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Nonhomogeneous boundary value problem for similar solutions of incompressible two-dimensional Euler equations
© Song et al.; licensee Springer. 2014
Received: 11 February 2014
Accepted: 27 June 2014
Published: 1 August 2014
In this paper we introduce the similar method for incompressible two-dimensional Euler equations, and obtain a series of explicit similar solutions to the incompressible two-dimensional Euler equations. These solutions include all of the twin wave solutions, some new singularity solutions, and some global smooth solutions with a finite energy. We also reveal that the twin wave solution and an affine solution to the two-dimensional incompressible Euler equations are, respectively, a plane wave and constant vector. We prove that the initial boundary value problem of the incompressible two-dimensional Euler equations admits a unique solution and discuss the stability of the solution. Finally, we supply some explicit piecewise smooth solutions to the incompressible three-dimensional Euler case and an example of the incompressible three-dimensional Navier-Stokes equations which indicates that the viscosity limit of a solution to the Navier-Stokes equations does not need to be a solution to the Euler equations.
MSC:35Q30, 76D05, 76D10.
where ; and denote the velocity and pressure, respectively. Though there is a large amount of physics and mathematics literature on the Euler and Navier-Stokes equations, many basic questions remain open.
Is an exact solution of the Euler equations explicitly given via solving the vortex equations (the weak Lax pair) to the Euler equations? Since the Lax pair has still only weak meaning, one cannot get the solutions to the Euler equations from those solutions of vortex equations by the Biot-Savart law. Thus whether the integrable two-dimensional Euler equations in some stronger sense are similar to those of the three-dimensional Euler equations is still an open question. In this paper we find a so-called similar method which can give some explicit smooth solutions to two-dimensional incompressible Euler equations (see Section 2). As applications of the similar method, a large amount of explicit twin wave solutions are constructed in Section 3.
There are various open problems in mathematics, such as: how to establish the global existence of smooth solutions, and how to establish the blow-up solution at least when the space dimension equals three (see ), and so on. The study of the incompressible Navier-Stokes equations has a long history. A deeper result on the weak solution was obtained by Caffarelli et al. in . On the blow-up problem of the incompressible Navier-Stokes equations, Tsai in  proved that the Leray self-similar solutions to (1.1) must be zero if they satisfy local energy estimates. So in Section 4 we discuss the method of determining the nonexistence of a non-constant affine solution to the two-dimensional Euler equations, which we can correctly obtain due to the similar method.
The blow-up problem of the compressible Navier-Stokes equation has been established by Xin (see ). He proved that any smooth solution to the multidimensional Navier-Stokes equation for polytropic fluids in the absence of heat conduction will blow up in finite time if the initial density is compactly supported (see ).
In Section 5, we prove that incompressible two-dimensional Euler equations under a class of initial boundary values has a unique solution for every bounded domain , and we discuss the stability of solutions in Section 6.
Since it is very hard to solve the Navier-Stokes equations in a three-dimensional space, we consider the two equations in the half space case. In Section 7, we construct some explicit smooth solutions to the incompressible three-dimensional Euler and Navier-Stokes equations and an example of the three-dimensional Navier-Stokes equations which indicates that a solution to the Navier-Stokes equations does not need to tend to a solution to the Euler equations in the continuous function space on the half space.
2 similar method in solving the Euler equations
We first have the following definition.
where and are smooth functions on , is a n-dimensional smooth vector function independent of t, and is a piecewise smooth vector function from to .
where , c is an arbitrary smooth function of t, h is an arbitrary smooth function of r.
Thus we have the following result.
Remark 2.3 To the best of our knowledge, there is little known of exact solutions to vortex equations, but they are not solutions to the two-dimensional Euler equations (2.6) except for the zero solution and they did not bring about any solution to the two-dimensional Euler equations (2.6) by the Biot-Savart law as they have a singularity (see [9, 10]), as seen by using VIM (see ), and by a Bäcklund transformation (see ) method. Notice that (2.5) correctly is a family of exact solutions to the two-dimensional Euler equations (2.6).
Remark 2.4 It is interesting to get many properties by choosing , .
is also a solution pair for any constant vectors . , is also a solution pair for any rotation matrices Q.
Then , however, for every and , we have but , where .
Then u is singular at and blows up at .
3 Twin wave solutions
In this section we give more explicit nonzero solutions by considering explicit twin wave solutions to the two-dimensional Euler equations. Here a twin wave solution has the form of . The twin wave solution is a similar solution. In fact, if we take , , , , , , , , , then . Inserting them into (2.6), we have the following theorem.
Theorem 3.1 If the pressure is independent of x, all twin wave solutions to the two-dimensional Euler equations will be given by , where v is any function of , and , , are arbitrary constants.
If , this may be interpreted as the equations in , .
where v is any function of , and , , are arbitrary constants. □
These are a global smooth twin wave solutions pair for any constant vectors . , are also a twin wave solutions pair for any rotation matrices Q.
Remark 3.3 These solutions in (3.8) are symmetric only in some domains. In particular, if , they are symmetry solutions for all , and if they are not symmetric for all and symmetric only at . These examples show that the difference between the velocity of flow and its wave speed has a finite energy over , i.e. .
are some twin wave solutions to (2.6), and they form a symmetry only at , or they are static.
are global smooth twin wave solutions to the Euler equations (2.6) with finite energy in any bounded domain, but with infinite energy over except for the static case. If the components of the wave speed are equal, then the system is static.
are some twin wave solutions with singularity to the Euler equations (2.6).
Remark 3.7 These solutions in (3.11) have a singularity on the line for every . In particular, we have the following result:
For every given time and arbitrary line , there exist some solutions with singularity over the line .
Example 3.8 According to , , is also a solution pair for any constant vectors . , , is also a solution pair for any rotation matrices Q. , is also a solution pair.
In this section we consider the explicit affine solution to the two-dimensional Euler equations. Here a solution is called an affine solution, if the is denoted by , . The affine solution indeed is a similar solution. In fact, this is the case: , , , , , , , , , , . We have the following result.
Theorem 4.1 All affine solutions must be twin wave solutions. Affine solutions to the two-dimensional Euler equations are constant vectors. That is to say there does not exist a non-constant affine solution to the two-dimensional Euler equations.
and we have the following result.
has a unique smooth solution , .
where n stands for the outward unit normal to Ω.
Therefore there exists a unique solution in the sense of , . The denseness of in implies the uniqueness of the solution in the sense of from . We can apply the same argument on the intervals , , etc., according to the uniform Gronwall lemma since . We obtain the uniqueness of the solution.
Thus we prove the uniqueness of the solution.
It clarifies that the uniqueness of the solution is possible even as if the right scope is chosen in Ω. □
6 Analysis of stability between the equations and its vortex equation
In this section we discuss the stability of the solution, respectively, in and for the problem (5.1).
where n stands for the outward unit normal to Ω.
as in the sense of and . So we reach the stability of the solution in finite time.
7 Explicit solution to -dimensional Navier-Stokes equation
We now give an improvement of the example in reference .
for some nonnegative integer numbers m and positive real numbers q, M.
Remark 7.2 This example indicates that the solution of the Navier-Stokes equation does not always tend to a solution of the Euler equation.
The third author is grateful to Professor Boling Guo and Professor Zhouping Xin for their support, and he also thanks Dr Jihui Wu for help in writing this paper. The work is supported by National Natural Science Foundation of China (No. 11161057).
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