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Register a new userAn application of global gradient estimates in Lorentz-Morrey spaces: The existence of stationary solution to degenerate diffusive Hamilton-Jacobi equations
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In historical mathematics and physics, the Kardar-Parisi-Zhang equation or a quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation in growing interface and universality classes, is also known by the different name as the quasilinear Riccati type equation. The existence of solutions to this type of equation under some assumptions and requirements, still remains an interesting open problem at the moment. In our previous studies cite{MP2018, MPT2019}, we obtained the global bounds and gradient estimates for quasilinear elliptic equations with measure data. There have been many applications are discussed related to these works, and main goal of this paper is to obtain the existence of a renormalized solution to the quasilinear stationary solution to the degenerate diffusive Hamilton-Jacobi equation with the finite measure data in Lorentz-Morrey spaces.
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Sharp temporal decay estimates are established for the gradient and time derivative of solutions to a viscous Hamilton-Jacobi equation as well the associated Hamilton-Jacobi equation. Special care is given to the dependence of the estimates on the viscosity. The initial condition being only continuous and either bounded or non-negative. The main requirement on the Hamiltonians is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r^p for r non-negatif and p positif and different from 1.
This paper is focused on the local interior $W^{1,infty}$-regularity for weak solutions of degenerate elliptic equations of the form $text{div}[mathbf{a}(x,u, abla u)] +b(x, u, abla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^infty$-norm for the solutions gradient in terms of its local $L^p$-norm. Specifically, we prove begin{equation*} | abla u|_{L^infty(B_{frac{R}{2}}(x_0))}^p leq frac{C}{|B_R(x_0)|}int_{B_R(x_0)}| abla u(x)|^p dx. end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{1,q}$-estimates (for $q>p$) for weak solutions to a much larger class of quasilinear elliptic equations.
We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ partial_t h(t,x)= uDelta h(t,x)+lambda V(| abla h(t,x)|) +sqrt{D}, eta(t,x), qquad xin{mathbb{R}}^d $$ in $dge 1$ dimensions. The forcing term $eta$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)ge 0$, one finds $V(| abla h|)ltimes | abla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article, a continuation of [24], is dedicated to a generalization of the PDE estimates obtained in the previous article to the case of a deposition rate $V$ with polynomial growth of arbitrary order at infinity, for which in general the Cole-Hopf transformation does not allow any more a comparison to the heat equation. The main tool here instead is the representation of $h$ as the solution of some minimization problem through the Hamilton-Jacobi-Bellman formalism. This sole representation turns out to be powerful enough to produce local or pointwise estimates in ${cal W}$-spaces of functions with locally bounded averages, as in [24], implying in particular global existence and uniqueness of solutions.
The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand
For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain $L^q$-estimates for the gradients of solutions, and for the lower order coefficients from a Kato-type class we show that the solutions are Lipschitz continuous with respect to the space variable.
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