Susumu Tanabé
We give an algebraic description of wave fronts that appear in strictly hyperbolic Cauchy problem. Concrete form of definig function of wave front issued from initial algebraic variety is obtained by the aid of the Gauss-Manin systems satisfied by Gel'fand -Leray integrals.
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We prove a general result that implies that very weak solutions to the Cauchy problem for the Navier-Stokes equations must be, in fact, Leray-Hopf solutions if only their initial data are (solenoidal) with finite kinetic energy.
We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-Δ)^α$ with exponent $α< \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and László Székelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.
In \cite{CJ}, the authors show that the Cauchy problem of the Navier-Stokes equations with damping $α|u|^{β-1}u(α>0,\;β\geq1)$ has global weak solutions in $L^2(\R^3)$. In this paper, we prove the uniqueness, the continuity in $L^2$ for $β>3$, also the large time decay is proved for $β\geq\frac{10}3$. Fourier analysis and standard techniques are used.
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray $\cite{jL34}$ showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer and Shnirelman obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about the behavior of solutio…
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of solutions for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer and Shnirelman obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli, Kohn and Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about behavior of weak s…
We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak solutions in the physical energy space. Our method relies on a new $\textit{a priori}$ estimate on the space-time norm in $L^{\f32}_{loc}$ of the Cauchy-Green strain tensor $τ=\F\F^\top$, or equivalently the $L^3_{loc}$ norm of the Jacobian of the flow map $\F$. It allows us to rule out possible concentrations of the energy due to deformations associated with the flow maps. Following the general compactness arguments d…
Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise
本文证明了受Stratonovich输运噪声扰动的三维Navier-Stokes方程柯西问题存在全局的概率强解析弱解，且解不唯一；可指定任意无散度平方可积初值或以高概率任意大的停时前动能，解具有Sobolev正则性但非Leray-Hopf。
Based on Leray's formulation of the Navier-Stokes equations and the conditions of the exact linear representation of the nonlinear problem found in this paper, a compact explicit expression for the exact operator solution of the Navier-Stokes equations is given. It is shown that the introduced linear operator for Leray's equations is the generator of one-parameter contraction semigroup. This semigroup yields the existence of a unique and smooth classical solution of the associated Cauchy problem of Navier-Stokes equations in space $\mathbb{R}^3$ under smooth initial conditions.
We consider the solutions of the Cauchy problem for a dyadic model of Euler equations. We prove global existence and uniqueness of Leray-Hopf solutions in a rather large class K that implies in particular global existence and uniqueness in l^2 for all initial positive conditions in l^2.
In this paper, we study the Cauchy problem of the 2D incompressible magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a strong solution in the Lei-Lin space $χ^{-1}(\mathbb{R}^2)$ with any initial data in $χ^{-1}(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$ is established. Furthermore, the uniqueness of the strong solution in $χ^{-1}(\mathbb{R}^2)$ and the Leray-Hopf weak solution in $L^2(\mathbb{R}^2)$ is proved.
The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global existence and blow-up as well as uniform bounds are reviewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry's version of weighted Gronwall's inequalities, and (vi) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirich…
Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces
本文研究了Navier-Stokes方程在临界空间中的初值问题的非唯一性，通过数值方法分析了轴对称初始数据的标度不变解，并探讨了大数据下初始数据反射对称性的破坏及非唯一性例子的建立。
On complex algebraic singularities of some genuinely nonlinear PDEs
本文研究一大类真正非线性 PDE 解的复代数奇性，分析无粘 Burgers 方程并提出相关猜想，借助映射 F 的不动点算法与数值试验支持解的唯一性。
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