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Register a new userOn the heat equation with drift in $L_{d+1}$
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N.V. Krylov
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In this paper we deal with the heat equation with drift in $L_{d+1}$. Basically, we prove that, if the free term is in $L_{q}$ with high enough $q$, then the equation is uniquely solvable in a rather unusual class of functions such that $partial_{t}u, D^{2}uin L_{p}$ with $p<d+1$ and $Duin L_{q}$.
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This paper is a natural continuation of cite{Kr_20_2} and cite{Kr_21_1} where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(mathbb{R}^{d+1})$ and some properties of their Greens functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Holder continuity of potentials, Fanghua Lin estimates and so on.
In subdomains of $mathbb{R}^{d}$ we consider uniformly elliptic equations $Hbig(v( x),D v( x),D^{2}v( x), xbig)=0$ with the growth of $H$ with respect to $|Dv|$ controlled by the product of a function from $L_{d}$ times $|Dv|$. The dependence of $H$ on $x$ is assumed to be of BMO type. Among other things we prove that there exists $d_{0}in(d/2,d)$ such that for any $pin(d_{0},d)$ the equation with prescribed continuous boundary data has a solution in class $W^{2}_{p,text{loc}}$. Our results are new even if $H$ is linear.
This paper is a natural continuation of cite{Kr_20_2}, where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(mathbb{R}^{d+1})$. Here we study some properties of these processes such as the probability to pass through narrow tubes, higher summability of Greens functions, and so on. The results seem to be new even if the diffusion is constant.
We prove the solvability of It^o stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in $L_{d+1}(mathbb{R}^{d+1})$. Actually, the powers of summability of the drift in $x$ and $t$ could be different. Our results seem to be new even if the diffusion is constant. The method of proving the solvability belongs to A.V. Skorokhod. Weak uniqueness of solutions is an open problem even if the diffusion is constant.
In this note, we obtain a version of Aleksandrovs maximum principle when the drift coefficients are in Morrey spaces, which contains $L_d$, and when the free term is in $L_p$ for some $p<d$.
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