本文選題：Naiver-Stokes方程　切入點(diǎn)：連續(xù)唯一性　出處：《華中科技大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：唯一性問題一直是偏微分方程研究中的重要問題.本文主要研究了兩類方程組的唯一性問題.第一部分研究Navier-Stokes方程組的解在Rn×[0,L]上的連續(xù)唯一性問題；第二部分則對由Maxwell方程衍化而來的Schrodinger方程組在有界區(qū)域DN映射下的系數(shù)唯一性做了論證.對Navier-Stokes方程解的數(shù)學(xué)理論研究一直是數(shù)學(xué)界和物理界的熱點(diǎn)問題.本文為簡化方程,特別地引入Navier-Stokes方程解的旋度ρ,通過變換將原方程約化為關(guān)于旋度ρ的方程.再對簡化后的方程選擇適當(dāng)?shù)臋?quán)函數(shù),在Rn×[0,L]上對ρ進(jìn)行帶權(quán)函數(shù)的估計(jì).在估計(jì)過程中,利用了反向唯一性方法克服了區(qū)域帶來的估計(jì)困難,最終得到ρ的加權(quán)函數(shù)估計(jì),進(jìn)而得到ρ在Rn×[0,L]上的連續(xù)唯一性,再結(jié)合已知引理最終得到Navier-Stokes方程的解在Rn×[0,L]上的連續(xù)唯一性.接下來討論Schrodinger方程組在有界區(qū)域DN映射下的系數(shù)唯一性問題,即為逆邊值問題.逆邊值問題來源于電阻抗成像技術(shù)的逆問題,將電阻抗成像技術(shù)的逆問題轉(zhuǎn)化為對Schrodinger方程的逆邊值問題的研究時(shí),對邊界的選取是此問題的重點(diǎn).本文首先將二階Schrodinger方程組轉(zhuǎn)化為四階方程,利用位勢有界的四階Schrodinger方程的Carleman估計(jì)和復(fù)幾何光學(xué)解的構(gòu)造,得到關(guān)于位勢q的積分恒等式.然后選取合適的實(shí)解析振蕩函數(shù),應(yīng)用微局部分析的理論,最終得到其系數(shù)的唯一性.
[Abstract]:The uniqueness problem has always been an important problem in the study of partial differential equations. In this paper, the uniqueness of two kinds of equations is studied. The first part is the continuous uniqueness of the solutions of Navier-Stokes equations on rn 脳 [0L]. In the second part, the uniqueness of the coefficients of the Schrodinger equations derived from the Maxwell equation is proved under the bounded domain DN mapping. The mathematical theory of the solution of the Navier-Stokes equation has always been a hot issue in the field of mathematics and physics. In particular, the curl 蟻 of the solution of the Navier-Stokes equation is introduced, and the original equation is reduced to the equation of the curl 蟻 by transformation. Then the proper weight function is chosen for the simplified equation, and the weighted function is estimated on the rn 脳 [0L]. In this paper, the inverse uniqueness method is used to overcome the difficulty of estimating the region, and finally the weighted function estimation of 蟻 is obtained, and then the continuous uniqueness of 蟻 on R n 脳 [0L] is obtained. Finally, combining the known Lemma, we obtain the continuous uniqueness of the solution of the Navier-Stokes equation on rn 脳 [0L]. Then we discuss the uniqueness of the coefficients of the Schrodinger equations under the bounded domain DN mapping. The inverse boundary value problem originates from the inverse problem of the electrical impedance imaging technology. The inverse problem of the electrical impedance imaging technique is transformed into the inverse boundary value problem of the Schrodinger equation. In this paper, the second order Schrodinger equations are first transformed into fourth-order equations, and the Carleman estimates of the potential bounded fourth-order Schrodinger equations and the construction of complex geometric optical solutions are used. The integral identity about potential Q is obtained, and then the proper real analytical oscillatory function is selected, and the uniqueness of its coefficient is obtained by applying the theory of differential local analysis.
【學(xué)位授予單位】：華中科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175.2

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