves problem on the corner domain \(\Omega _t\) as the following system \(\text{(WW) }\) of velocity v and pressure P:
Here, (1.1) from \(\text{(WW) }\) is Euler’s equation where, \(\textbf{g}=-g \mathbf{e_z}\) is the vertical gravity vector; (1.2) describes the incompressibility and irrotationality; (1.3) is the condition of the pressure on the free surface in the case with surface tension, where \(\sigma \) is the coefficient of surface tension and \(\kappa \) is the mean curvature of \(\Gamma _t\) (see Section 1.2); (1.4) is the classical kinematic condition on the free surface \(\Gamma _t\) with \(D_t\) the material derivative, Meanwhile, (1.5) describes that the velocity along the fixed bottom \(\Gamma _b\) is always tangential, where \(n_b\) is the unit outward normal vector of \(\Gamma _b\). These equations and conditions are standard in water waves, see [45, 46].
In particular, (1.6) gives the conditions at contact points, which come from [58]. We denote by \(v_i\) the upward tangential components of the velocity at the corner points along \(\Gamma _b\):
Here \(\omega _s\) is the stationary contact angle decided by the materials of the bottom and the fluid (see [76]), and \(\beta _c\) denotes the effective friction coefficient. This condition shows that the slip velocity is dominated by the unbalanced Young stress, and it is indeed an effective variation of Young’s law (1805) for stationary contact angles [76]. In fact, this kind of conditions are commonly seen, see [11, 15, 27, 62].
Before presenting our results, we recall briefly earlier works on the well-posedness of classical water waves problem, where one has smooth surfaces \(\Gamma _t\) satisfying \(\Gamma _t\cap \Gamma _b=\emptyset \).
We recall results on the local well-posedness. When the fluid is irrotational, some early works like Nalimov [55], Yosihara [74, 75] and Craig [22] established the two-dimensional local well-posedness with small initial data in Sobolev spaces. In the late 1990s, Wu [68, 69] proved for the first time the local well-posedness with general initial data in Sobolev spaces and showed that the Taylor sign condition
held on \(\Gamma _t\) as long as \(\Gamma _t\) was not self-intersecting. Iguchi, Tanaka and Tani [38] and Iguchi [36] proved the local well-posedness in two-dimensional case respectively. Later on, Lannes [45] derived the local well-posedness of the gravity water waves under Zakharov formulation, which is convenient to link with approximate models. Alazard, Burq and Zuily in [2,3,4,5] used paradifferential operators and Strichartz estimates to study the problem in a low-regularity space. On the other hand, when the fluid is rotational, Christodoulou and Lindblad [18] proved a priori estimates based on the geometry of the moving domain. Lindblad [50] obtained the existence of solutions using Nash-Moser iteration. In 2007, Coutand and Shkoller [20] used Lagrangian coordinates to show the local well-posedness. Shatah and Zeng [60, 61] adopted a geometric point of view to reformulate the problem and prove the local well-posedness, while Beyer and Günther [12, 13] used a similar geometric approach to study the irrotational flow. Zhang and Zhang [77] proved the local well-posedness for rotational flow using a framework of Clifford analysis introduced by Wu [69]. For more references see [7,8,9, 46, 54, 56, 57, 59, 67, 77] etc..
For the global well-posedness of small data, Wu [72] and Germain, Masmoudi and Shatah [28] proved the global three-dimensional existence of gravity water waves respectively using different approaches. One can check [6, 25, 33,34,35, 39, 66, 71, 73] e.t.c. and their references for more results on gravity or capillary-gravity water waves,. Meanwhile, there are also some works concerning geometric singularities on the free surfaces. The authors in [16] proved the existence of a wave which is given initially as the graph of a function and then can overturn at a later time. Later on, the authors in [17] showed the existence for some “splash" or “splat” singularities. This result was extended to three-dimensional case and some other models in [21].
Compared to the rich literature on the well-posedness of classical water waves, the research on the well-posedness of water waves problem with non-smooth boundaries (we call it “non-smooth water waves”) just started several years ago and there are a lot of open questions. In general, there are two kinds of non-smooth water waves problems: The first kind of problem has contact points (or contact lines) between the free surface and the bottom, i.e. \(\Gamma _t\cap \Gamma _b\ne \emptyset \); The other kind contains crests or cusps on the free surface, i.e. the surface is Lipschitz. We would like to mention that in the case with large crest angles, the famous Stokes waves can be dated back to papers by G. Stokes [63, 64] which obtained traveling-wave solutions with limit crest angle \(2\pi /3\). Obviously, the main difference here compared to classical water waves lies in the corners on boundaries. As a result, the analysis involving the corners (i.e. domain singularities) becomes the key point in the non-smooth water waves.
Now, we are in a position to state an informal version of the main result. To begin with, we introduce the following compatibility conditions at \(t=0\):
When we reduce system \(\text{( }WW)\) into some initial-boundary value problem essentially related to the mean curvature and the pressure (see (3.15), (5.30) and (6.6)), these compatibility conditions are needed in the linearized system (Section 6.3) in a natural way as long as we try to solve this problem.
Theorem 1.1
Let the initial data belong to a suitable space and the initial contact angles \(\omega _{i0}\in (0, \pi /2)\) for \(i=l, r\). If the compatibility conditions (1.8) are satisfied, there exists a time interval depending on the initial data such that system \(\text{(WW) }\) is locally well-posed in a suitable space.
Remark 1.1
The suitable space is defined by \(\Sigma _h\) in Section 6.3 for some good unknowns built upon the mean curvature, the pressure and the position information of contact points. In fact, the proof of Theorem 1.1 is divided into three parts. In Section 4 and Section 5, we obtain a priori estimates (see Theorem 4.1 and Theorem 5.1). In the last section, we construct approximate solutions to prove the well-posedness based on the a priori estimates. The precise statement of local well-posedness for a geometric form of \(\text{( }WW)\) is given in Theorem 6.1 (Section 6.3).
Remark 1.2
When there is no surface tension, the authors of [19, 40] studied the case with contact angles. They assume that the wall \(\Gamma _b\) is vertical and the contact angle \(\omega \in (0, \pi /4)\). Then, by a symmetric extension, they reduced the problem to the crest case with the crest angle less than \(\pi /2\). As a result, [19, 40] do not contain the case of limit Stokes waves with the “ \(2\pi /3\)” crest. In that case, one needs a contact angle of \(\pi /3\) even with the help of a symmetric extension. In this paper, we only require that the contact angles \(\omega _i\in (0, \pi /2)\), which brings us some useful experiences to deal with domain singularities in a more general case for water waves.
Remark 1.3
Noticing that
we can simply have
with some polynomial F. Therefore, the time evolution of contact angles \(\omega _i=\omega _i(t)\) depends on the derivatives of v and the free surface \(\Gamma _t\). A continuity argument in time shows that as long as the initial contact angles \(\omega _{i0}\in (0, \pi /2)\), we have \(\omega _i(t)\in (0, \pi /2)\) in a short time interval.
Remark 1.4
The compatibility conditions (1.8) seem to be complicated, but they can be satisfied according to the proof of our main theorem. In fact, when one considers the initial data \((v_0, \Gamma _{t0})\) for system \(\text{( }WW)\), it suffices to find the corresponding initial data for essentially the mean curvature \(\kappa \) (and its time derivative), the pressure P and the positions of contact points (see the initial data in Section 6.3). According to the linear system (6.7) in the iteration, compatibility conditions are required there which are linear, localized and derived from (1.8). Therefore, one can find corresponding initial values for this linear system satisfying these compatibility conditions. As a result, based on the proof of the existence of the solution to system (6.6) in Theorem 6.1, we can show that conditions (1.8) are satisfied thanks to the linear system (6.7).
We recall some works concerning the local well-posedness of non-smooth water waves. In the case where there are crests or cusps on the free surface (the “crest” case), Alazard, Burq and Zuily [5] study a special case (without surface tension) when the contact angle is equal to \(\pi /2\) (the right angle), where they used symmetric and periodic extension to turn this problem into a classical smooth periodic case. A breakthrough in this subject is made by Kinsey and Wu, see [40, 70]. They focus on gravity water waves where the crest angle is less than \(\pi /2\). The main difficulty is that the free surface is a non-\(C^1\) interface with angled crest and the Taylor sign \(-\nabla _{n_t}P\) degenerates at the crest point. To be more precise, they start with reducing the water waves problem into the following equation in [40, 70]:
there \({\mathfrak {a}}=-\nabla _{n_t}P\) is the Taylor sign. When \({\mathfrak {a}}\) degenerates at the crest point, the above system will loss its hyperbolicity, and classical analysis does not apply any more. To solve the problem, Kinsey and Wu flatten the domain with a Riemann mapping, and some singular weights appear naturally in their equation. As a result, they introduce some weighted Sobolev spaces accordingly for energy estimates to deal with these singular weights. Based on these works, Agrawal [1] show that these singularities are “rigid", which means that the angle of these crests can not change in time. Very recently, Córdoba, Enciso and Grubic [19] study a similar case with cusps and crests without gravity, where the angles of these crests are less than \(\pi /2\) and change in time.
For the other case where there are contact angles (that is \(\Gamma _t\cap \Gamma _b\ne \emptyset \)), things become different from the “crest” case. First, the corners appear due to intersections of the free surface and the bottom (or wall). Second, there are different boundary conditions in corresponding elliptic systems compared to the crest case. In fact, various boundary conditions may have no big difference if we only focus on the elliptic theory, but there will be a series of consequences in water waves when boundary conditions change. For example, the evolution of the free surface is different from the crest case. Moreover, when there is surface tension, boundary conditions at the contact points as (1.6) are needed in order to close the system, and dissipations appear at the contact points too (See Section 4 and [52]).
In the case with contact angles, de Poyferré [23] prove a priori estimates in bounded n-dimensional corner domains without surface tension. The contact angle is assumed to be small to ensure sufficient Sobolev regularity near the corner. Meanwhile, under a similar assumption of small contact angles, we obtain the local well-posedness in a two-dimensional corner domain (beach type) with surface tension, see [52, 53]. Meanwhile, we notice that [23] and [52, 53] use similar geometry formulations introduced in [60].
To explain why small contact angles are needed in [23, 52, 53], we look at a typical mixed-boundary system in water waves:
In fact, the elliptic theory on corner domains is well known already, see for example [30, 43, 44]. Generally, one still has variational solution \(u\in H^1(\Omega _t)\) if the right-side functions lie in proper Sobolev spaces. If one seeks for \(H^2\) and above regularities, singularity decompositions are needed naturally for the solution, which decompose the solution into a singular part \(u_s\) (i.e. not good enough) near the corners and a regular part \(u_r\):
For the mixed-boundary system above, the most singular part in \(u_s\) is like \(r^{\pi /2\omega }\), where r is the radius with respect to the corner point and \(\omega \) is the contact angle. Consequently, when \(\omega \) is small enough, the singular part \(u_s\) will be good enough so that we will have enough regularities from elliptic systems as in classical water waves to close the energy estimates. One can also find a singularity decomposition for v in Proposition 2.3 (Section 2).
In contrast, when the contact angle is larger or more general, the idea of taking small angles to improve regularities in [23, 52, 53] does not work any more. Meanwhile, there is no obvious weighted space to use due to the structure of the water waves problem.
