Solution of the Navier–Stokes problem

A new a priori estimate for solutions to Navier–Stokes equations is derived. Uniqueness and existence of these solutions in ${\mathbb{R}}^3$ for all $t>0$ is proved in a class of solutions locally differentiable in time with values in $H^1\left({\mathbb{R}}^3\right)$, where $H^1\left({\mathbb{R}}^3\right)$ is the Sobolev space. By the solution a solution to an integral equation is understood. No smallness restrictions on the data are imposed.     

The regularity of a semilinear elliptic system with quadratic growth of gradient

In this paper, we study semilinear elliptic systems with critical nonlinearity of the form

(0.1)$\mathrm{\Delta }u=Q(x,u,\mathrm{?}u),$

for $u:{\mathbb{R}}^n\to {\mathbb{R}}^K$, Q has quadratic growth in ?u. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When $n=2$, such a system does not have smooth regularity in general for $W^{1,2}$ weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for $n=2$) and F. Béthuel (for $n\ge 3$), assert that a $W^{1,n}$ weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1), that a $W^{1,n}$ weak solution of the system is smooth for $n\ge 3$. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a $W^{2,n/2}$ weak solution of such system is always smooth, for $n\ge 5$. We also construct various examples, and these examples show that our regularity results are optimal in various sense.     

The Cauchy problem for a generalized Camassa–Holm equation

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces $H^s$, $s>3/2$ for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in $H^r$-topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

Global existence and decay estimate of solutions to magneto-micropolar fluid equations

We are concerned with magneto-micropolar fluid equations (1.3)–(1.4). The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the magneto-micropolar-Navier–Stokes (MMNS) system, we obtain global existence and large time behavior of solutions near a constant states in ${\mathbb{R}}^3$. Appealing to a refined pure energy method, we first obtain a global existence theorem by assuming that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms ${\dot{H}}^{?s}$ $(0\le s<\frac{3}{2})$ or homogeneous Besov norms ${\dot{B}}_{2,\infty }^{?s}$ $(0<s\le \frac{3}{2})$, we obtain the optimal decay rates of the solutions and its higher order derivatives. As an immediate byproduct, we also obtain the usual $L^p?L^2$ $(1\le p\le 2)$ type of the decay rates without requiring that the $L^p$ norm of initial data is small. At last, we derive a weak solution to (1.3)–(1.4) in ${\mathbb{R}}^2$ with large initial data.     

Proof of the Hénon–Lane–Emden conjecture in R3

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on $S^{N?1}$, we prove this conjecture is true for space dimension $N=3$; which also implies the single elliptic equation has no positive classical solutions in ${\mathbb{R}}^3$ when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for ${\mathbb{R}}^3$.     

SL(m,C)-equivariant and translation covariant continuous tensor valuations

The space of continuous, $\mathrm{SL}(m,\mathbb{C})$-equivariant, $m\ge 2$, and translation covariant valuations taking values in the space of real symmetric tensors on ${\mathbb{C}}^m?{\mathbb{R}}^{2m}$ of rank $r\ge 0$ is completely described. The classification involves the moment tensor valuation for $r\ge 1$ and is analogous to the known classification of the corresponding tensor valuations that are $\mathrm{SL}(2m,\mathbb{R})$-equivariant, although the method of proof cannot be adapted.     

Regularity theory for Ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

We develop interior $W^{2,p,\mu }$ and $W^{2,\text{BMO}}$ regularity theories for $L^n$-viscosity solutions to fully nonlinear elliptic equations $T(D^{2}u,x)=f(x)$, where T is approximately convex at infinity. Particularly, $W^{2,\text{BMO}}$ regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of $D^{2}T(M)$ are at least $?C{6M6}^{?(1+{\sigma }_0)}$ as $M\to \infty$. $W^{2,\text{BMO}}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W^{2,\text{BMO}}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.     

Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium $\mathrm{\Omega }?{\mathbb{R}}^N$, $N\ge 2$. The governing equation may be degenerate of p-Laplace type, with $1<p\le N$. We prove optimal $L^{\infty }$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.     

Uniqueness of critical points and maximum principles of the singular minimal surface equation

We consider a smooth solution $u>0$ of the singular minimal surface equation $\sqrt{1+|Du{|}^2}\text{ div}(Du/\sqrt{1+|Du{|}^2})=\alpha /u$ defined in a bounded strictly convex domain of ${\mathbb{R}}^2$ with constant boundary condition. If $\alpha <0$, we prove the existence a unique critical point of u. We also derive some $C^0$ and $C^1$ estimates of u by using the theory of maximum principles of Payne and Philippin for a certain family of Φ-functions. Finally we deduce an existence theorem of the Dirichlet problem when $\alpha <0$.     

The dual code of any (δ+αu2)-constacyclic code over F2m[u]∕〈u4〉 of oddly even length

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $R={\mathbb{F}}_{2^m}\left[u\right]∕〈u^4〉={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+u^2{\mathbb{F}}_{2^m}+u^3{\mathbb{F}}_{2^m}$ $(u^4=0)$ which is a finite chain ring, and $n$ is an odd positive integer. For any $\delta ,\alpha \in {\mathbb{F}}_{2^m}^{\times }$, an explicit representation for the dual code of any $(\delta +\alpha u^2)$-constacyclic code over $R$ of length $2n$ is given. And some dual codes of $(1+u^2)$-constacyclic codes over $R$ of length 14 are constructed. For the case of $\delta =1$, all distinct self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of length $2n$ are determined.