Related papers: Euler Equations in Sobolev conormal spaces

We study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [Lindblad H.,…

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the…

In this paper we establish well posedness of the Neumann problem with boundary data in $L^2$ or the Sobolev space $\dot W^2_{-1}$, in the half space, for linear elliptic differential operators with coefficients that are constant in the…

Presented are two results on the formation of finite time singularities of solutions to the compressible Euler equations in two and three space dimensions for isentropic, polytropic, ideal fluid flows. The initial velocity is assumed to be…

We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for…

We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local…

We consider the $\alpha$-Euler equations on a bounded three-dimensional domain with frictionless Navier boundary conditions. Our main result is the existence of a strong solution on a positive time interval, uniform in $\alpha$, for…

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions…

We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in $(t,x_1)$ for $a^{ij}$, $1 \leq i,j \leq d$, $(i,j) \neq…

We continue the~study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a~paper by Gol$'$dshtein and Troyanov. As in this paper, our study is based on relations between $L_{q,p}$-cohomology…

We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion's directions. The result is that u is also…

Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we construct global weak solutions to the three-dimensional incompressible Euler equations which are zero outside of a finite time interval and have velocity in the…

This paper focuses on the study of the density-dependent incompressible Euler equations in space dimension $d=2$, for low regularity (\textsl{i.e.} non-Lipschitz) initial data satisfying assumptions in spirit of the celebrated Yudovich…

We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity $\bv_0$, logarithmic density $\rho_0$, and specific vorticity \(w_0\), which satisfy $(\bv_0, \rho_0,…

We consider integrals in the sense of Choquet with respect to the $\delta$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<\delta\le n$. In…

In this short paper we generalize the classical inequality between the norms in Lebesgue spaces of the functions and its derivatives, which in the multidimensional case are called Sobolev's inequalities, on the many popular classes pairs of…

These lecture notes contain an extended version of the material presented in the C.I.M.E. summer course in 2017, aiming to give a detailed introduction to the metric Sobolev theory. The notes are divided in four main parts. The first one is…

We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $\det Du = f$, where $f$ is integrable and bounded away from zero. In particular, we take $f\in L^p$, where $p > 1$, or in $L\log…

We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange…

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has…