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We consider a second-order nonlinear degenerate parabolic equation in the case, where the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov-Carrillo) of the Cauchy problem can be nonunique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy sub- and supersolutions, in the case where at least one of the initial functions is periodic. The obtained results are a generalization of the results known for conservation laws to the parabolic case., UDC 517.957 CONTENTS 1. Introduction 988 2. Auxiliary Statements 990 3. Main Results 1000 References 1007 1. Introduction In the half-space [product] = [R.sub.+] * [R.sup.n], [R.sub.+] = (0, +[infinity]), [...]

When the velocity shear between the two plasmas separated by Earth's magnetopause is locally super‐Alfvénic, the Kelvin‐Helmholtz (KH) instability can develop. A crucial role is played by the interplanetary magnetic field (IMF) orientation, which can stabilize the velocity shear. Although, in a linear regime, the instability threshold is equally satisfied during both northward and southward IMF orientations, in situ measurements show that KH instability is preferentially excited during the northward IMF orientation. We investigate this different behavior by means of a mixing parameter which we apply to two KH events to identify both boundaries and the center of waves/vortices. During the northward orientation, the waves/vortex boundaries have stronger electrons than ions mixing, while the opposite is observed at their center. During the southward orientation, instead, particle mixing is observed predominantly at the boundaries. In addition, stronger local ion and electron non‐thermal features are observed during the northward than the southward IMF orientation. Specifically, ion distribution functions are more distorted, due to field‐aligned beams, and electrons have a larger temperature anisotropy during the northward than the southward IMF orientation. The observed kinetic features provide an insight into both local and remote processes that affect the evolution of KH structures. Plain Language Summary: Due to the velocity shear layer generated by the solar wind flowing past the Earth's magnetosphere, large surface Kelvin‐Helmholtz (KH) waves and vortices can be formed at the magnetopause. These waves and vortices play a crucial role in transporting the solar wind particles through the magnetopause into the magnetosphere, where the particles form a so‐called low‐latitude boundary layer (LLBL). The particle transport occurs due to stretching and twisting of the magnetic field lines by the KH waves/vortices, which result in plasma mixing and diffusion through the magnetopause. It appears that spacecraft observe the KH waves/vortices more often during northward orientations of the interplanetary magnetic field (IMF). During northward IMF, the induced high‐latitudes reconnection thicken the preexisting LLBL and lower the density gradient at the magnetopause, thus favoring KH instability. Conversely, higher density jump and dayside reconnection, during southward IMF, can suppress the instability development and disrupt the KH vortices. To clarify these differences in the KH wave/vortex appearance under different IMF directions, we compare the wave/vortex and particle properties during both IMF orientations. We employ a mixing parameter, which helps identify specific regions of KH waves/vortices and investigate their kinetic signatures, thus providing an insight into KH evolution. Key Points: Two Kelvin‐Helmholtz events during northward and southward interplanetary magnetic field (IMF) orientations are compared using a mixing‐parameterHigher mixing and local non‐thermal features due to field‐aligned ion beams during the northward IMF are observedKinetic features of Kelvin‐Helmholtz structures can identify both local and remote processes affecting the instability evolution [ABSTRACT FROM AUTHOR]

We establish the asymptotic convergence of an entropy solution, periodic with respect to the spatial variable, of a degenerate nonlinear parabolic equation to a traveling wave. It is shown that, on a segment containing the essential values of the limit profile, the flow function is linear (with the angular coefficient equal to the traveling wave velocity) and the diffusion function is constant. Bibliography: 12 titles. Dedicated to Nina Nikolaevna Uraltseva, UDC 517.95 1 Introduction In the half-plane n = [R.sub.+] x R, where [R.sub.+] = (0, +[infinity]>), we consider the parabolic equation [u.sub.t] + [phi] [(u).sub.x] - g[(u).sub.xx] = 0, [...]

It is shown that bounded sequences satisfying nonlinear differential constraints, strongly precompact under an exact condition of nondegeneration of these conditions. The proof is based on new localization principles for ultraparabolic H-measures with continuous indices. Bibliography: 16 titles., Dedicated to the memory of Vasilii Vasil'evich Zhikov UDC 517.95 1 Introduction Let Q be an open subset in [R.sup.n]. We assume that [phi](x, u) [member of] [L.sup.2.sub.loc]([OMEGA], C(R, [R.sup.n])) [...]