We want to show here the main ingredients of this paper. Similarly as in our previous works [52, 53], we still adopt the geometric formulation from Shatah and Zeng [60, 61]. In fact, we rewrite \((\hbox {WW})\) into an equation for \({\mathfrak {J}}=\nabla {\mathfrak {K}}_{{\mathcal {H}}}\) with boundary conditions, where \({\mathfrak {K}}\) is the modified mean curvature on \(\Gamma _t\) and \({{\mathcal {H}}}\) means the harmonic extension in \(\Omega _t\) (see Section 3, and we only present a simpler form for this equation here),
where R is some remainder part.
The trouble here is that singularities from the domain \(\Omega _t\) affect directly the regularity of the solution to \(\text{(WW) }\), which means singularities always appear in related elliptic systems even if the boundary conditions are good enough in Sobolev spaces (see for example [30]). More precisely, the natural norm \(\Vert {{\mathcal {A}}}^{k}f \Vert _{L^2(\Omega _t)}+\Vert f\Vert _{L^2(\Omega _t)}\) with \(k\geqq 1\) arising from this equation above is not equivalent with \(\Vert f \Vert _{H^{3k}(\Omega _t)}\) due to larger contact angles, and apparently “singular parts" are contained in this norm.
Due to this kind of singularities from elliptic systems, we only have limited regularities for some quantities (for example, the velocity v) in Sobolev spaces. Compared to [52, 53], we must make full use of the maximal regularity for each quantity (especially for v) while the contact angles \(\omega _i\in (0, \pi /2)\). To do this, some delicate estimates together with singular parts from singularity decompositions (see [30, 51]) are carefully used. Meanwhile, it is also very important to gain more information from the structure of \(\text{(WW) }\).
The main part of the (lower-order) energy functional is defined as
which gives us the estimate (see Section 4.1)
with P(E(t)) the positive-coefficient polynomial of E(t). This means that the free surface \(\Gamma _t\) still has enough regularity and v is Lipschitz.
In our previous work [52, 53], one main part from the remainder term R in (1.9) is about the higher-order derivative terms of \(P_{v,v}\), where \(P_{v, v}\) is defined by an elliptic system with mixed-boundary conditions. According to the elliptic theory on corner domains (see for example [30]), the system of \(P_{v,v}\) only gives \(P_{v,v}\) limited regularity around \(H^2(\Omega _t)\), when the contact angles are less than \(\pi /2\). To improve its regularity, we modify the definition of \(P_{v, v}\) in this paper to have a Neumann-boundary system, see (3.1). In fact, thanks to the elliptic theory, solutions of Neumann-boundary system (or Dirichlet system) may become more regular than solutions of mixed-boundary problem in Sobolev spaces, while the right-side data have the same regularity. As a result, we obtain a bit more regularity from the elliptic system of \(P_{v,v}\) (we have \(P_{v,v}\in H^3(\Omega _t)\) indeed), which is important in the energy estimates.
Moreover, another main part of R lies in the higher-order derivative terms of the velocity v. Here, we point out that compared to [52, 53], v loses some regularity due to the existence of corners. Fortunately, \(D_t v\) and \(D_t^2 v\) have the same regularities as before. As a result, when we deal with v, sometimes we need to use material derivatives \(D_t\) instead of spatial derivatives. Meanwhile, we also need to apply singularity decompositions to v and its potential \(\phi \) in the estimates, see for example Lemma 4.5.
For the higher-order energy estimates, we use the material derivative \(D_t\) instead of \({{\mathcal {A}}}^{1/2}\) from [60]. The higher-order energy is defined as
In fact, one will see that \(D_t\) is convenient to use when there are contact points. For example, taking \(D_t\) on elliptic systems does not violate boundary conditions, while it will change boundary conditions if one takes spatial derivatives. Moreover, one will find in the higher-order energy estimate that taking \(D_t\) leads to better regularities for some quantities than their own regularities (like \(D_t v\) for the velocity v). However, due to the singularities of the boundary (contact points), it is not as convenient as before to turn these regularities into spatial regularities, which explains the reason why we need to choose very carefully for “a suitable space” in Theorem 1.1. As long as the energy estimates for \(E(t), E_1(t)\) are finished, we use the equation of \({\mathfrak {J}}\) to gain more spatial regularities.
we now mention some other related works. Lannes and Métivier [48] studied the Green-Naghdi system in a beach-type domain, which is a shallow-water model of the water waves problem. Lannes [47] studied the floating-body problem and proposed a new formulation that can be easily generalized in order to take into account the presence of a floating body. Lannes and Iguchi [37] proved some sharp results for initial boundary value problem with a free boundary arising in wave-structure interaction, and it contains the floating problem in the shallow water regime. In addition, Guo and Tice [31] showed a priori estimates for the contact line problem in the case of the stokes equations. Later on, Tice and Zheng proved the local well-posedness of the contact line problem in two-dimensional Stokes flow, see [65]. In 2020, Guo and Tice [32] proved a priori estimates for the contact line problem for two-dimensional Navire-Stokes flow. For Darcy’s flow, one can see [41, 42].
1.1 Organization of the Paper
In Section 2, we present various useful lemmas including singularity decompositions and estimates for elliptic systems. In Section 3, we derive the equation for the good unknown \({\mathfrak {J}}\) from \((\hbox {WW})\) with modified curvature \({\mathfrak {K}}\) and modified pressure \(P_{v,v}\). The lower-order energy is constructed and the energy estimate is proved in Section 4, where estimates for various quantities like \(\Gamma _t\), v, \(P_{v,v}\) are proved. Moreover, we consider the higher-order energy estimate using \(D_t\) in Section 5. In Section 6, we present the precise main theorem in our paper and show the local well-posedness.
1.2 Notations
- X stands for a point in 
. \(p_l, p_r\) are the left and right contact points. \(n_j (j=t, b)\) are the unit outward normal vectors on \(\Gamma _j\), and \(\tau _j\) are the corresponding unit tangential vectors obeying the right-hand rule with \(n_j\).
- \(\sigma \) is the surface tension coefficient. \(\beta _c\) is the effective friction coefficient determined by interfacial widths, interactions between the fluid and the bottom, and the normal stress contributions.
- \(\chi _{\omega }\) is a characteristic function of contact angles:
- \(\chi _i\) (\(i=l, r\)) are cut-off functions near the corner points \(p_i\):
with some small \(r_0>0\).
-
\(f|_c=\chi _l (f|_{p_l})+\chi _r (f|_{p_r})\) stands for taking values of f at the corner points.
-
\({{\mathcal {S}}}_{t,i}\) are straightened sector of \(\Omega _t\) with radius \(r_0>0\) near the corner points \(p_i\).
-
\(D_t=\partial _t+\nabla _v\) is the material derivative.
-
\(M^*\) denotes the transport of a matrix M.
-
\(w^\perp \) on \(\Gamma _t\): \(w\cdot n_t\) for a vector \(w\in T_X\Gamma _t\).
-
\(w^\top \) on \(\Gamma _t\): \((w\cdot \tau _t)\,\tau _t\). Sometimes we also use \(w^\top \) on \(\Gamma _b\) with a similar definition.
-
\(\Pi \) is the second fundamental form on \(\Gamma _t\), where \(\Pi (w)=\nabla _w n_t\in T_X\Gamma _t\) for \(w\in T_X\Gamma _t\).
-
\(\kappa =tr \Pi =\nabla _{\tau _t}n_t\cdot \tau _t\) is the mean curvature of the surface \(\Gamma _t\).
-
We define on \(\Gamma _t\) that \({{\mathcal {D}}}w={{\mathcal {D}}}_{\tau _t}w=(\nabla _{\tau _t}w)^\top =(\nabla _{\tau _t}w\cdot \tau _t\big )\tau _t\) for a vector \(w\in T_X\Gamma _t\).
-
\({{\mathcal {D}}}^2f(\tau _1, \tau _2)=D^2f(\tau _1,\tau _2)-\big (\Pi (\tau _1)\cdot \tau _2\big )\nabla _{n_t}f\) for any two vector \(\tau _1, \tau _2\in T_X\Gamma _t\).
-
\(\Delta _{\Gamma _t}\) is the Beltrami-Laplace operator on \(\Gamma _t\):
$$\begin{aligned} \Delta _{\Gamma _t}f={{\mathcal {D}}}^2f(\tau _t, \tau _t)=\nabla _{\tau _t}\nabla _{\tau _t}f-\nabla _{(\nabla _{\tau _t}\tau _t)^\top }f. \end{aligned}$$- \({{\mathcal {H}}}(f)\) or \(f_{{\mathcal {H}}}\) is the harmonic extension for some function f on \(\Gamma _t\), which is defined by the elliptic system
$$\begin{aligned} \left\{ \begin{array}{ll} \Delta {{\mathcal {H}}}(f)=0\qquad \hbox {in}\quad \Omega _t,\\ {{\mathcal {H}}}(f)|_{\Gamma _t}=f,\quad \nabla _{n_b}{{\mathcal {H}}}(f)|_{\Gamma _b}=0. \end{array}\right. \end{aligned}$$- \({{\mathcal {N}}}=\nabla _{n_t}{{\mathcal {H}}}\) is the Dirichlet-Neumann operator on \(\Gamma _t\). - \(\Delta ^{-1}(h,g)\) stands for the solution u to the system
$$\begin{aligned} \left\{ \begin{array}{ll} \Delta u=h\qquad \hbox {in}\quad \Omega _t\\ u|_{\Gamma _t}=0,\qquad \nabla _{n_b}u|_{\Gamma _b}=g. \end{array} \right. \end{aligned}$$- The Sobolev norm \(H^s\) for the boundary \(\Gamma _t\) or \(\Gamma _b\) is defined by local coordinates and local graphs. - \({\tilde{H}}^{1/2}(\Gamma _j)\) (\(j=t,b\)) (see [30]) is a subspace of \(H^{1/2}(\Gamma _j)\) related to corner domains
$$\begin{aligned} {\tilde{H}}^{1/2}(\Gamma _j)=\Big \{u\in \dot{H}^{1/2}(\Gamma _j)\Big | \,\rho _i^{-1/2}u\in L^2(\Gamma _j),\ i=l, r\Big \}\end{aligned}$$where \(\dot{H}^{1/2}(\Gamma _j)\) is the closure of \(\mathscr {D}(\Gamma _j)\) in \(H^{1/2}(\Gamma _j)\), and \(\rho _i=\rho _i(X)\) (\(i=l, r\)) is the distance (arc length) between the point \(X\in \Gamma _j\) and the end \(p_i\). We define the norm
$$\begin{aligned} \Vert u\Vert ^2_{{\tilde{H}}^{1/2}(\Gamma _j)}=\Vert u\Vert ^2_{H^{1/2}(\Gamma _j)}+\int _{\Gamma _j} \rho _l^{-1}|u|^2{\text {d}}X+\int _{\Gamma _j} \rho _r^{-1}|u|^2{\text {d}}X. \end{aligned}$$- \({\tilde{H}}^{-1/2}(\Gamma _j)\) stands for the dual space of \({\tilde{H}}^{1/2}(\Gamma _j)\). For more details, see [30]. - We define \(P_{w,v}\) (with \(w\ne v\)) by the following Neumann system:
$$\begin{aligned} \left\{ \begin{array}{ll} \Delta P_{w,v}=-tr (\nabla w \nabla v),\qquad \hbox {in}\quad \Omega _t\\ \nabla _{n_t} P_{w,v}|_{\Gamma _t}=C_{w,v}(t)-(w\cdot \tau _t) \nabla _{\tau _t} v\cdot n_t, \qquad \nabla _{n_b} P_{w,v}|_{\Gamma _b}=w\cdot \nabla _v n_b. \end{array}\right. \nonumber \\ \end{aligned}$$(1.10)Here \(C_{w,v}\) is a function of t satisfying the compatibility condition
$$\begin{aligned} |\Gamma _t|C_{w, v}(t)= -\int _{\Omega _t}tr (\nabla w \nabla v){\text {d}}X+\int _{\Gamma _t}(w\cdot \tau _t) \nabla _{\tau _t} v\cdot n_t {\text {d}}s-\int _{\Gamma _b}w\cdot \nabla _v n_b {\text {d}}s. \end{aligned}$$- \(P_{v,v}\) is defined in (3.1) from Section 3. - C(a) stands for a positive constant C depending on a quantity a. P(E(t)) stands for a polynomial of the energy E(t) with positive constant coefficients.
