1. Introduction
In this work we consider the Cauchy problem for the tsunami wave equation governed by the shallow water equations. Namely, for , we study the Cauchy problem
(1.1) 
where
is a vector valued function. Our model is a general case of a well known physical model when
is real valued. In this particular case, denotes the water depth and represents the free surface displacement. Let us start by the description of the physical motivation.Tsunamis are a series of traveling waves in water induced by the displacement of the sea floor due to earthquakes or landslides. Three stages of tsunami development are usually distinguished: the generation phase, the propagation of the waves in the open ocean (or sea) and the propagation near the shoreline. Since the wavelengths of tsunamis are much greater than the water depth, they are often modelled using the shallow water equations. The most common model used to describe tsunamis (see, for instance [Kun07], [DD07], [Ren17], [RS08], [RS10], [DDORS14], [ADD19] and the references therein) is
(1.2) 
where is a source term related to the formation of a localized disturbance in the first stage of the tsunami life. When analysing the system at the final stages, that is for , the source term can be neglected and a homogeneous equation can be considered instead:
(1.3) 
where the initial free surface displacement and the initial velocity can be described by known functions of the spacial variable, i.e.
(1.4) 
In the present paper, we are interested in the final stages of the tsunami development. So, we consider the latter model, and for the sake of simplicity we take as the initial time instead of . That is, we consider
(1.5) 
where we allow the water depth coefficient to be discontinuous or even to have less regularity. The singularity of can be interpreted as sudden changes in the water depth caused by the interaction of the wave with complicated topographies of the sea floor such as bays and harbours.
While from a physical point of view this is a natural setting, mathematically we face a problem: If we are looking for distributional solutions, the term does not make sense in view of Schwartz famous impossibility result about multiplication of distributions [Sch54]. In this context the concept of very weak solutions was introduced in [GR15], for the analysis of second order hyperbolic equations with irregular coefficients and was further applied in a series of papers [ART19], [RT17a] and [RT17b] for different physical models, in order to show a wide applicability. In [MRT19, SW20] it was applied for a damped wave equation with irregular dissipation arising from acoustic problems and an interesting phenomenon of the reflection of the original propagating wave was numerically observed. In all these papers the theory of very weak solutions is dealt for timedependent equations. In the recent works [Gar20, ARST20a, ARST20b, ARST20c], the authors start to study the concept of very weak solutions for partial differential equations with spacedepending coefficients.
It is shown there, that this notion is very well adapted for numerical simulations when a rigorous mathematical formulation of the problem is difficult in the framework of the classical theory of distributions. Furthermore, by the theory of very weak solutions we can talk about uniqueness of numerical solutions to differential equations. So, here we consider the Cauchy problem (1.5) and prove that it has a very weak solution.
Moreover, since numerical solutions are useful for predicting and understanding tsunami propagation, many numerical models are developed in the literature, we cite fore instance [Behr10, LGB11, RHH11, BD15]. As a second task in the present paper we do some numerical computations, where we observe interesting behaviours of solutions.
2. Main results
For , we consider the Cauchy problem
(2.1) 
where ; is singular and positive in the sense that there exists such that for all we have . The following lemma is a key of the proof of existence, uniqueness and consistency of a very weak solution to our model problem. It is stated in the case when is a regular vectorfunction.
In what follows we will use the following notations. By writing for functions and , we mean that there exists a positive constant such that . Also, we denote
In addition, we introduce the Sobolev space by
Theorem 2.1.
Let be positive. Assume that and . Then, the unique solution to the Cauchy problem (2.1
), satisfies the estimates
(2.2) 
for all .
In addition, assume that , and . Moreover, if for all . Then, the solution satisfies the estimate
(2.3) 
for all , where .
Proof.
We multiply the equation in (2.1) by and we integrate with respect to the variable , to obtain
(2.4) 
where denotes the inner product of the Hilbert space and is the imaginary unit, such that . After short calculations, we easily show that
(2.5) 
and
(2.6) 
Then, from (2.4), we get the energy conservation formula
(2.7) 
By taking in consideration that can be estimated by
(2.8) 
for all , it follows that
(2.9) 
and
(2.10) 
for all .
In the last inequality, using that the left hand side can be estimated by
(2.11) 
and that is positive, we get for all the estimate
(2.12) 
Let us estimate . By the fundamental theorem of calculus we have that
(2.13) 
Taking the norm in (2.13) and using (2.9) to estimate , we arrive at
(2.14) 
Now, let us assume that , and . We note that, if solves the Cauchy problem
(2.15) 
then solves
(2.16) 
Then, using the estimates (2.9) and (2.10), we get
(2.17) 
(2.18) 
where for all , we estimated by
(2.19) 
To get the estimate (2.3), we need the following result.
Lemma 2.2.
Proof.
Using the assumption that are bounded from below, that is,
for all , we get
It proves the lemma. ∎
2.1. Existence of a very weak solution
In what follows, we consider the Cauchy problem
(2.22) 
with singular coefficients and initial data. Now we want to prove that it has a very weak solution. To start with, we regularise the coefficients and the Cauchy data and by convolution with a suitable mollifier , generating families of smooth functions , and , that is
(2.23) 
and
(2.24) 
where
(2.25) 
The function is a Friedrichsmollifier, i.e. , and .
Assumption 2.3.
In order to prove the well posedness of the Cauchy problem (2.22) in the very weak sense, we ask for the regularisations of the coefficients and the Cauchy data , to satisfy the assumptions that there exist such that
(2.26) 
for and
(2.27) 
Remark 2.1.
We note that making an assumption on the regularisation is more general than making it on the function itself. We also mention that such assumptions on distributional coefficients, are natural. Indeed, we know that for we can find and functions such that, . The convolution of with a mollifier gives
(2.28) 
and we easily see that the regularisation of satisfy the above assumption. Fore more details, we refer to the structure theorems for distributions (see, e.g. [FJ98]).
Definition 1 (Moderateness).

