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Register a new userGlobal existence of a weak solution to unsaturated poroelasticity
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In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biots quasi-static consolidation model. The coupled, elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media obtained under similar assumptions as usually considered for Richards equation. In this work, the existence of a weak solution is established using regularization techniques, the Galerkin method, and compactness arguments. The final result holds under non-degeneracy conditions and natural continuity properties for the non-linearities. The assumptions are demonstrated to be reasonable in view of geotechnical applications.
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This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.
In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $dot{H}^{-alpha}(mathbb{R}^{3})$ or $dot{H}^{-alpha}(mathbb{T}^{3})$ with $0<alphaleq 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $tgeq0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlovi{c} and Staffilani on (SIMA, [1])in which $alpha$ is restricted to $0<alpha<frac{1}{4}$.
In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key element of our proof is the control of a particular defect measure associated to the pressure which avoids the use of the eective ux. Using this new tool, we solve an open problem namely global existence of solutions {`a} la Leray for such a system without assuming any restriction on the anisotropy amplitude. It provides a exible and natural way to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.
In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving fractional Laplacian with a Hardy term: begin{equation} label{eq0.1} (-Delta)^{s}u-{gamma} {frac{u}{|x|^{2s}}}= {frac{{|u|}^{ {2^{*}_{s}}(beta)-2}u}{|x|^{beta}}}+ big [ I_{mu}* F_{alpha}(cdot,u) big](x)f_{alpha}(x,u), u in {dot{H}}^s(R^{n}) end{equation} where $s in(0,1)$, $0leq alpha,beta<2s<n$, $mu in (0,n)$, $gamma<gamma_{H}$, $I_{mu}(x)=|x|^{-mu}$, $F_{alpha}(x,u)=frac{ {|u(x)|}^{ {2^{#}_{mu} }(alpha)} }{ {|x|}^{ {delta_{mu} (alpha)} } }$, $f_{alpha}(x,u)=frac{ {|u(x)|}^{{ 2^{#}_{mu} }(alpha)-2}u(x) }{ {|x|}^{ {delta_{mu} (alpha)} } }$, $2^{#}_{mu} (alpha)=(1-frac{mu}{2n})cdot 2^{*}_{s} (alpha)$, $delta_{mu} (alpha)=(1-frac{mu}{2n})alpha$, ${2^{*}_{s}}(alpha)=frac{2(n-alpha)}{n-2s}$ and $gamma_{H}=4^sfrac{Gamma^2(frac{n+2s}{4})} {Gamma^2(frac{n-2s}{4})}$. We show that problem (ref{eq0.1}) admits at least a weak solution under some conditions. To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings begin{equation} label{eq0.2} {dot{H}}^s(R^{n}) hookrightarrow {L}^{2^*_{s}(alpha)}(R^{n},|y|^{-alpha}) hookrightarrow L^{p,frac{n-2s}{2}p+pr}(R^{n},|y|^{-pr}) end{equation} where $s in (0,1)$, $0<alpha<2s<n$, $pin[1,2^*_{s}(alpha))$, $r=frac{alpha}{ 2^*_{s}(alpha) }$; We also establish an improved Sobolev inequality. By using mountain pass lemma along with an improved Sobolev inequality, we obtain a nontrivial weak solution to problem (ref{eq0.1}) in a direct way. It is worth while to point out that the improved Sobolev inequality could be applied to simplify the proof of the main results in cite{NGSS} and cite{RFPP}.
The aim of this paper is to prove the existence of almost global weak solutions for the unsteady nonlinear elastodynamics system in dimension $d=2$ or $3$, for a range of strain energy density functions satisfying some given assumptions. These assumptions are satisfied by the main strain energies generally considered. The domain is assumed to be bounded, and mixed boundary conditions are considered. Our approach is based on a nonlinear parabolic regularization technique, involving the $p$-Laplace operator. First we prove the existence of a local-in-time solution for the regularized system, by a fixed point technique. Next, using an energy estimate, we show that if the data are small enough, bounded by $varepsilon >0$, then the maximal time of existence does not depend on the parabolic regularization parameter, and the behavior of the lifespan $T$ is $gtrsim log (1/varepsilon)$, defining what we call here {it almost global existence}. The solution is thus obtained by passing this parameter to zero. The key point of our proof is due to recent nonlinear Korns inequalities proven by Ciarlet & Mardare in $mathrm{W}^{1,p}$ spaces, for $p>2$.
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