2 Preliminaries
To get started, we recall some useful estimates for Sobolev space.
Lemma 2.1
(Sobolev Embeddings) One has the inclusion
for \(1/q=1/p-s/n\) and \(\Omega \) any bounded open subset of 
with a Lipschitz boundary. Moreover, one also has
for \(t\leqq s, q\geqq p\) such that \(s-n/p=t-n/q\).
Lemma 2.2
(Product estimates) (1) For functions \(f\in H^{1/2}(\Omega _t)\) and \(g\in H^1(\Omega _t)\cap L^\infty (\Omega _t)\), one has the product estimate
with a constant C independent of f, g;
(2) For functions \(f, g\in L^\infty (\Gamma _t)\cap H^{1/2}(\Gamma _t)\), one has
with a constant C independent of f, g.
Proof
(1) In fact, one firstly extends f, g to be defined on the full plane 
with a control of their corresponding norms. Secondly, one can apply standard para-product analysis to prove the estimate on 
. The details are omitted here. (2) The proof is similar to the proof of (1). \(\quad \square \)
We quote some Hardy inequalities here.
Lemma 2.3
(Hardy inequalities) (1)([14, Corollary 2.3]) Let \(f\in H^1(0, d)\cap C^0(0, d)\) with \(d>0\) and \(f(0)=0\). Then there exists a positive constant \(C=C(\epsilon , d)\) such that
for \(\epsilon \in (1/2, 1)\);
(2) (Fractional-order version, see [26, 29]) For a number \(\varepsilon \in (0, 1)\) and any function 
with \(f(0)=0\), there exists a positive constant \(C=C(\varepsilon )\) such that
Lemma 2.4
(Traces on \(\Gamma _t\) or \(\Gamma _b\), [52, Theorem 5.3]) The maps
have unique continuous extensions as operators from \(H^{s}(\Omega _t)\) onto \(\Pi ^1_{i=1} H^{s-i-1/2}(\Gamma _j)\) for \(s>3/2\).
Moreover, one has the estimate:
Next, we present some some special trace theorems on corner domains involving \({\tilde{H}}^{1/2}(\Gamma _j)\) and \({\tilde{H}}^{-1/2}(\Gamma _j)\) (\(j=t, b\)).
Lemma 2.5
Assume that \(u|_{\Gamma _t}=f\) with \(f|_{p_i}=0\) (\(i=l,r\)) for a function \(u\in H^1(\Omega _t)\). Then one has \(f\in {\tilde{H}}^{1/2}(\Gamma _t)\) and
The case on \(\Gamma _b\) holds similarly.
Proof
First, one has \(f\in H^{1/2}(\Gamma _t)\) immediately by Lemma 2.4. Second, noticing \(f|_{p_i}=0\) and applying Lemma 2.3(2) with some straightening localizations near \(p_i\), one can see that \(f\in {\tilde{H}}^{1/2}(\Gamma _t)\) with the desired estimate. Moreover, a lemma similar to Lemma 5.5 [52] can be follow \(\quad \square \)
Lemma 2.6
[52, Lemma 5.6] Let \(u\in H^{1/2}(\Gamma _j)\) (\(j=t, b\)), then \(\nabla _{\tau _j} u\) belongs to \({\tilde{H}}^{-1/2}(\Gamma _j)\) and satisfies the estimate
We now recall some elliptic systems and estimates including singular decompositions in corner domains. First, for the mixed-boundary system
we quote directly the following variational result:
Lemma 2.7
[52, Lemma 5.9] For a given function \(f\in H^{1/2}(\Gamma _t)\), the system
admits a unique solution \(f_{{\mathcal {H}}}={{\mathcal {H}}}(f)\in H^1(\Omega _t)\), and there holds
The next proposition shows the existence and estimate for solutions in \(H^2(\Omega _t)\). Notice that when the contact angles are below \(\pi /2\), no singularity appears.
Proposition 2.1
Let \(h\in L^2(\Omega _t)\), \(f\in H^{3/2}(\Gamma _t)\), \(g\in H^{1/2}(\Gamma _b)\) and \(\Gamma _t \in H^{5/2}\) be given in \(\text{(MBVP) }\). The contact angles \(\omega _i\in (0, \pi /2)\). Then there exists a unique solution \(u\in H^2(\Omega _t)\) to \(\text{(MBVP) }\). Moreover, one has
with the constant \(C=C(\Vert \Gamma _t\Vert _{H^{5/2}})\).
Proof
This proposition is a direct conclusion from Proposition 5.1, Lemma 5.2 and Theorem 5.3 in [51]. \(\quad \square \)
Second, we consider the Neumann-boundary system
satisfying the compatibility condition
The following results for \(H^3\)-and-below case are needed in our paper:
Proposition 2.2
(1) Let \(h\in L^2(\Omega _t)\), \(f\in L^2(\Gamma _t)\), \(g\in L^2(\Gamma _b)\) in \(\text{(NBVP) }\). Then there exists a unique (up to an additive constant) variational solution \(u\in H^1(\Omega _t)\) to \(\text{(NBVP) }\) satisfying
(2) Let \(h\in H^s(\Omega _t)\), \(f\in H^{1/2+s}(\Gamma _t)\), \(g\in H^{1/2+s}(\Gamma _b)\) in \(\text{(NBVP) }\). The contact angles \(\omega _i\in (0, \pi /2)\). Then there exists a unique (up to an additive constant) solution \(u\in H^3(\Omega _t)\) to \(\text{(NBVP) }\) satisfying
for a constant \(s\in [0,1]\).
Proof
We only write a sketch for the proof here, since one can find similar details in our previous papers [51,52,53]. First, the existence of the variational solution \(u\in H^1(\Omega _t)\) can be proved directly by a standard variation procedure based on Lemma 4.4.3.1 [30]. Note that here one doesn’t require that the contact angles are below \(\pi /2\). Second, one needs to notice that, there is no singular part since the most singular part for \(\text{(NBVP) }\) behaves like \(r^{\pi /\omega }\) near the corners (see for example [30]). Therefore, one can have directly \(H^2\) estimate from Theorem 3.2.5 [10] and Proposition 5.1 [51], Theorem 4.3.1.4 [30]. Similarly to the proof for Proposition 5.13 [51], one has \(H^3\) estimate without singular part. As a result, the desired estimate in (2) can be proved by an interpolation. \(\quad \square \)
Remark 2.1
For the first estimate in Proposition 2.2, when one has additionally
the norm \(\Vert u\Vert _{L^2(\Omega _t)}\) can be deleted from the right side during the proof.
Moreover, we present the \(H^4\) singularity decomposition and estimate.
Proposition 2.3
Let \(h\in H^2(\Omega _t)\), \(f\in H^{5/2}(\Gamma _t)\), \(g\in H^{5/2}(\Gamma _b)\) and \(\Gamma _t \in H^{4}\) in \(\text{(NBVP) }\). The contact angles \(\omega _i\in (0, \pi /2)\). Then there exists a unique (up to an additive constant) solution \(u\in H^3(\Omega _t)\) to \(\text{(NBVP) }\) such that
and the regular part \(u_r\in H^4(\Omega _t)\). Here the cut-off functions \(\chi _\omega , \chi _i\) are defined in the notation part. \(T_i\in H^{4}(\Omega _t)\) are boundary-straightening diffeomorphisms from \(\Omega _t\) onto the sectors \({{\mathcal {S}}}_{t,i}\) near the corner points \(p_i\) (from [51]), and \(S_i= r^{\pi /\omega _i}s(\theta )\) with r the radius with respect to \(p_i\) in \({{\mathcal {S}}}_{t, i}\) and \(s_i(\theta )\) some fixed sine or cosine functions.
Moreover, one has estimates for the singular coefficients \(c_i\) (\(i=l,r\)) and the regular part \(u_r\):
with the constant \(C=C(\Vert \Gamma _t\Vert _{H^{4}})\).
Proof
Similarly to the proofs of Proposition 5.17, Proposition 5.18 [51] for the mixed-boundary problem, the proof can be checked and follows line by line thanks to [30], so we omit the details here. \(\quad \square \)
Remark 2.2
A direct conclusion from this proposition is that when the contact angles \(\omega _i\in (0,\pi /3]\), the solution u to \(\text{(NBVP) }\) with the same right side will be in \(H^4(\Omega _t)\) with the corresponding estimate.
Remark 2.3
Based on the proposition above and estimate (9.15) in [24], when the contact angles \(\omega _i\in (\pi /3, \pi /2)\), one can have a more delicate and also natural estimate for \(\text{(NBVP) }\) with \(h\in H^{1+\epsilon }(\Omega _t)\), \(f\in H^{3/2+\epsilon }(\Gamma _t)\), \(g\in H^{3/2+\epsilon }(\Gamma _b)\) and \(\Gamma _t \in H^{4}\):
where \(\epsilon \in (0, \pi /\omega -2)\subset (0,1)\).
In the end, we recall some useful expressions and commutators from [52, 53, 60].
- \(D_tn_t\) and \(D_t\tau _t\). One has
- \([D_t,{{\mathcal {H}}}]\). One has for a smooth function f on \(\Gamma _t\) that
where \(\Delta ^{-1}(h,g)\) and \({{\mathcal {H}}}\) are defined in the notation part.
- \([D_t, {{\mathcal {N}}}]\). One has
- \([D_t,\Delta _{\Gamma _t}]\). For a smooth function f on \(\Gamma _t\), there holds
- \([D_t,\,\Delta ^{-1}]\). We have
with
- \([D_t, \nabla _{\tau _t}]\). Direct computations lead to
3 Reformulation of the Problem
In this section, we derive the equation for a good unknown \({\mathfrak {J}}\), which is slightly different from the quantity \(J=\nabla \kappa _{{{\mathcal {H}}}}\) introduced in [60] and used in our previous papers [52, 53].
To begin with, we define \(P_{v, v}\) by the Neumann-boundary system
where \(C_{v,v}(t)\) satisfies
Moreover, due to the non-uniqueness of the variational solution to this Neumann problem, we assume that \(P_{v, v}\) satisfies
In this way, we will have a unique solution to (3.1) in Sobolev space.
Let the pressure P be decomposed into
where \({\mathfrak {K}}\) is the modified curvature on \(\Gamma _t\) defined by
Now, we can define the new quantity
and we are ready to derive its equation from \(\hbox {(WW)}\). In fact, since the derivation for the equation of \({\mathfrak {J}}\) is much similar to the derivation for J in [52], many details are omitted here.
