Regularity of global solution to atmospheric circulation equations with humidity effect
© Luo; licensee Springer 2012
Received: 1 June 2012
Accepted: 14 November 2012
Published: 5 December 2012
In this article, the regularity of the global solutions to atmospheric circulation equations with humidity effect is considered. Firstly, the formula of the global solutions is obtained by using the theory of linear operator semigroups. Secondly, the regularity of the global solutions to atmospheric circulation equations is presented by using mathematical induction and regularity estimates for the linear semigroups.
MSC:35D35, 35K20, 35Q35.
Keywordsglobal solution regularity atmospheric circulation equations humidity effect
Partial differential equations (1.1)-(1.7) are presented in atmospheric circulation with humidity effect . Atmospheric circulation is one of the main factors affecting the global climate so it is very necessary to understand and master its mysteries and laws. Atmospheric circulation is an important mechanism to complete the transports and balance of atmospheric heat and moisture and the conversion between various energies. Moreover, it is also the important result of these physical transports, balance and conversion. Thus, it is of necessity to study the characteristics, formation, preservation, change and effects of the atmospheric circulation and master its evolution law, which is not only the essential part of human’s understanding of nature, but also a helpful method of changing and improving the accuracy of weather forecasts, exploring global climate change, and making effective use of climate resources.
The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system. The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, and a lot of them cannot be produced by laboratory experiments. The atmosphere or the ocean or the couple atmosphere and ocean can be viewed as initial and boundary value problems [2–5], or an infinite dimensional dynamical system [6–8]. We deduce atmospheric circulation models which are able to show the features of atmospheric circulation and are easy to be studied from the very complex atmospheric circulation model based on the actual background and meteorological data, and we present global solutions of atmospheric circulation equations with the use of the T-weakly continuous operator . In , the steady state solutions to atmospheric circulation equations with humidity effect are studied. A sufficient condition of the existence of the steady state solutions to atmospheric circulation equations is obtained, and the regularity of the steady state solutions is verified. In this article, we investigate the regularity of the solutions to atmospheric circulation equations (1.1)-(1.7).
The paper is organized as follows. In Section 2, we present preliminary results. In Section 3, we present the formula of the solution to the atmospheric circulation equations. In Section 4, we obtain the regularity of the solutions to equations (1.1)-(1.7).
denotes the norm of the space X, and C, are variable constants.
Lemma 2.1  (Theory of linear elliptic equations)
where depends on n, p, λ, Ω and -norm or -norm of the coefficient functions.
where depends on μ, n, k, α, Ω.
where , is unknown.
Lemma 2.6 
Assume is T-weakly continuous and satisfies:
where , are constants, (), , is a seminorm of , ,
where depends only on T, , , and .
If is Frechét differentiable, then the regular solution can be presented under some condition.
for all .
Lemma 2.8 
where is the domain of . By the semigroup theory of linear operators (Pazy ), we know that is a compact inclusion for any .
For sectorial operators, we also have the following properties which can be found in .
is bounded for all and ,
- (3)for each , is bounded and
- (4)the -norm can be defined by(2.9)
- (5)if ℒ is symmetric, for any , we have
3 Formula of global solutions
where is an analytic semigroup generated by L, and is a Leray projection.
Therefore, L generates the analytic semigroup .
4 Regularity of global solution
for all .
for any and . Then (2.7) holds.
Proof We prove the theorem using mathematical induction.
If , , , then , . Using Theorem 4.1, we find that .
Then and .
Thus, . Then in Eq. (4.5). By the ADN theory, . Thus, and .
which implies . Then in Eq. (4.5). Using the ADN theory, and . Thus, and . Then and from the formula (3.1).
Then and .
which implies . Then in Eq. (4.8). It follows from the linear elliptic equation . Thus, and .
which implies . We obtain that in Eq. (4.8). Then from the theory of linear elliptic equations. Thus, . From the formula (3.1), and .
Then and .
which implies . Then in Eq. (4.11). Thus, from the theory of linear elliptic equations. Then and .
which implies . We see in Eq. (4.11). Then from the theory of linear elliptic equations. Thus, . We have and from the formula (3.1).
If , , and , then and . From the hypothesis of mathematical induction, we see .
Then and .
Then and .
Then and .
The proof is completed. □
Since the differentiability of time and of space can be transformed into each other, we obtain
for , where l, r, α, β are positive integers satisfying and .
The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The project is supported by the National Natural Science Foundation of China (11271271), the NSF of Sichuan Science and Technology Department of China (2010JY0057) and the NSF of Sichuan Education Department of China (11ZA102).
- Luo, H: Global solution of atmospheric circulation equations with humidity effect (submitted)
- Ma T, Wang SH: Phase Transition Dynamics in Nonlinear Sciences. Springer, New York; 2012.
- Ma T: Theories and Methods in Partial Differential Equations. Science Press, China; 2011. in Chinese
- Phillips NA: The general circulation of the atmosphere: a numerical experiment. Q. J. R. Meteorol. Soc. 1956, 82: 123-164. 10.1002/qj.49708235202View Article
- Rossby CG: On the solution of problems of atmospheric motion by means of model experiment. Mon. Weather Rev. 1926, 54: 237-240. 10.1175/1520-0493(1926)54<237:OTSOPO>2.0.CO;2View Article
- Lions JL, Temam R, Wang SH: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 1992, 5(2):237-288. 10.1088/0951-7715/5/2/001MathSciNetView Article
- Lions JL, Temam R, Wang SH: On the equations of the large-scale ocean. Nonlinearity 1992, 5(5):1007-1053. 10.1088/0951-7715/5/5/002MathSciNetView Article
- Lions JL, Temam R, Wang SH: Models for the coupled atmosphere and ocean. (CAO I,II). Comput. Mech. Adv. 1993, 1(1):5-54.MathSciNet
- Luo H: Steady state solution to atmospheric circulation equations with humidity effect. J. Appl. Math. 2012. doi:10.1155/2012/867310
- Evens LC: Partial Differential Equations. Am. Math. Soc., Providence; 1998.
- Temam R CBMS-NSF Regional Conference Series in Applied Mathematics. In Navier-Stokes Equation and Nonlinear Functional Analysis. SIAM, Philadelphia; 1983.
- Temam R: Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam; 1979.
- Pazy A Appl. Math. Sci. 44. In Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin; 2006.
- Ma T, Wang SH: Stability and Bifurcation of Nonlinear Evolution Equations. Science Press, China; 2007. in Chinese
- Ma T, Wang SH Nonlinear Science Ser. A 53. In Bifurcation Theory and Applications. World Scientific, Singapore; 2005.
- Temam R Applied Mathematical Science 68. In Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd edition. Springer, New York; 1997.View Article
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.