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We consider a nonlinear, spatially nonlocal initial value problem in one space dimension on that describes the motion of surface quasigeostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth solution under a convergence condition on the multilinear expansion of the nonlinear term in the equation, and, for sufficiently smooth and small initial data, we prove that the solution is global.
We study a class of degenerate elliptic operators (which is a slight extension of the sums of squares of real-analytic vector fields satisfying the Hörmander condition). We show that, in dimensions and , for every operator in such a class and for every distribution such that is real-analytic, the analytic singular support of , , is a “negligible” set. In particular, we provide (optimal) upper estimates for the Hausdorff dimension of . Finally, we show that in dimension , there exists an operator in such a class and a distribution such that is of dimension .
We consider the time-harmonic scalar wave equation in junctions of several different periodic half-waveguides. In general this problem is not well-posed. Several papers propose radiation conditions, i.e., the prescription of the behavior of the solution at the infinities. This ensures uniqueness—except for a countable set of frequencies which correspond to the resonances—and yields existence when one is able to apply Fredholm’s alternative. This solution is called the outgoing solution. However, such radiation conditions are difficult to handle numerically. In this paper, we propose so-called transparent boundary conditions which enable us to characterize the outgoing solution. Moreover, the problem set in a bounded domain containing the junction with these transparent boundary conditions is of Fredholm type. These transparent boundary conditions are based on Dirichlet-to-Neumann operators whose construction is described in the paper. Contrary to the other approaches, the advantage of this approach is that a numerical method can be naturally derived in order to compute the outgoing solution. Numerical results illustrate and validate the method.
We will consider regularity criteria for the Navier–Stokes equation in mixed Lebesgue sum spaces. In particular, we will prove regularity criteria that only require control of the velocity, vorticity, or the positive part of the second eigenvalue of the strain matrix, in the sum space of two scale-critical spaces. This represents a significant step forward, because each sum-space regularity criterion covers a whole family of scale-critical regularity criteria in a single estimate. In order to show this, we will also prove a new inclusion and inequality for sum spaces in families of mixed Lebesgue spaces with a scale invariance that is also of independent interest.
We are concerned with the size of the regular set for weak solutions to the Navier–Stokes equations. It is shown that if a weighted norm of initial data is finite, the suitable weak solutions are regular in a set above a space-time hypersurface determined by the degree of the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math.35:6 (1982), 771–831) in various aspects. Our main tool is an -regularity theorem in terms of initial data, which is of independent interest. As applications, we also study energy concentration near a possible blow-up time and regularity for forward discretely self-similar solutions.
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