First, recall the equation for the curvature \(\kappa \) from [52],
or equivalently,
Applying \(D_t\) to (3.6) and using (3.4), we have
with
Moreover, direct computations almost the same as in [52] lead to
where the remainder terms are
with
and
Here we note that the leading-order terms in \(A_1, A_2\) are like \({\mathfrak {J}},D_t{\mathfrak {J}},\nabla v,\nabla D_tv\).
Substituting (3.7) into (3.8) and applying Euler’s equation, we can have after some more computations that
with
Here, one observation is that since \(\nabla _{n_t} P_{v,v}\big |_{\Gamma _t} =C_{v,v}(t)\) from (3.1), we find immediately in \(R_0\) that
Moreover, we use the following Hodge decomposition from [52, 60]:
such that \({{\mathcal {D}}}_t {\mathfrak {J}}\) satisfies
and \({{\mathcal {P}}}_{{\mathfrak {J}},v}\) satisfies the Neumann-boundary system
with
Here the assumption
holds similarly to guarantee the uniqueness as for \(P_{v,v}\) system.
This implies immediately
As a result, we finally conclude the equation for \({\mathfrak {J}}\) in the form based on (3.12),
where the third-order elliptic operator \({{\mathcal {A}}}\) is defined in the same way as in [52, 60],
for any smooth-enough function w defined on \(\Omega _t\), and the remainder term \({{\mathcal {R}}}\) is
Moreover, one can find the definition of \(\nabla P_{v,v}|_c\) in the notation part.
Remark 3.1
Instead of (3.12), we write the equation of \({\mathfrak {J}}\) in a more complicated form (3.15). This happens because of technical reasons. In fact, \({{\mathcal {D}}}_t {\mathfrak {J}}\) is more handy to use in the energy estimates, while as a price we have the remainder \({{\mathcal {P}}}_{{\mathfrak {J}},v}\) part to deal with. Meanwhile, the part \(v\cdot (\nabla P_{v,v}|_c)\) is added to derive better estimate for \(D_tP_{v,v}\), see Lemma 4.7.
4 Lower-Order Energy Estimates
The lower-order energy E(t) is defined as
where we write
Moreover, recalling from [52, 53] that our \(\text{(WW) }\) system has some dissipation at corner points, we have the following dissipation term at corner points
Theorem 4.1
Let the contact angles \(\omega _i\in (0,\pi /2)\). Assume that \(E(t), \int _0^T F(t){\text {d}}t\) are both bounded above in [0, T] for some \(T>0\). Then the following a priori estimate holds for system \(\text{(WW) }\):
there \(P(\cdot )\) is some polynomial with positive constant coefficients depending on \(\sigma , \beta _c\).
One can see immediately that, our energy is defined mainly in forms of \({\mathfrak {J}}\). As a result, we need to related (v, P), \(\Gamma _t\) and other quantities to \({\mathfrak {J}}\) firstly before we prove this energy estimate.
4.1 Dependence on E(t)
We show in this part that all the quantities related to our problem can be controlled by E(t). To start with, the following proposition focuses on the estimates for \(P_{v,v}, n_t, {\mathfrak {K}}_{{\mathcal {H}}}, {\mathfrak {J}}\):
Lemma 4.1
Assuming that \(E(t)\in L^\infty [0,T]\) for some \(T>0\), one has the following estimates:
and
Proof
(1) The first inequality. Applying Proposition 2.2 with an interpolation to \(P_{v,v}\) system (3.1), we get
For the right-hand side, we have firstly by Lemma 2.1 that
Second, it’s straightforward to find
and
where (3.2) is used and the variational estimate above is proved similarly as Proposition 2.2(1).
Consequently, summing up all these inequalities above, we conclude that
(2) The second inequality. In fact, the key point of the proof lies in the estimates for mean curvature \(\kappa \) and \({\mathfrak {K}}=\sigma \kappa -P_{v,v}\).
To begin with, one can see immediately that \({\mathfrak {K}}_{{\mathcal {H}}}\) satisfies the system
Applying Proposition 2.2(2), we find
where interpolation for \({\mathfrak {K}}_{{\mathcal {H}}}\) is applied to \(\Vert {\mathfrak {J}}^\perp \Vert _{L^2}\).
Moreover, from the definition of \({\mathfrak {K}}\) and Lemma 2.7 we have
which together with the previous inequality leads to
This implies immediately the desired \(H^{3/2}\) estimate for \({\mathfrak {J}}\).
For the estimate for \(n_t\), we find by Lemma 2.4, (4.3) and part (1) that
As a result, the proof is finished thanks to the fact that
\(\square \)
Lemma 4.2
Assuming that \(E(t)\in L^\infty [0,T]\) for some \(T>0\), one has the following estimates for v:
Meanwhile, one also has
Proof
- \(H^1\) estimate for \({{\mathcal {P}}}_{{\mathfrak {J}}, v}\). From system (3.14) of \({{\mathcal {P}}}_{{\mathfrak {J}}, v}\), we can have the following variational estimate by Proposition 2.2(1) and Remark 2.1:
which Together with Lemma 4.1 thus leads to the desired estimate for \({{\mathcal {P}}}_{{\mathfrak {J}}, v}\).
- \(v^\perp \) estimate. In fact, it is straightforward to get by Lemma 2.4 that
Rewriting \(\nabla \kappa _{{\mathcal {H}}}\) in terms of \({\mathfrak {J}}, \nabla P_{v,v}\) leads to
Applying Lemma 4.1, we have
On the other hand, we rewrite (3.5) into
which together with the inequality above for \(\Vert D_t \kappa \Vert _{H^{1/2}(\Gamma _t)}\) and the estimate for \({{\mathcal {P}}}_{{\mathfrak {J}}, v}\) imply
Here we use (3.6) to obtain
As a result, the estimate for \(v^\perp \) depends on the estimates for \(\Vert D_tP_{v, v}\Vert _{H^1(\Omega _t)}, \Vert v\Vert _{H^2(\Omega _t)}\).
- \(H^2\) estimate for v. Recalling system (4.10) for velocity potential \(\phi \) with \(v=\nabla \phi \), we have by Proposition 2.2 that
Bringing the above estimate into (4.5) and applying interpolations to \(\Vert v^\perp \Vert _{H^{3/2}(\Gamma _t)}\), we arrive at
Therefore, it remains to deal with \(\Vert D_tP_{v, v}\Vert _{H^1(\Omega _t)}\).
- \(D_tP_{v,v}\) estimates. From the definition of \(P_{v, v}\), we derive the system for \(D_tP_{v,v}\):
with
It is easy to see from Euler’s equation that
and similar expression can be derived for \(D_t(v\cdot \nabla _v n_b)\) on \(\Gamma _b\).
Moreover, we have
where \(\frac{\text {d}}{{\text {d}}t}|\Gamma _t|=\int _{\Gamma _t}\big (v^\perp \kappa +\nabla _{\tau _t}(v\cdot \tau _t)\big ){\text {d}}s\). So we obtain by Lemma 4.1, (4.3) and (4.4) that
To prove the \(H^1\) estimate for \(D_tP_{v,v}\), we use the following form of variation equation:
Substituting the expressions in (4.7) into the equality above, we can deal with the integrals one by one, where Lemma 2.1 is applied. For example, for the highest-order terms of v on the boundary, we use Green’s Formula and Lemma 2.1 to find
The other integrals above can be handled similarly and the details are omitted.
As a result, we obtain by Lemma 4.1 the following desired estimate:
Consequently, applying (4.6) leads to the estimates for \(v^\perp \) and v. \(\quad \square \)
Remark 4.1
We know immediately from the proof of Lemma 4.2 the following estimate:
Moreover, we also have
Based on these lemmas above, we are ready to show some higher-order estimates.
Lemma 4.3
Let \(E(t)\in L^\infty [0,T]\) for some \(T>0\), then one has
Proof
For \(P_{v,v}\), similar arguments as in the proof of Lemma 4.1 lead to
Applying Lemma 4.2, we have the desired estimate.
Again, similar arguments as (4.4) in the proof of Lemma 4.1 lead directly to the estimate
which implies
Therefore, combining this with the estimate for \(P_{v,v}\), the proof is finished. \(\quad \square \)
We also present more estimates for v.
Lemma 4.4
Assuming that \(E(t)\in L^\infty [0,T]\) for some \(T>0\), one has
Proof
First, one recalls Euler’s equation to get that
and applying Lemma 4.3 leads to the estimate for \(D_tv\).
Second, taking \(D_t\) on Euler’s equation leads to the following estimate:
Using Lemma 4.2 again, we can finish the estimate for \(D^2_t v\). \(\quad \square \)
We now consider the following Neumann system for velocity potential \(\phi \):
where in order to obtain the uniqueness we chose \(\phi \) to satisfy
without loss of generality.
We show by singularity decompositions that \(\nabla v\) lies in \(L^\infty (\Omega _t)\), which is a key ingredient in the following estimates:
Lemma 4.5
Assume that \(E(t)\in L^\infty [0,T]\) for some \(T>0\), then there exists a unique \(\phi \in H^3(\Omega _t)\) to system (4.10) and one finds the singular decomposition
where the regular part \(\phi _r\in H^4(\Omega _t)\), and the singular part is expressed in the same way as \(u_s\) in Proposition 2.3. Moreover, one has
Proof
In fact, applying Proposition 2.3, we know immediately about the existence of \(\phi \in H^3(\Omega _t)\) and the singular decomposition. Therefore, we have
It only remains to show the estimate for \(\nabla v\). Since we have
where the singular part exists when \(\omega _i\in (\pi /3, \pi /2)\). In this case, we find by Proposition 2.3 that
with \(\pi /\omega _i-2\in (0,1)\). Consequently, we obtain
thanks to a direct variational estimate
Moreover, we also have by Proposition 2.3 the following estimate:
As a result, we find that
which together with Lemma 4.2 lead to the desired estimate. \(\quad \square \)
As long as we have \(\Vert \nabla v\Vert _{L^\infty (\Omega _t)}\) estimate, we are able to deal with more higher-order estimates.
Lemma 4.6
Let \(E(t)\in L^\infty [0,T]\) for some \(T>0\), one has the following estimate:
Proof
- \(P_{v,v}\) estimate. In fact, similar arguments as in the proof of Lemma 4.1 show that
Consequently, applying Lemmas 4.2 and 4.5 leads to the desired estimate.
- Estimate for \(P_{{\mathfrak {J}}, v}\). Apply Proposition 2.2, we have
Using Lemmas 2.1, 2.2, 4.1, 4.2 and 4.5, we can finish the esitmate.
- \(H^{3/2}\) estimate for \(D_tP_{v,v}\). Following the proof of Lemma 4.2 and using Lemma 4.2 again to improve estimates for v, one can easily see that we have the desired estimate. \(\quad \square \)
We now give a high-order estimate for \(D_t P_{v, v}\).
Lemma 4.7
Assuming \(E(t)\in L^\infty [0,T]\) for some \(T>0\), one has
Proof
We denote by
A direct computation using (4.7) leads to
Applying Proposition 2.2 to system (4.12), we have
We need to deal with the terms on the right side above. First, thanks to (4.8), Lemmas 4.2, 4.3 and 4.5, it is straightforward to check that
Notice that here we need to use Lemma 2.2 for the product estimate of \(\Vert \nabla ^2{\mathfrak {K}}_{{\mathcal {H}}}\nabla v\Vert _{H^{1/2}(\Omega _t)}\) from \(\Vert D_t tr(\nabla v \nabla v)\Vert _{H^{1/2}(\Omega _t)}\):
The estimate for \(\Vert \nabla ^2 P_{v,v}\nabla v\Vert _{H^{1/2}(\Omega _t)}\) also follows in a similar way.
