A simplified version of the abstract Cauchy-Kowalewski theorem with weak singularities
A simplified version of the abstract Cauchy-Kowalewski theorem with weak singularities
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具有弱奇点的抽象 Cauchy-Kowalewski 定理的简化版本
DOI：
10.1090/s0273-0979-1990-15962-2
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发表时间：
1990
影响因子：
1.3
通讯作者：
R. Caflisch
中科院分区：
文献类型：
--
作者：
R. Caflisch
给出了Nirenberg[9]、Ovsjannikov[12]、Nishida[10]、Baouendi和Goulaouic[3]和Asano[1]的抽象Cauchy-Kowalewski定理的简化版。新版本需要关于方程形式的更具体的信息，但恢复了一个更强的结果，因为存在区域在迭代的每一步都不会被迫缩小，并且存在区域的边界允许弱奇点。Cauchy-Kowalewski定理是偏微分方程解析解的基本存在性定理，其抽象形式[1,3,9,10,12]可以应用于涉及非局部算子的方程，如水波方程[8]、流体动力极限[11]的玻尔兹曼方程、零粘度极限[2]的不可压缩流体方程和涡片方程[4,6,13]。摘要[1,3,9,10,12]中的Cauchy-Kowalewski定理的证明是“Nash-Moser型”的，它要求在迭代的每一步都损失存在域的大小（但不使用牛顿迭代）。本文的目的是提出一个不使用“NashMoser型”证明的抽象Cauchy-Kowalewski定理的新版本。存在域在迭代的每一步都不会缩小，并且在存在域的边界处允许存在轻微的奇点。这个新版本的主要意义在于，更简单的证明使其更适合新的应用，并且对存在区域的更多控制允许解决具有奇点的问题。新版本的定理确实需要更多关于方程结构的信息。然而，这些新的假设是在1989年5月30日和1989年7月12日被编辑们收到的。1980年数学学科分类（1985年修订）。主要35 a10。由空军科学研究办公室在URI拨款860352下支持的研究。©1990美国数学学会0273-0979/90 $1.00+ $。许可证或版权限制可能适用于再分发；参见https://www.ams.org/journal-terms-of-use 496 RUSSEL E. CAFLISCH的定理对大多数应用都是有效的。对于[4- 6,8,13]中的应用，假设很容易验证；对[2,11]中更复杂的应用程序的验证尚未完成。这个新版本的抽象CauchyKowalewski定理在[7]中用于构造具有奇点的涡片解。结果在Banach空间B族中表示，当0 < P < Po>，范数||•||，使得Bp， c Bp和||w|| < ||w|| /对于0 < P < P < pQ。考虑问题(1)ut = A(u,t), (2) w(0) = 0，其中u = u(t)，作为空间B族中的微分方程。定理的陈述也使用了一个线性算子D，它连续地将B映射到B，对于任意0 < p < p < p0。另外，假设0 < /？< 1和定义规范(3)M u | | | =一口{\ \ u (t) \ \ p + (p0-p-t) \ \ Du p (t) \ \}。0<p<p0 0<t<p -p假设A满足以下条件：存在正常数e和R，使得(i) A（', t）映射BpC\{u: \\u\\ <R，}到B，连续在u和t中，对于0<p<p < pQ t。（ii）如果w， w, Du, DweBp，与\\u\ p，<R, \\w\ p,<R，则\\A(u,t)-A(w,t)\\p {<e{(\ + \\Du, Dw\ p)\\u w\\p + \\D(\w w)\\p}, (5) \\DA(u9t)-DA(w,t)\\p <e (p Py {(l + \\Du， Z)ti；| | pOII«^ 11 / + \ \ D (u w) \ \ p} 9 0 < p < p < pQ t。(3)为0 < p < p0t,(6)华盛顿大学,t) \ \ p < e {pz-p-t) 9 (7) \ \ DA (Q, t) \ \ < e (p0-p-t)。许可或版权限制可能适用于再分发；CAUCHY-KOWALEWSKI定理497这些假设有如下解释：算子D通常为D /dz。然后(4)表示A是u乘以uz的函数，(5)是A导数的柯西估计。函数空间B将由解析函数组成，这些函数在{z: | Imz \ < p}中有界，范数如||w|| = sup| Imz, \u\（或下面（25）中的相关范数）。最后(6)和(7)允许A在| Imz| = pQ t上有一个奇点。我们也假设e很小，通过对t或u进行缩放，就等价于短时间存在或对于小的非线性。为节省空间，已使用“||/,g i | = ||/|| + ||g||”。定理。对于任意正R和p0和任意0 < /？< 1存在数e0 > 0和0< a < R，使得对于0< e < e0和对于a满足假设(i)-(iii)，系统(1),(2)有解u(t) G B对于0< p < p0 t且\\\ \\\ \\\ \\\ t< a。换句话说，在B中存在解u(t)且\\u(t)\\<a,(8)对于0<t< p0证明。解决(1),
A simplified version of the abstract Cauchy-Kowalewski theorem of Nirenberg [9], Ovsjannikov [12], Nishida [10], Baouendi and Goulaouic [3], and Asano [1] is presented. The new version requires more specific information on the form of the equation but recovers a stronger result in that the region of existence is not forced to shrink at each step of an iteration and that weak singularities are allowed along the boundary of the region of existence. The Cauchy-Kowalewski theorem is the basic existence theorem for analytic solutions of partial differential equations and in its abstract form [1, 3, 9, 10, 12] can be applied to equations that involve nonlocal operators, such as the water wave equations [8], the Boltzmann equation in the fluid dynamic limit [11], the incompressible fluid equations in the zero-viscosity limit [2] and the vortex sheet equations [4-6, 13]. The proof of the abstract Cauchy-Kowalewski theorem in [1, 3, 9, 10, 12] is of "Nash-Moser type" in that it requires a loss in the size of the existence region at each step of an iteration (but without use of