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Nguyen Cong Phuc

Publications and source records attributed to Nguyen Cong Phuc.

26 records · Page 2Linked to original sources
arXiv2014

Global Lorentz and Lorentz-Morrey estimates below the natural exponent for quasilinear equations

Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the form $$\text{div}\,\mathcal{A}(x, \nabla u)=\text{div}\, |{\bf f}|^{p-2}{\bf f},$$ where $\text{div}\,\mathcal{A}(x, \nabla u)$ is modelled after the $p$-Laplacian, $p>1$. The estimates are global over bounded domains that satisfy a mild exterior uniform thickness condition that involves the $p$-capacity. The vector field datum ${\bf f}$ is allowed to have low degrees of integrability and thus solutions may not have finite $L^p$ energy. A higher integrability result at the boundary of the ground domain is also obtained for infinite energy solutions to the associated homogeneous equations.

math.AP↗
arXiv2014

The Navier-Stokes equations in nonendpoint borderline Lorentz spaces

It is shown both locally and globally that $L_t^{\infty}(L_x^{3,q})$ solutions to the three-dimensional Navier-Stokes equations are regular provided $q\not=\infty$. Here $L_x^{3,q}$, $0<q\leq\infty$, is an increasing scale of Lorentz spaces containing $L^3_x$. Thus the result provides an improvement of a result by Escauriaza, Seregin and {\v S}verák ((Russian) Uspekhi Mat. Nauk {\bf 58} (2003), 3--44; translation in Russian Math. Surveys {\bf 58} (2003), 211--250), which treated the case $q=3$. A new local energy bound and a new $ε$-regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray-Hopf weak solutions in $L_t^{\infty}(L_x^{3,q})$, $q\not=\infty$, is also obtained as a consequence.

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arXiv2013

Stationary Navier-Stokes Equations with Critically Singular External Forces: Existence and Stability Results

We show the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces belonging to a critical space. To the best of our knowledge, this is the largest critical space that is available up to now for this kind of existence. This result can be viewed as the stationary counterpart of the existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with initial data in BMO$^{-1}$. The stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup.

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arXiv2012

Quasilinear equations with source terms on Carnot groups

In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane--Emden type with measure data on a Carnot group $\mathbb G$ of arbitrary step. The quasilinear part involves operators of the $p$-Laplacian type $Δ_{\mathbb G,\,p}\,$, $1<p<\infty$. These results are based on new a priori estimates of solutions in terms of nonlinear potentials of Th. Wolff's type. As a consequence, we characterize completely removable singularities, and prove a Liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub-Laplacian ($p=2$) on the Heisenberg group.

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arXiv2010

Boundary behavior p-harmonic functions in the Heisenberg group

We study the boundary behavior of nonnegative p-harmonic functions which vanish on a portion of the boundary of a domain in the Heisenberg group H^n. Our main results are: 1) An estimate from above which shows that, under suitable geometric assumptions on the relevant domain, such a p-harmonic function vanishes at most linearly with respect to the sub-Riemannian distance to the boundary. 2) An estimate from below which shows that for a (Euclidean) C^{1,1} domain, away from the characteristic set, such a p-harmonic function vanishes exactly like the distance to the boundary. By combining 1) and 2) we obtain a comparison theorem stating that, at least away from the characteristic set, any two such p-harmonic functions must vanish at the same rate.

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arXiv2008

Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces

We consider various types of Hardy-Sobolev inequalities on a Carnot-Carathéodory space $(\Om, d)$ associated to a system of smooth vector fields $X=\{X_1, X_2,...,X_m\}$ on $\RR^n$ satisfying the Hörmander's finite rank condition $rank Lie[X_1,...,X_m] \equiv n$. One of our main concerns is the trace inequality \int_{\Om}|ϕ(x)|^{p}V(x)dx\leq C\int_{\Om}|Xϕ|^{p}dx,\qquad ϕ\in C^{\infty}_{0}(\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\Om$, and $1<p<+\infty$. Under sharp geometric assumptions on the domain $\Om\subset \Rn$ that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities.

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arXiv2005

Quasilinear and Hessian equations of Lane-Emden type

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: $ -Δ_p u = u^q + μ$ and $F_k[-u] = u^q + μ$, $u > 0$, on $R^n$, or on a bounded domain $Ω$. Here $Δ_p$ is the p-Laplacian, and $F_k[u]$ is the $k$-Hessian defined as the sum of $k\times k$ principal minors of the Hessian matrix $D^2 u$; $μ$ is a nonnegative measurable function (or measure) on $Ω$. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data $μ\in L^s (Ω)$, $s>1$. Such results are deduced from our existence criteria with the sharp exponents $s = n(q-p+1)/pq$ for the first equation, and $s = n(q-k)/2kq$ for the second one. Furthermore, a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff's potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpelainen and Maly, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of Monge-Ampere type.

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