Second, for the boundary terms, we only need to take care of the following estimate from \(\Vert \nabla _{n_t} v\cdot (\nabla P_{v, v}-\nabla P_{v,v}|_{c})\Vert _{H^1(\Gamma _t)}\):
Here the estimate for the second term in the equality above holds thanks to (4.11) and Proposition 2.3.
To handle the first term in (4.13), using a straightening differeomorphism as \(T_i\) from Proposition 2.3 and (4.11), we know immediately that
Moreover, applying Lemma 2.3 leads to
so we obtain by an interpolation the inequality
As a result, combining this with (4.14), we conclude that
Substituting this estimate into (4.13), we finally obtain
In the end, the estimate for \(\Vert \nabla _{n_b} v\cdot (\nabla P_{v, v}-\nabla P_{v,v}|_{c})\Vert _{H^1(\Gamma _b)}\) follows in a similar way, and the proof can be finished. \(\quad \square \)
In the end, we give the estimate of \(\int _{\Omega _t} \nabla D_t{{\mathcal {P}}}_{\mathfrak {J}, v}\cdot D_t \mathfrak {J}{\text {d}}X\).
Lemma 4.8
Let \(E(t)\in L^\infty [0,T]\) for some \(T>0\). Then, we have the following estimate:
Proof
Recalling system (3.14), we get
Therefore, we have, directly that
which leads to the following equality:
To finish the proof, the key lies in the analysis for the first integral. The remainder part is controlled by P(E(t)) thanks to Lemma 4.1, Lemma 4.2, Remark 4.1 and Lemma 4.6.
For the first integral on the right side of the equality above, we have by Green’s Formula that
where the remainder lower-order terms can be controlled by P(E(t)) in a similar way as before.
Now we deal with the terms in (4.16) one by one. To begin with, we write from the definition of \(C_{{\mathfrak {J}},v}\) in system (3.14) that
where the lower-order terms can be controlled again and hence the details are omitted. Applying Green’s Formula again and using the decompositions
we find
Here, for the last three terms on \(\Gamma _b\), thanks to the assumption that \(n_b\) is constant near the contact points \(p_i\) and \(v\cdot n_b|_{\Gamma _b}=0\), we know that \(\nabla _{\tau _b} v\cdot n_b=-v\cdot \nabla _{\tau _b}n_b\) and \(\nabla _v n_b=(\nabla _v n_b\cdot \tau _b)\tau _b\) vanish near \(p_i\). Moreover, we also know from the definition of \({\mathfrak {K}}_{{\mathcal {H}}}\) that
Consequently, applying Lemmas 2.5 and 2.6, we have
and
For the first integral in (4.17), applying Green’s Formula again leads to
so this term is controlled by P(E(t)) again.
Therefore, summing up all these estimates above and going back to (4.17), we arrive at
with the lower-order terms controlled by P(E(t)). This leads to the following estimate
For the moment, we can go back and deal with the other terms in (4.16). In fact, using similar arguments as above, we also conclude that
Therefore, the proof is finished. \(\quad \square \)
4.2 Boundary Terms at the Contact Points
Lemma 4.9
We have the following equation on \(\Gamma _b\):
then there holds at the contact points \(p_i\) (\(i=l, r\)) that
where
satisfy
Proof
The proof follows the proof of Lemma 7.1 in [52]. For the estimates, we apply Lemmas 2.1, 4.3, 4.5 and 4.7. Here the differences compared to the proof of Lemma 7.1 in [52] lie in that we have different \({\mathfrak {J}}\) and
in \(r_c\) due to the definition of \(P_{v,v}\). \(\quad \square \)
4.3 Proof of Theorem 4.1
We are ready to show the a priori energy estimate in Theorem 4.1. In fact, taking \(L^2(\Omega _t)\) inner product with \({{\mathcal {D}}}_t J-\nabla {{\mathcal {H}}}( D_t P_{v,v} - v\cdot \big (\nabla P_{v,v}|_{c})\big )\) on both sides of (3.15), we have
We deal with these integrals above one by one. To get started, for the first term on the left side, we rewrite it as
For the second term on the left side of (4.19), we have by Green’s Formula that
For the first term on the right side of (4.20), one deduces from Hodge decomposition (3.13) and integration by parts as in [53] that
where we apply Lemma 4.9 at corner points.
For the second term in (4.20), one has in a similar way as above that
where Lemma 2.1 and Lemma 4.7 are used.
As a result, going back to (4.19), we summerise that
Moreover, direct estimates similarly to those in [53] lead to the estimate
so we obtain
In order to finish the energy estimate, we still need to deal with the integral on the right side involving the remainder term \({{\mathcal {R}}}\) in (3.15).
- Estimates for the part \(R_0-\sigma \nabla {{\mathcal {H}}}(R_1)\) in \({{\mathcal {R}}}\).
Lemma 4.10
For the remainder term \(R_0\) defined in (3.12), we have the estimate
Proof
Recalling from (3.12), we know that
where \(R_1\) and \(A_1,A_2,A_3\) are defined in (3.7) and (3.9), (3.10), (3.8) respectively.
For the term \(\nabla {{\mathcal {H}}}(n_t\cdot \Delta _{\Gamma _t}\nabla P_{v,v})\), we have
where the boundary condition \(\nabla _{n_t}P_{v,v}|_{\Gamma _t}=C_{v,v}(t)\) and Lemma 4.6 are applied.
For the term \([D_t, \nabla {{\mathcal {H}}}]( D_t P_{v,v} )\), direct computations using (2.4) and similar arguments as in the proof of Lemma 4.2 lead to
The estimates for the other terms follow from lemmas in the previous section and can be done similarly as [53], so we omit the details here. \(\quad \square \)
- The part \(\sigma \nabla {{\mathcal {H}}}(R_1 )\) in \({{\mathcal {R}}}\). In fact, the integral from the right side of (4.21) is rewritten as
On the other hand, noticing from (3.7) that \(R_1\) contains terms like \(\nabla ^2 v,\nabla n_t, \kappa \) and \(\nabla ^2 P_{v, v}\) and using (4.11) and lemmas in the previous subsections, we have
The details in the estimate above are omitted and we only note that for \(\partial ^2 v\) terms in \(R_1\), we can have the following estimate thanks to (4.11):
Similarly but more easily, we also obtain
In addition, for the part \(\int _{\Gamma _t}R_1 D_t {\mathfrak {J}}\cdot n_t {\text {d}}s\), we put \(D_t\) out of the integral:
Here we need to take care of the terms in \(\int _{\Gamma _t}D_t R_1\,({\mathfrak {J}}-{\mathfrak {J}}|_c){\text {d}}s\) and \(\int _{\Gamma _t}R_1\,({\mathfrak {J}}-{\mathfrak {J}}|_c)D_t{\text {d}}s\). In fact, similarly to the analysis in (4.14), the key terms like \(\Vert \partial ^2 v ({\mathfrak {J}}-{\mathfrak {J}}|_c)\Vert _{L^2(\Gamma _t)}\) can be handled as follows:
there the constant \(\delta \in (0,1)\) is chosen to satisfy
Consequently, we derive
where all the lower-order terms are controlled by P(E(t)).
As a result, we conclude that
with the lower-order terms controlled by P(E(t)).
- The terms \(D_t\nabla {{\mathcal {P}}}_{{\mathfrak {J}},v}+D_t\nabla {{\mathcal {H}}}\big (v\cdot (\nabla P_{v,v}|_c)\big )\) in \({{\mathcal {R}}}\). First, one has directly from Lemma 4.8 that
where the lower-order terms are all controlled by P(E(t)).
Second, for the integral
applying Green’s formula and similar calculations as in (4.16) show directly that it can be controlled by P(E(t)), and we omit the details.
Therefore, we conclude that
where all the lower-order terms are controlled by P(E(t)).
Summing up all these estimates related to \({{\mathcal {R}}}\) above and going back to (4.21), we obtain the following estimate:
Integrating on both sides with respect to time on [0, t], we have
Replacing the first integral with \(\Vert {{\mathcal {D}}}_t {\mathfrak {J}}\Vert ^2_{L^2(\Omega _t)}\), we arrive at the inequality
where the constant C depends on \(\sigma , \beta _c\).
We now deal with the last three integrals on the right side above one by one. First, for the terms in \(R_1\), similarly as in (4.22), we can have by careful estimates with interpolations that
and
where \(1>\delta >3-\pi /\omega \) when \(\omega \in (\pi /3, \pi /2)\).
Therefore, we can prove by interpolations that there exist \(\delta _0\in (1/2, 1)\) and \(\varepsilon \) small enough such that
Moreover, the following estimates can also be proved in a similar but easier way:
As a result, summing up the estimates above, we are able to conclude that
In the end, to close the energy estimates, we deal with \(E_l(t)\). First, we show by direct calculations using Euler’s equation and lemmas from the previous section that
for \(k=1,2\). Then we take the square root on both sides of the inequalities above and apply an interpolation between \(H^1(\Omega _t), H^2(\Omega _t)\) as well as Lemma 4.1 to obtain
Second, the estimate for \(\Vert \Gamma _t\Vert _{H^{5/2}}\) can be derived in a similar way as above. Consequently, we have
Combining this estimate with the estimate above for E(t), we finish the lower-order energy estimates.
5 Higher-Order Time-Derivative Energy Estimates
In this section, we prove higher-order energy estimates with respect to \(D_t\), which is needed in the local well-posedness part. To start with, we define the energy functional
and the dissipation
The main result of this section is as follows:
Theorem 5.1
Let the contact angles \(\omega _i\in (0,\pi /2)\) and \(E(t), \int _0^T F(t)dt, E_1(t), \int _0^T F_1(t)dt\) be bounded above in [0, T] for some \(T>0\). Then the following higher-order a priori estimate holds
To prove this theorem, we begin with a higher-order equation for \({\mathfrak {J}}\) and more delicate estimates involving \(D_t\) based on Section 4.1. With these preparations, we are able to finish the energy estimate in the last subsection of this part.
5.1 The Higher-Order Equation for \({\mathfrak {J}}\)
Firstly, we recall system (3.12) and rewrite it as follows:
where
and \(h_v\) contains all the second-order terms of v in \(R_1\) which come from the commutator \([D_t, n_t\Delta _{\Gamma _t}]\cdot v\). More precisely, we have
As a result, \(R_1+h_v\) only contains lower-order derivatives like \(\partial v, \partial n_t, \kappa \).
Acting \(D_t\) on both sides of (3.12), we obtain
with the right side
Here we use
with all these commutators from the end of Section 2, and
where \(P_{w, v}\) is defined in (1.10).
5.2 More Preliminary Estimates.
Based on Section 4.1, we are going to prove higher-order estimates for different quantities using our higher-order energy.
- Higher-order estimates for \(\mathfrak {K}_{{{\mathcal {H}}}}\) and \(D_t{\mathfrak {J}}\). To get started, we recall system (4.1) to find
Applying Proposition 2.2 and lemmas in Section 4.1, we have
Here we notice that the regularity of \(D_t\mathfrak {K}_{{\mathcal {H}}}\) is constrained due to the regularity of v. To get a higher-order estimate, we need to get rid of the worst part.