Newton iteration). The purpose of this paper is to present a new version of the abstract Cauchy-Kowalewski theorem that does not use a "NashMoser type" proof. The domain of existence does not shrink at each step of the iteration, and in addition mild singularities are allowed at the boundary of the existence region. The main significance of this new version is that the simpler proof makes it more adaptable to new applications and that more control over the region of existence allows solution of problems with singularities. The new version of the theorem does require more information on the structure of the equation. However these new hypotheReceived by the editors May 30, 1989 and, in revised form, July 12, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 35A10. Research supported in party by the Air Force office of Scientific Research under URI grant 860352. © 1990 American Mathematical Society 0273-0979/90 $1.00+ $.25 per page 495 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 496 RUSSEL E. CAFLISCH ses are valid for most of the applications of the theorem. For the applications in [4-6, 8, 13] the hypotheses are easily verified; verification for the more complicated applications in [2, 11] has not yet been completed. This new version of the abstract CauchyKowalewski theorem is used in [7] for construction of vortex sheet solutions with singularities. The result is stated in a family of Banach spaces B for 0 < P < Po> with norm || • || such that Bp, c Bp and ||w|| < ||w|| / for 0 < p < p < pQ . Consider the problem (1) ut = A(u,t), (2) w(0) = 0, with u = u(t), as a differential equation in the family of spaces B . The statement of the theorem also uses a linear operator D which maps B , continuously into B for any 0 < p < p < p0 . In addition suppose that 0 < /? < 1 and define the norm (3) M u | | |= sup {\\u(t)\\p + (p0-p-t) \\Du(t)\\p}. 0<p<p0 0<t<p0-p Assume that A satisfies the following conditions: There are positive constants e and R such that (i) A(', t) maps BpC\{u: \\u\\ < R, } into B , continuously in u and t, for 0 < p < p < pQ t. (ii) If w, w, Du, DweBp, with \\u\\p,<R, \\w\\p,<R, then \\A(u,t)-A(w,t)\\p {) <e{(\ + \\Du, Dw\\p)\\u w\\p + \\D(u w)\\p}, (5) \\DA(u9t)-DA(w,t)\\p < e(p Py {(l + \\Du, Z)ti;||pOII« ^11/ + \\D(u w)\\p,}9 for 0 < p < p < pQ t. (iii) For 0 < p < p0t, (6) UW,t)\\p<e{pz-p-t)p 9 (7) \\DA(Q,t)\\ <e(p0-p-t)-. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SIMPLIFIED ABSTRACT CAUCHY-KOWALEWSKI THEOREM 497 These hypotheses have the following interpretation: The operator D will usually be d/dz. Then (4) says that A acts like a function of u times uz, and (5) is the Cauchy estimate on the derivative of A . The function space B will consist of functions that are analytic and bounded in {z : | Im z\ < p} with a norm like ||w|| = sup| Imz, \u\ (or the related norm in (25) below). Finally (6) and (7) allow A to have a singularity on | Imz| = pQ t. It will also be assumed that e is small, which, by rescaling of t or u, is equivalent to existence for a short time or for small nonlinearity. The notation | | / , g i | = ||/|| + ||g|| has been used to save space. Theorem. For any positive R and p0 and any 0 < /? < 1 there are numbers e0 > 0 and 0 < a < R, such that for 0 < e < e0 and for A satisfying assumptions (i)-(iii), the system (1), (2) has a solution u(t) G B for 0 < p < p0 t with \\\ u \\\ < a. In other words there is a solution u(t) in B with \\u(t)\\<a, (8) for 0<t< p0Proof. Solve (1),