A net of functions , is said to be moderate, if there exist such that

A net of functions , is said to be moderate, if there exist such that

A net of functions , is said to be moderate, if there exist such that

A net of functions from is said to be moderate, if there exist such that
for all .
We note that if for and , then the regularisations for of the coefficients and , of the Cauchy data, are moderate in the sense of the last definition.
Definition 2 (Very weak solution).
The net is said to be a very weak solution to the Cauchy problem (2.22), if there exist

moderate regularisations of the coefficients , for ,

moderate regularisation of ,

moderate regularisation of ,
such that solves the regularised problem
(2.29) 
for all , and is moderate.
Theorem 2.4 (Existence).
Proof.
The nets , for and , are moderate by assumption. To prove the existence of a very weak solution, it remains to prove that the net , solution to the regularised Cauchy problem (2.29), is moderate. Using the estimates (2.2), (2.3) and the moderateness assumptions (2.26) and (2.27), we arrive at
for all . This concludes the proof. ∎
In the next sections, we want to prove uniqueness of the very weak solution to the Cauchy problem (2.22) and its consistency with the classical solution when the latter exists.
2.2. Uniqueness
Let us assume that we are in the case when very weak solutions to the Cauchy problem (2.22) exist.
Definition 3 (Uniqueness).
We say that the Cauchy problem (2.22), has a unique very weak solution, if for all families of regularisations , , , and , of the coefficients , for and the Cauchy data , , satisfying
and
we have
for all , where and are the families of solutions to the related regularised Cauchy problems.
Theorem 2.5 (Uniqueness).
Proof.
Let , be regularisations of the coefficients , for and the Cauchy data , , and let assume that they satisfy
and
Let us denote by , where and are the solutions to the families of regularised Cauchy problems, related to the families and . Easy calculations show that solves the Cauchy problem
(2.30) 
where
(2.31) 
By Duhamel’s principle (see, e.g. [ER18]), we obtain the following representation
(2.32) 
for , where is the solution to the homogeneous problem
(2.33) 
and solves
(2.34) 
Taking the norm on both sides in (2.32) and using (2.2) to estimate and , we obtain
(2.35) 
Let us estimate . We have
In the last inequality, we used the product rule for derivatives and the fact that and can be estimated by and , respectively. We have by assumption that for all , the net is moderate. The net is also moderate as a very weak solution. Thus, there exists such that
(2.36) 
(2.37) 
On the other hand, we have that
and
Comments
There are no comments yet.