Similarly as in Lemma 4.7, we consider a good quantity \(D_t\mathfrak {K}_{{\mathcal {H}}}-v\cdot (\nabla \mathfrak {K}_{{\mathcal {H}}}|_{c})\) which satisfies
As a result, similar arguments as in (4.22) applied for the part \(\nabla v\cdot \big (\nabla {\mathfrak {K}}_{{\mathcal {H}}}-\nabla \mathfrak {K}_{{\mathcal {H}}}|_{c}\big )\), we can have
where \(\Vert D_t{\mathfrak {J}}^\perp \Vert _{L^2(\Gamma _t)}\) is handled by (5.5).
On the other hand, we deal with \(D_t{\mathfrak {J}}\). First, we know directly from (5.5) that
Second, we notice that
Similarly as in (4.14), we apply Lemma 2.2(1) and (4.11) to obtain the estimate
while recall that \({{\mathcal {S}}}_{t,i}\) are straightened sector of \(\Omega _t\) with radius \(r_0\) near the corners.
Therefore, combining this estimate with (5.6), we have
and also
- Estimates for \(D_tP_{v,v}\). Checking system (4.7) carefully and applying lemmas from Section 4.1, one has immediately
where we use arguments similar to (4.13) on boundary terms.
- Estimates for v. To begin with, applying Remark 2.3, we obtain for a small number \(\epsilon \in (0, \pi /\omega _i-2)\) (when \(\omega _i\in (\pi /3, \pi /2)\)) that
Moreover, we improve the regularity of \(v^\perp \). In fact, similarly as in (4.5), we have thanks to (5.5),(5.9) and (5.10) the estimate
- Estimates for \(D^2_tv, D^3_tv\). First, acting \(D_t\) on both sides of Euler’s equation leads to
Applying Lemma 4.2, Lemma 4.6, (5.7), and (5.8), we find
Next, we give the estimates of \(D_t^3 v\). Before that, we first prove the following lemma:
Lemma 5.1
Let \(P_{\mathfrak {J}, v}, P_{\nabla P_{\mathfrak {J}, v}, v}\) and \(P_{{{\mathcal {D}}}_t \mathfrak {J}, v}\) be defined by (1.10). Then there hold
and
Proof
First, we deal with \(D_t P_{\mathfrak {J}, v}\). Recalling system (4.15), we have by Proposition 2.2, (4.17), (5.5), (5.8) and lemmas from Section 4.1 that
Second, checking the definition (1.10) and applying Lemma 2.2 (1) and Lemma 4.6, we can have for \( P_{\nabla P_{\mathfrak {J}, v}, v}\) the following estimate immediately:
Moreover, a similar argument also leads to
In the end, for the part \(D_t^2 P_{v,v}\), we deduce from (4.7) the following system:
with
Applying (5.16), (5.12) and checking term by term, we obtain the variational estimate
and the proof is finished. \(\quad \square \)
Now, we are in a position to give the estimate for \(D_t^3 v\). In fact, acting \(D_t^2\) on both sides of Euler’s equation, we derive by the previous lemma and lemmas from Section 4.1 that
- Estimate for \(v^\perp \). We obtain by the definition of \(\mathfrak {J}\) that
and applying (3.5) and (5.5) implies
Recalling (3.5) again, we know
so we obtain by (5.15) and (5.17) the estimate
- Some more higher-order estimates for \({\mathfrak {K}}_{{\mathcal {H}}}\), v and \({\mathfrak {J}}\).
Lemma 5.2
Assuming that \(E(t), E_1(t)\in L^\infty [0,T]\) for some \(T>0\), we have the following estimate:
Proof
(1) Higher regularity for \({\mathfrak {K}}_{{\mathcal {H}}}\). To begin with, we use Euler’s equation to rewrite (3.7) as follows:
where we notice that \(\Delta _{\Gamma _t}(\nabla _{n_t} P_{v,v})|_{\Gamma _t}=0\) thanks to the definition of \(P_{v,v}\) and \((R_1+h_v)\) contains only \(\partial v\) terms instead of \(\partial ^2 v\) terms.
Meanwhile, we know from (5.15) and (5.17) that
so checking term by term in the equation above, we have immediately
On the other hand, we know from (4.11) that (when \(\omega _i\in (\pi /3, \pi /2)\))
where we note \(\alpha _i=\pi /\omega _i-1\in (1,2)\), \(a_{i,k}\) (\(i=l, r\)) contain \(n_t, \tau _t, \partial n_t, \partial \tau _t\) and the singular coefficient from (4.11). Moreover, \(a_{i,3}(\partial v_r\circ T^{-1}_i)\) is linear with respect to \(\partial v_r\circ T^{-1}_i\), where we recall that \(v_r=\nabla \phi _r\in H^3(\Omega _t)\) and \(\Vert v_r\Vert _{H^3(\Omega _t)}\) is controlled by P(E(t)).
As a result, we find
which implies immediately that
Summing up the estimates above for \(\Vert \Delta _{\Gamma _t}J^\perp -h_v\Vert _{H^{1/2}(\Gamma _t)}\) and \(\Vert h_v\Vert _{L^p(\Gamma _t)}\), we obtain
and this leads to
Applying Lemma 2.1, we finally show that
with \(\varepsilon =1-1/p\).
Consequently, applying Proposition 2.2, we have the desired estimate for \(\Vert {\mathfrak {K}}_{{\mathcal {H}}}\Vert _{H^{3}(\Omega _t)}\). Notice that we have in fact the estimate by Remark 2.3:
Moreover, we apply Lemma 2.4 to have \({\mathfrak {K}}=\sigma \kappa -P_{v,v}\in H^{5/2+\varepsilon }(\Gamma _t)\) with the estimate
(2) \(H^2\) estimate for \(D_tv\). Recalling the Euler’s equations and \(P_{v,v}\) estimate from Lemma 4.6, we get
(3) \(L^\infty \) estimates. Applying Remark 2.3 and (5.21) with the same \(\varepsilon =1-1/p\) as above lead immediately to
Applying Remark 2.3 to \(P_{v,v}\), we obtain
Using (4.11) and a similar argument as in (1), we have
which implies
In the end, apply Euler’s equation again leads to
which finishes the proof. \(\quad \square \)
- Estimate for \(D^2_tP_{v,v}\). Based the above estimates, we firstly improve the estimate for \(D_tP_{v,v}\). Using system (4.12) and Lemma 5.2, we improve the estimate in Lemma 4.7 and get
Here we use Lemma 2.2 (2) for the boundary terms like \(\Vert \nabla _{n_t} v\cdot (\nabla P_{v, v}-\nabla P_{v,v}|_{c})\Vert _{H^{3/2}(\Gamma _t)}\), which are handled similarly as in the proof for (5.8).
Next, we derive the equation of \(D_t(D_t P_{v, v}-v\cdot (\nabla P_{v,v}|_{c}))\). To simplify the notation, we define
and we rewrite (5.22) as
Direct computations lead to the following system for \(D_t{\mathcal {P}}_{t,1}\):
Moreover, we define
and we modify this system above as in Lemma 4.7 into a new system for \({\mathcal {P}}_{t,2}\) as follows:
Thanks to Lemma 4.7, (5.12) and (5.23), it is straightforward to show that
- The boundary condition for \(D_t^2 \mathfrak {J}\) at corner points.
Lemma 5.3
We have at the contact points \(p_i (i=l,r)\) the equations
where the remainder terms
with \(R_{c1}\) defined in Lemma 4.9. Moreover, if holds for \(i=l, r\) that
Proof
First, using \(D_t\) directly on both sides of the equations in Lemma 4.9, one has the desired equations immediately. Next, apply lemmas from Section 4.1, Lemma 5.2 (5.21) and (5.27), the estimates follow. Moreover, we notice that the higher-order term \(\nabla ^2_{\tau _t}v^\perp (\tau _t\cdot {\mathfrak {J}})\) from \( D_t(\nabla _{\tau _t} n_t\cdot \mathfrak {J})\tau _b\) in \(R_{c,h,i}\) can be controlled thanks to (5.11), where we use
\(\square \)
5.3 Proof of Theorem 5.1
At this moment, we are finally ready to prove the higher-order energy estimate in Theorem 5.1. To begin with, we rewrite equation (5.3) of \({{\mathcal {D}}}_t^2 \mathfrak {J}\) into
where
In fact, this complicated form is used due to similar technical reasons as (3.15), see Remark 3.1, where the terms \(\nabla {{\mathcal {H}}}({{\mathcal {P}}}_{t,2})\), \(D_t h_v\) are added on the left side to improve the estimates.
Next, we apply \(L^2(\Omega _t)\) inner product with \({{\mathcal {D}}}^2_t \mathfrak {J} -\nabla {{\mathcal {H}}}({{\mathcal {P}}}_{t,2})\) on both sides of (5.30) to get
Consequently, for any \(t'\in [0,T]\) if holds that
5.3.1 Left Side of (5.31)
In this subsection, we prove estimates for the second term on the left side of (5.31).
Proposition 5.4
One has for some \(\delta _0\in (0,1)\) the following estimate:
Proof
Applying Green’s Formula, we have
where
We deal with these integrals above one by one. First, integrating by parts and taking one \(D_t\) out of the first integral in (5.32), we get
Here, Lemma 5.3 is applied at the corner terms,
so one has, by (5.21) and lemmas from Section 4.1, that
On the other hand, since \(D_tds=(v^\perp \kappa +\nabla _{\tau _t}(v\cdot \tau _t)\big ){\text {d}}s\), we directly obtain the estimate:
Moreover, a direct computation using (5.29) shows that
Therefore, thanks to (5.8), Lemma 5.2 and checking term by term, we have the following estimate for the last two integrals in (5.33):
As a result, the proof is finished as long as we have Lemma 5.5 and Lemma 5.6. \(\quad \square \)
We deal with the remainder integrals in (5.32) in the next two lemmas.
Lemma 5.5
One has for the second term of the right hand of (5.32) that
Proof
To begin with, one recalls the definition of \(h_v\) in (5.2) to obtain
We only deal with the first integral in the integral above, since the second one can be handled in a similar way.
In fact, it is straightforward to see by (5.29) that
The estimates for \(I_1, I_2\) are proved in the following lines.
- Estimates of \(I_1\). First, direct computations lead to
We deal with \(I_{1i}\) one by one. In fact, applying integration by parts, we have
Using (5.10) and similar arguments as to those in (4.22) for \(\partial ^2v\) and \(v^\perp \) terms and applying Lemma 5.3 to \(D^2{\mathfrak {J}}|_{p_i}\), we derive
For \(I_{11}\), since
where
contains higher-order terms of v, and the remainder part contains products like \(\partial D_tv\,\partial ^2v\) and other lower-order terms and can be controlled by (5.8), (5.12), Lemma 5.2 and similar arguments as in (4.22).
Therefore, we arrive at
To finish the estimate for \(I_{11}\), we firstly take care of the first term \(\nabla _{\tau _t}D^2_t\nabla _{\tau _t}v\) in \(h_{v,2}\). In fact, we use integration by parts on \(\Gamma _t\) to obtain
Similar arguments as to those in(4.22) and applying (5.8), (5.12), we derive
Meanwhile, the other terms in \(I_{11}\) can also be handled similarly, so we conclude
For \(I_{12}\), we have
where the first integral can be handled as above. For the second integral, we know from Lemma 2.5 that
so we apply Lemmas 2.5, 2.6, (5.8) and 5.2 to find
As a result, we summarize that
Moreover, similar arguments as for \(I_{12}\), we also have
Together with all these estimates above for \(I_{11}\) to \(I_{14}\), we can go back to \(I_1\) expression and integrate on both sides with respect to time t on [0, T] to find
Moreover, we have for the first integral on the right side above the following estimate
Similar arguments as in (5.34) and checking carefully on the highest-order terms, we derive
with the number \(\delta _0\in (0,1)\) as above.
