36 research outputs found
On variants of -measures and compensated compactness
We introduce new variant of -measures defined on spectra of general
algebra of test symbols and derive the localization properties of such
-measures. Applications for the compensated compactness theory are given. In
particular, we present new compensated compactness results for quadratic
functionals in the case of general pseudo-differential constraints. The case of
inhomogeneous second order differential constraints is also studied
On one criterion of the uniqueness of generalized solutions for linear transport equations with discontinuous coefficients
We study generalized solutions of multidimensional transport equation with
bounded measurable solenoidal field of coefficients . It is shown that
any generalized solution satisfies the renormalization property if and only if
the operator , in the Hilbert space
is an essentially skew-adjoint operator, and this is
equivalent to the uniqueness of generalized solutions. We also establish
existence of a contractive semigroup, which provides generalized solutions, and
give a criterion of its uniqueness
On the Cauchy problem for scalar conservation laws on the Bohr compactification of
We study the Cauchy problem for a multidimensional scalar conservation law on
the Bohr compactification of . The existence and uniqueness of entropy
solutions are established in the general case of merely continuous flux vector.
We propose also the necessary and sufficient condition for the decay of entropy
solutions as time
On decay of entropy solutions to multidimensional conservation laws
Under a precise genuine nonlinearity assumption we establish the decay of
entropy solutions of a multidimensional scalar conservation law with merely
continuous flux
On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws
We propose a new sufficient non-degeneracy condition for the strong
precompactness of bounded sequences satisfying the nonlinear first-order
differential constraints. This result is applied to establish the decay
property for periodic entropy solutions to multidimensional scalar conservation
laws
On long time behavior of periodic entropy solutions of a degenerate non-linear parabolic equation
We prove the asymptotic convergence of a space-periodic entropy solution of a
one-dimensional degenerate parabolic equation to a traveling wave. It is also
shown that on a segment containing the essential range of the limit profile the
flux function is linear (with the slope equaled to the speed of the traveling
wave) and the diffusion function is constant
On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: global well-posedness and decay property
We study the Cauchy problem for a multidimensional scalar conservation law
with merely continuous flux vector in the class of Besicovitch almost periodic
functions. The existence and uniqueness of entropy solutions are established.
We propose also the necessary and sufficient condition for the decay of almost
periodic entropy solutions as time
On decay of almost periodic viscosity solutions to Hamilton-Jacobi equations
We establish that a viscosity solution to a multidimensional Hamilton-Jacobi
equation with a convex non-degenerate hamiltonian and Bohr almost periodic
initial data decays to its infimum as time .Comment: arXiv admin note: text overlap with arXiv:1707.0014
On some properties of entropy solutions of degenerate non-linear anisotropic parabolic equations
We prove existence of the largest and the smallest entropy solutions to the
Cauchy problem for a nonlinear degenerate anisotropic parabolic equation.
Applying this result, we establish the comparison principle in the case when at
least one of the initial functions is periodic. In the case when initial
function vanishes at infinity (in the sense of strong average) we prove the
long time decay of an entropy solution under exact nonlinearity-diffusivity
condition.Comment: arXiv admin note: text overlap with arXiv:1904.01370,
arXiv:1910.0873
To the theory of entropy sub-solutions of degenerate non-linear parabolic equations
We prove existence of the largest entropy sub-solution and the smallest
entropy super-solution to the Cauchy problem for a nonlinear degenerate
parabolic equation with only continuous flux and diffusion functions. Applying
this result, we establish the uniqueness of entropy solution with periodic
initial data. The more general comparison principle is also proved in the case
when at least one of the initial functions is periodic