Consequently, we have the following estimate for \(I_1\):
- Estimates of \(I_2\). The integral \(I_2\) can be handled in the same way as \(I_1\). We simply conclude that
In the end, combing (5.35) with (5.36), the proof is finished. \(\quad \square \)
Next, we deal with \(I_R\).
Lemma 5.6
One has
Proof
First, to simplify the notations in the first integral of \(I_R\), we denote by
Thanks to the definition of \(P_{w, v}\) (see (1.10)), we have
By (5.5), (5.8), Lemma 5.2 and (5.27), we show immediately
Now we are ready to have estimate of \(I_R\). For the first integral in \(I_R\), we integrate by parts to derive
where (5.8), (5.11), (5.29), Lemma 5.3 and (5.38) are used.
Moreover, we have from (5.2) and (5.38) that
where similar analysis as in the proof of the previous lemma is applied.
On the other hand, the second integral in \(I_R\) can also be handled by a similar argument as above. As a result, the proof is finished. \(\quad \square \)
5.3.2 Right Side of (5.31)
We firstly deal with the integral involving \({{\mathcal {R}}}_2\).
Lemma 5.7
One has for some \(\delta _0\in (0,1)\) the following estimate:
Proof
Recalling from (5.30), we deal with the terms in \({{\mathcal {R}}}_2\) one by one.
- Estimates of \(D_t \big (\tilde{R}_0-\nabla {{\mathcal {H}}}( D^2_t P_{v,v} ) \big ) \). In fact, one knows directly from (5.1) that
where recall that \(R_1+h_v\) only contains lower-order derivatives like \(\partial v, \partial n_t, \kappa \) and \(A_1\) to \(A_3\) are defined in (3.9)–(3.11).
As a result, checking term by term on \(\tilde{R}_0- \nabla {{\mathcal {H}}}( D^2_t P_{v,v} ) \), one finds that it contains \(\partial D_t v\), \(D_tJ\), \(\partial v\) and other lower-order terms, so acting \(D_t\) on \(\tilde{R}_0- \nabla {{\mathcal {H}}}( D^2_t P_{v,v} ) \) and applying lemmas in Sections 4.1 and 5.2 lead to the following estimate:
- Estimates of \([\nabla , D_t]{{\mathcal {H}}}(\Delta _{\Gamma _t}\mathfrak {J}^\bot -h_v) \). It is straightforward to show by (5.19) and Lemma 2.7 that
- Estimates of \(\nabla [{{\mathcal {H}}}, D_t](\Delta _{\Gamma _t}\mathfrak {J}^\bot -h_v) \). Recalling (2.4) and using Lemma 2.7, (5.19) and applying variational estimates as in page 33 [52] imply that
- Estimates of \(\nabla {{\mathcal {H}}}[\Delta _{\Gamma _t}, D_t]\mathfrak {J}^\bot \). To estimate this term, we use similar arguments as in the proof of Lemma 5.5.
First, using Green’s Formula, we obtain
These two integrals can be handled in a similar way as before, and here we only give the details for the first part in the second integral above. Moreover, the analysis for \(\int _{\Omega _t} \nabla {{\mathcal {H}}}[\Delta _{\Gamma _t}, D_t]\mathfrak {J}^\bot \cdot \nabla {{\mathcal {H}}}({{\mathcal {P}}}_{t,2}){\text {d}}X\) can be done in a similar and easier way thanks to (5.27), and hence estimates for all these remainder parts are omitted.
Recalling (5.4) and the definition of \(C_P\), we rewrite the first part in the second integral above as follows:
Similarly as the estimates for \(I_1\) in the proof of Lemma 5.5, one has for \(I_3\) that
where we use (5.20), Lemma 5.2 and Lemma 5.3 to have
with \(1>\delta >3-\pi /\omega _i\) for \(\omega _i\in (\pi /3,\pi /2)\) and
As a result, we have
Since similar arguments can be applied to \(I_4\) and other integrals, we conclude directly
- Estimates of \(D_t( \nabla P_{\nabla P_{\mathfrak {J}, v}, v} )\). Recalling the definition of \(P_{\mathfrak {J}, v}\) and \(P_{\nabla P_{\mathfrak {J}, v}, v} \) by (1.10), one has firstly the estimate (5.13). Moreover, one can take \(D_t\) on the system of \(P_{\nabla P_{\mathfrak {J}, v}, v}\) as in (4.7) to obtain the system for \(D_tP_{\nabla P_{\mathfrak {J}, v}, v}\).
Consequently, checking term by term and applying Lemma 4.6, Lemma 5.1, one derives
- Estimates of the last term in \({{\mathcal {R}}}_2\). Thanks to (2.4), (5.12), (5.27) and Lemma 5.1, we have
In the end, summing up all these estimates above, the proof is finished. \(\quad \square \)
At this moment, it remains to handle the last integral on the right side of (5.31). In fact, similar arguments as in the proof of Lemma 4.8, we can conclude the following estimate:
5.3.3 The End of the Higher-Order Energy Estimate
Summing up all the estimates in the previous two subsections, we finally arrive at:
Thanks to (5.27) and Lemma 5.1, we have for a number \(\delta _1>0\) small enough such that
Therefore, we’ve finished the proof of Theorem 5.1.
6 Well-Posedness of System \(\text{(WW) }\)
In this section, we use Picard iteration to prove the existence of solutions to \(\text{(WW) }\). The main idea is the same as [53], although some necessary modifications are needed and presented here. As a result, we only show the skecth of the proof. For more details, see [53].
6.1 Definitions of Surfaces and Domains
Following [53, 61], we introduce a map \(\Phi _{S_t}\) on the boundary \(S_t=\Gamma _t\cup \Gamma _b\) to fix the moving domain \(\Omega _t\). To start with, we choose a reference domain \(\Omega _*\) with upper surface \(\Gamma _{t*}\) and bottom \(\Gamma _{b*}\), which can be taken as the initial domain \(\Omega _0\) without loss of generality. The contact points of \(\Omega _*\) are denoted by \(p_{i*}\) (\(i=l, r\)) and the other notations follow similarly.
we define a unit upward vector field \(\mu \in H^s(\Gamma _{*}, {{\mathcal {S}}}^1)\) with some large s satisfying
for some fixed constant \(c_0\in (0,1)\). Here we notice that the conditions above hold at \(p_{l*}, p_{r*}\).
Applying the Implicit Function Theorem, there exists a small constant \(d_0>0\) such that the map
is an \(H^s\) diffeomorphism from its domain to a neighborhood of \(\Gamma _*\).
Consequently, this map identifies each upper surface \(\Gamma _t\) near \(\Gamma _{t*}\) with a unique function
and we can define the following map
Meanwhile, we can use the function \(d_{\Gamma _t}(p)\) as the expression of the upper surface \(\Gamma _t\), and we have at the corner points that
Moreover, \(\Phi _{S_t}\) can be extended to the entire boundary \(S_*=\Gamma _{t*}\cup \Gamma _{b*}\). Consequently, we obtain the map on \(S_*\):
Using the harmonic extension, we define the following map on \(\Omega _*\):
Here \({{\mathcal {H}}}_*\big (\Phi _{S_t}-Id_{S_*}\big )\) is the harmonic extension of \(\Phi _{S_t}-Id_{S_*}\) satisfying
6.2 Recovery of the Velocity
When the domain \(\Omega _t\) is defined by \({{\mathcal {T}}}_S\), we can define the velocity v by the free surface function \(d_{\Gamma _t}\). To begin with, the kinematic condition on \(\Gamma _t\) in \(\text{(WW) }\) is rewritten into
So we obtain
Due to the assumption that the velocity v is irrotational, we define v by
with \(\phi \) satisfying
where
Moreover, we define as in [53] that
where \(v^{*}= D\Phi _{S_t}^{-1}(v^\top \circ \Phi _{S_t} -\partial _t d_{\Gamma _t} \mu ^{\top } )\). A direct computation shows that
for a function f on \(\Gamma _t\).
6.3 The Modified Formulation and the Precise Form of the Main Theorem
Before we construct the approximate solutions, we derive a new equation modified from the previous sections, which turns out to be more convenient in this part.
First, we define \({\mathfrak {K}}_a\) based on the definition of \(\mathfrak {K}\):
for some constant \(a>0\), where \(\kappa \) can be expressed by \(d_{\Gamma _t}\) and \(P_{v,v}\) is defined by (3.1). So we have
Applying (3.7) and using \({\mathfrak {K}}_a\) instead of \({\mathfrak {K}}\), we obtain
where \(R_{a,0}\) is defined by
Acting \(\nabla {{\mathcal {H}}}\) on both sides of the above equation, we get the equation of \({\mathfrak {J}}_a\):
where
and recall that \(h_v\) is defined in (5.2) and comes from \(R_1\).
Meanwhile, the condition (4.18) at the contact points are rewritten as
with
Next, we consider how to recover the free surface and the domain from \(\mathfrak {K}_a\), which is slightly different from [53]. In [53], we use the equation of \({\mathfrak {N}}_a={{\mathcal {N}}}(\kappa +a \,d_{\Gamma _t}\circ \Phi _{S_t}^{-1})\) together with the boundary information \(d_i=d_{\Gamma _t}|_{p_{i*}}\) of \(d_{\Gamma _t}\) to recover the free surface \(d_{\Gamma _t}\), so the system of \((\mathfrak {K}_a, d_l, d_r)\) is needed In this paper, we use the quantity \({\mathfrak {K}}_a\) instead of \({\mathfrak {N}}_a\), where an extra \(P_{v,v}\) is added here in (6.4). Moreover, we will need to use the equation and norms of \({\mathfrak {J}}_a\) in the iteration scheme, where \({\mathfrak {K}}_a\) can be retrieved. In fact, we have \({\mathfrak {J}}_a^\perp ={\mathfrak {J}}_a\cdot n_t={{\mathcal {N}}}{\mathfrak {K}}_a\) on \(\Gamma _t\). Therefore, to identify \({\mathfrak {K}}_a\), we look at the Neumann-boundary elliptic system of \({{\mathcal {H}}}({\mathfrak {K}}_a)\) with the compatibility condition \(\int _{\Gamma _t}{\mathfrak {J}}_a^\perp {\text {d}}s=0\). We know immediately that there exists a unique solution \({{\mathcal {H}}}({\mathfrak {K}}_a)\) up to an additive constant to this system. As a result, as long as we have \({\mathfrak {J}}_a\), we obtain \(\mathfrak {K}_a\). (One can also check Lemma 2.5 in [53].)
Consequently, to recover \(d_{\Gamma _t}\) (which is the key to recover the water-waves system), we need the system of \(({\mathfrak {K}}_a, P_{v,v}, d_l, d_r)\). As long as we have proved the existence of the solution to this system, we obtain immediately the following quantity:
As a result, the the desired function \(d_{\Gamma _t}\) can be solved directly from these quantities above in a similar way as in Proposition 4.2 [53], and then we can finally recover our water-waves system \(\text{(WW) }\).
Based on the analysis above, we need to give the boundary condition of \(d_{\Gamma _t}\) which is deduce from (6.1) (for more details, see (4.27) in [53]). In fact, one has the evolution equations for \(d_i(t)=d_{\Gamma _t}(p_{i*})\) (\( i=l, r\)),
where
We rewrite the equation for \(P_{v,v}\) by (4.7),
where \(\Delta ^{-1}_N\) means solving the Neumann-boundary system (4.7) with \(\int _{\Omega _t}D_tP_{v,v}{\text {d}}X=0\), and the right-side functions are
and
As a result, we sum up the system of \(({\mathfrak {K}}_a, P_{v,v}, d_l, d_r)\) as follows:
Based on these preparations above, we are finally ready to state our precise form of Theorem 1.1. We start with introducing the space \(\Sigma \) for given \(T, L>0\):
where the norm
Meanwhile, according to the higher-order energy \(E_h(t)\), we also define the space (for given \(L_1>0\))
by the norm
The initial data is given by
Now we can present the local well-posedness theorem.
Theorem 6.1
Assume that the initial data 
and initial contact angles \(\omega _{i0}\in (0, \pi /2) \) for \(i=l, r\). When the compatibility conditions (1.8) at \(t=0\) are satisfied for \(k=0,1,2,3\), there exists a unique solution \(({\mathfrak {K}}_a, P_{v,v}, d_l, d_r)\in \Sigma _h\) to system (6.6). Moreover, system (6.6) is locally well-posed with \(({\mathfrak {K}}_a, P_{v,v}, d_l, d_r)\) depending continuously on the initial data in \(\Sigma \).
6.4 Iteration Scheme
In this subsection, we present the iteration scheme. First of all, we set the initial boundary \(S_0=S_*\) without loss of generality. To simplify the notations, we denote by
when no confusion will be made.
When we have \(({\mathfrak {K}}^{k}_a, P^{k}_{v,v}, d^{k}_l, d^{k}_r)\), the linear system of \(({\mathfrak {K}}^{k+1}_a, P^{k+1}_{v,v}, d^{k+1}_l, d^{k+1}_r)\) for the iteration scheme is set to be
Here we use the superscript k on \(R_{a, 0}\) (for example) to denote that all the quantities there are obtained using \(({\mathfrak {K}}^{k}_a, P^{k}_{v,v}, d^{k}_l, d^{k}_r)\).
Moreover, we point out that the velocity v in the definition of \(R^k_{a, 0}\) is given by
while in the other quantities we use \(v^k\) defined by (6.1)–(6.3). This happens due to the difference of regularities using these two definitions, which can be seen already in the previous sections.
Besides, the initial data is given by
As a result, a similar proof as the proof of Proposition 5.1 in [53], we show the existence of the solution \(({\mathfrak {K}}^{k+1}_a, P^{k+1}_{v,v}, d^{k+1}_l, d^{k+1}_r)\) to the linear system (6.7)–(6.8). The details for the proof are omitted.
Proposition 6.1
Let 
, \(\omega _{i0}\in (0, \pi /2)\) and \(({\mathfrak {K}}^{k}_a, P^{k}_{v,v}, d^{k}_l, d^{k}_r)\) be given correspondingly. Moreover we assume that the conditions for the corner points from (6.7) hold at \(t=0\). Then there exists a small \(T>0\) such that the system (6.7)–(6.8) has a unique solution on [0, T].
6.5 Uniform Estimates
Now we are ready to give the uniform estimates for the linear system (6.7)–(6.8). To begin with, we define the energy functional for 
as
where \(E^{k+1}_{low}(t)\) and \(E^{k+1}_{high}(t)\) are defined by
and
Moreover, the dissipation \(F^{k+1}(t)\) is
where
Meanwhile, we define some more auxiliary functions. Recalling from (5.25), here \({{\mathcal {P}}}^k_{t,1}\) and \({{\mathcal {P}}}^k_{t,2}\) are defined by
The following proposition is our main result on the uniform estimates:
Proposition 6.2
Let \(\big (\bar{\mathfrak {J}}_{a, 0}, \bar{\mathfrak {J}}_{a, 1}, {\bar{P}}_{v, v, 0}, d_{i,0}, d_{i, 1}\big )\) and \(({\mathfrak {K}}^{k}_a, P^{k}_{v,v}, d^{k}_l, d^{k}_r)\) be given as in Proposition 6.1. Then there exists constants \(T>0\) small enough and \(A>0\) large enough such that when \(a\geqq A\), the inequality below holds
Proof
Since the main steps of the proof follow Theorems 4.1 and 5.1, we only present the sketch of the proof here.
First, we consider the basic energy estimates \(E^{k+1}_l\), where we only focus on the estimates for \({\mathfrak {K}}_a\) or \({\mathfrak {J}}_a\) and the other estimates follow from Lemma 4.6 and [53].
To begin with, acting \((\Phi ^k_{S_t})^{-1}\) on both sides of (6.7), one has
which implies that
Using the same arguments as in Section 4 and Section 5, we have
Next, we define \(P_{J^{k+1}, v^k}\) on \(\Omega _t^k\) by system (3.14). Taking the \(L^2(\Omega ^k_t)\) inner product of (6.9) with \(D_t{\mathfrak {J}}^{k+1}_a +\nabla P_{J^{k+1}, v^k}\), one has
Following the energy estimates in Section 4, there exists a constant A large enough such that when \(a\geqq A\), one concludes
On the other hand, for the higher-order energy \(E_h^{k+1}(t)\), we get similarly as (5.30) the equation
where \(R^k_{a, 1}\) is given by
Compared to the energy estimate of Theorem 5.1, the main difference lies in the term \(D_t ([\nabla {{\mathcal {H}}}, D_t^2]\mathfrak {K}^{k+1}_a)\) which contains a term like \(D_t(\partial D_t v^k)\partial {{\mathcal {H}}}(\mathfrak {K}^{k+1}_a)\). In Theorem 5.1, since v satisfies Euler’s equation, we have the estimate for \(D_t^2\nabla v\). But here in the iteration scheme, \(D_t^2\partial v^k\) acts like \(\partial \partial _t^3 d_{\Gamma _t}\), which can not be controlled by the energy. Therefore, we put \(D_t ([\nabla {{\mathcal {H}}}, D_t^2]\mathfrak {K}^{k+1}_a)\) together with \(D^2_t \mathfrak {J}_a^{k+1}\) to go back to the form \(D_t^2 \mathfrak {K}^{k+1}_a\):
Taking the \(L^2(\Omega ^k_t)\) inner product of the above equation with \(\nabla {{\mathcal {H}}}(D_t^2 \mathfrak {K}^{k+1}_a) -\nabla {{\mathcal {H}}}({{\mathcal {P}}}^k_{t,2}) \), we derive
For the second integral on the left side of the above equation, we have by Green’s Formula that
and proceeding in a similar way as before, we obtain
On one hand, since the commutator \([D_t,\,{{\mathcal {N}}}]\) is already expressed in (2.5), one can conclude that
Consequently, we have
On the other hand, using similar arguments as in Lemma 5.5, we get
In the end, using similar arguments as in Section 5.3, we can show that when T small enough, the following estimate holds
Therefore, by a bootstrap argument, we can prove the desired result. \(\quad \square \)
6.6 Cauchy Sequence and Going Back to \(\text{(WW) }\)
In this part, we are finally in a position to prove that the sequence of \((\mathfrak {K}^{k}_a, P^k_{v,v}, d^{k}_l, d^{k}_r)\) is indeed a Cauchy sequence. In fact, we know from the previous subsection that
where \(C>0\) is a constant depending on E(0).
To simplify the notation, we denote by
and
Using (6.9) and rewriting it similarly as (3.15), we have the equation of \(\delta _{\overline{{\mathfrak {J}}}^{k}_a} \):
where we note
and
Besides, similar but simpler equations for \((\delta _{P^k_{v,v}}, \delta _{d^k_l}, \delta _{d^k_r})\) can be derived, and we omit the details here.
Moreover, we define the energy of the difference according to the definition of \(\Sigma \) as
Before we consider the convergence, we need to deal with \(D_{R^k}\) first.
Lemma 6.1
The right side of (6.11) satisfies the following estimate:
with the positive constant C depending on E(0).
Proof
Here, we only give the outline of the proof, and one can see similar details in [53]. First, we consider the estimate for \(\big ((\partial _t+v^k_*\cdot \nabla )^2-(\partial _t+v^{k-1}_*\cdot \nabla )^2\big )\overline{{\mathfrak {J}}}^k_a\). Using similar arguments as Proposition 4.12 and Corollary 4.13 in [53] and applying (6.7) and (6.10), we can have
Second, for the term \(({{\mathcal {A}}}(d_{\Gamma _t})-{{\mathcal {A}}}(d_{\Gamma ^{k-1}_t}))(\overline{{\mathfrak {J}}}^{k}_a-\overline{g}^k)\), we notice by (5.19) that
As a result, by similar arguments as in the proof of Proposition 6.3 [53], we have
In the end, similar arguments as in Lemma 4.6 [53], we can have
Combining all these estimates above, the proof is finished. \(\quad \square \)
Now, we are able to conclude about the convergence result.
Proposition 6.3
The sequence \((\mathfrak {K}^{k}_a, P^k_{v, v}, d^{k}_l, d^{k}_r)\) is a Cauchy sequence.
Proof
We follow the steps in Theorem 4.1 and Section 6.3 in [53] to conclude that there exists a constant T small enough and A large enough such that when \(a\geqq A\), we have
As a result, this implies immediately that \((\mathfrak {K}^{k}_a, P^k_{v, v}, d^{k}_l, d^{k}_r)\) is convergent. \(\quad \square \)
We are finally in a position to finish the proof for Theorem 6.1.
Proof of Theorem 6.1
We only present the sketch for the proof here. In fact, since we have proved in Proposition 6.3 that \((\mathfrak {K}^{k}_a, P^k_{v, v}, d^{k}_l, d^{k}_r)\in \Sigma \) is a Cauchy sequence, we know immediately that there exists \((\mathfrak {K}_a, P_{v, v}, d_l, d_r)\in \Sigma \) satisfying
As a result, one can show in a standard way that \((\mathfrak {K}_a, P_{v, v}, d_l, d_r)\) satisfies system (6.6). Moreover, one also has \((\mathfrak {K}_a, P_{v, v}, d_l, d_r)\in \Sigma _h\) in the proof of Proposition 6.3. \(\quad \square \)
In the end, we go back to our water-waves system \(\text{(WW) }\). In fact, thanks to system (6.6) for \((\mathfrak {K}_a, P_{v, v}, d_l, d_r)\in \Sigma \), we derive the mean curvature \(\kappa \), which defines the free surface. Based on the knowledge of \(\Gamma _t\), we also obtain v by Section 6.2. Therefore, using similar arguments as in Section 6.4 [53] and thanks to discussions in Section 6.3, we can finally retrieve the solution (v, P) to the water-waves system \(\text{(WW) }\).
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Acknowledgements
The authors would like to thank Chongchun Zeng for very fruitful discussions. The author Mei Ming is supported by NSFC no.12071415. The author Chao Wang is supported by NSFC No. 12071008.
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Ming, M., Wang, C. Local Well-Posedness of the Capillary-Gravity Water Waves with Acute Contact Angles. Arch Rational Mech Anal 248, 72 (2024). https://doi.org/10.1007/s00205-024-02019-2
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DOI: https://doi.org/10.1007/s00205-024-02019-2